Datasets:

euclaise

/

mathoverflow-accepted

like 3

https://mathoverflow.net/questions/664

15

Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs? That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some $c>0$, with no monochromatic complete subgraph of size $k$?

https://mathoverflow.net/users/416

Can one make Erdős's Ramsey lower bound explicit?

I believe the answer is "no"; the best known constructions only give no clique or independent set of size about $2^\sqrt{n}$ in a graph with $2^n$ vertices. Bill Gasarch has a page on the subject [here](http://www.cs.umd.edu/~gasarch/const_ramsey/const_ramsey.html), although I don't know how frequently it updates.

5

https://mathoverflow.net/users/382

667

https://mathoverflow.net/questions/652

22

[This question](https://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it) reminded me of a possibly stupid idea that I had a while back. On page 2 of [this paper](http://arXiv.org/abs/math/0502016v1), while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely interesting comment: > > Euclid misses a great opportunity here: if he stated the principle > > > “The extremity of an extremity is empty”, > > > he could be considered as the discoverer of the > > > BASIC EQUATION OF HOMOLOGICAL ALGEBRA: d^2 = 0. > > > Ever since I read this, I've had a suspicion that the equation "d^2 = 0" of homological algebra is somehow related to the equation "epsilon^2 = 0" of (first-order) calculus (as in Newton)\*, since the latter equation can be interpreted as saying "a very very small quantity is zero" which at least superficially seems similar to "the extremity of an extremity is empty". I once explained my suspicion to Dan Erman over beers, and he responded by asking another question: Can we do some sort of homological algebra using the equation d^n = 0 rather than d^2 = 0? Perhaps if d^2 = 0 can be related to first-order calculus, then d^3 = 0 can be related to second-order calculus, and so on... I don't really have a specific question to ask -- I just thought I might put this idea out there. Maybe someone can tell me why this idea is stupid, or why it is not stupid. --- \*or the ring of dual numbers k[epsilon]/(epsilon^2) if you're an algebraist or an algebraic geometer.

https://mathoverflow.net/users/83

Homological algebra and calculus (as in Newton)

I saw the notion of an "n-complex" once in this preprint by Peter Olver: <http://www.math.umn.edu/~olver/a_/hyper.pdf> I only ever studied sections 5 and 6 of this paper, so I don't know what he's actually doing in the other sections (he introduces "hypercomplexes" which contain "n-complexes" as subcomplexes in section 7). The introduction mentions that he is interested in higher-order versions of de Rham complexes.

5

https://mathoverflow.net/users/321

670

https://mathoverflow.net/questions/672

11

I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a bit boring (mainly specific case work, seemingly arbitrary curves, etc.). This is in stark contrast with say, Modern Graph Theory by Bollobas: plenty of fun problems. A friend told me he had experienced the same with other Algebraic Geometry books. And my lecturer told us that this might be related to the nature of algebraic geometry. Indeed, so much theory is needed before being able to properly analyze the most basic problems... Thoughts? Counter-examples (i.e. introductory books with many fun problem)?

https://mathoverflow.net/users/416

Are good introductory/pedagogical problems in algebraic geometry rare?

I highly recommend Fultons book "Algebraic curves" It's available on his [webpage](http://www.math.lsa.umich.edu/~wfulton/) It's a very good introduction, and in the first chapter there are 54 exercises.

11

https://mathoverflow.net/users/135

675

https://mathoverflow.net/questions/674

37

I have a few elementary questions about cup-products. Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an explicitly computable map on cochains? Second, how do they relate to cup-products in algebraic topology? In general, are there connections between cup-products and other mathematical constructions that may provide more intuition into them?

https://mathoverflow.net/users/344

What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?

You can identify group cohomology with $Ext\_{kG}(k,k)$ where $k$ is your base ring, and $kG $ is the group algebra. Here, the cup product given by Yoneda product (which there's some material on [here](https://en.wikipedia.org/wiki/Ext_functor#Ring_structure_and_module_structure_on_specific_Exts)). You should think Ext as being a generalization of Hom (it measures maps that should be there and aren't, and higher order things along the same lines), and this product as a generalization of composition. Almost anything anyone ever calls cohomology is Ext of something with something, and products come up very naturally this way. Cohomology of manifolds is Ext of the sheaf of locally constant k valued functions with itself, and the cup product again is just Yoneda product (all the explicit models for cohomology (simplicial, De Rham, Cech) are all just different resolutions of the this sheaf).

21

https://mathoverflow.net/users/66

679

https://mathoverflow.net/questions/592

9

I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the cusps so that the natural projection maps obtained by forgetting the level structure are log-etale (at least away from primes dividing the order of your level structure). I can have some rough intuition about how this happens over a field of characteristic zero, but not integrally. Can anybody explain this or give me a reference for this structure? Additionally, has anybody worked out the appropriate integral ring of modular forms with logarithmic structure in some cases, similar to the Deligne-Tate calculation of modular forms over Z?

https://mathoverflow.net/users/360

Logarithmic structures on moduli of elliptic curves over Z

I thing Kato's log purity theorem gives you this. See, for instance, Theorem B in Mochizuki's "Extending Families of Curves over Log Regular Schemes." I think all you need is that the cusps form a normal crossings divisor on X(1) [if you're worried about X(1) being a stack rather than a scheme, you can start with a bit of extra level structure coprime to the primes you're interested in] and then your map Y(N) -> Y(1) is tamely ramified, which tells you that the normalization X(N) of X(1) in Y(N) carries a canonical log-structure in which the map X(N) -> X(1) is log-etale.

4

https://mathoverflow.net/users/431

693

https://mathoverflow.net/questions/696

22

This is probably quite easy, but how do you show that the Euler characteristic of a manifold *M* (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to the self intersection of *M* in the diagonal (of *M* × *M*)? The few cases which are easy to visualise (ℝ in the plane, S1 in the torus) do not seem to help much. The [Wikipedia article about the Euler class](http://en.wikipedia.org/wiki/Euler_class) mentions very briefly something about the self-intersection and that does seem relevant, but there are too few details.

https://mathoverflow.net/users/362

Euler characteristic of a manifold and self-intersection

The normal bundle to $M$ in $M\times M$ is isomorphic to the tangent bundle of $M$, so a tubular neighborhood $N$ of $M$ in $M\times M$ is isomorphic to the tangent bundle of $M$. A section $s$ of the tangent bundle with isolated zeros thus gives a submanifold $M'$ of $N \subset M\times M$ with the following properties: 1) $M'$ is isotopic to $M$. 2) The intersections of $M'$ with $M$ are in bijection with the zeros of $s$ (and their signs are given by the indices of the zeros). The desired result then follows from the Hopf index formula.

22

https://mathoverflow.net/users/317

700

https://mathoverflow.net/questions/691

41

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [commutative] ring and a simplicial [commutative] ring?

https://mathoverflow.net/users/83

Simplicial objects

One could say many things about this, and I hope you get many replies! Here are some remarks, although much of this might already be familiar or obvious to you. In some vague sense, the study of simplicial objects is "homotopical mathematics", while the study of objects is "ordinary mathematics". Here by "homotopical mathematics", I mean the philosophy that among other things say that whenever you have a set in ordinary mathematics, you should instead consider a space, with the property that taking pi\_0 of this space recovers the original set. In particular, this should be done for Hom sets, so we should have Hom spaces instead. This is formalized in various frameworks, such as [infinity-categories](http://ncatlab.org/nlab/show/%28infinity,1%29-category), [simplicial model categories](http://ncatlab.org/nlab/show/simplicial+model+category), and [A-infinity categories](http://ncatlab.org/nlab/show/A-infinity-category). Here "space" can mean many different things, in these examples: infinity-category, simplicial set, or chain complex respectively. For intuition, it helps to think of a simplicial object as an object with a topology. For example, a simplicial set is like a topological space, a simplicial ring is like a topological ring etc. The precise statements usually takes the form of a Quillen equivalence of model categories between the simplicial objects and a suitable category of topological objects. Simplicial sets are Quillen equivalent to compactly generated topological spaces, and I think a similar statement holds if you replace sets by rings, although I am not sure if you need any hypotheses here. If you like homological algebra, it helps to think of a simplicial object as analogous to a chain complex. The precise statements are given by various generalizations of the Dold-Kan correspondence. For simplicial rings, they should correspond to chain complexes with a product, more precisely DGAs. Again, one has to be a bit careful with the precise statements. I think the following is true: Simplicial commutative unital k-algebras are Quillen equivalent to connective commutative differential graded k-algebras, provided k is a Q-algebra. A remark about the word "simplicial": A simplicial object in a category C is a functor from the Delta category into C, but for almost all purposes the Delta category could be replaced with any test category in the sense of Grothendieck, see this [nLab post](http://ncatlab.org/nlab/show/geometric+shapes+for+higher+structures) for some discussion which doesn't use the terminology of test categories. Since you used the tag "derived stuff" I guess you are already aware of Toen's derived stacks. Some of his articles have introductions which explain why one would like to use simplicial rings instead of rings. See in particular his really nice [lecture notes](http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf) from a course in Barcelona last year. I tried to write a [blog post](http://homotopical.wordpress.com/2009/05/20/homotopical-categories-and-simplicial-sheaves/) on some of this a while ago, there might be something useful there, especially relating to motivation from algebraic geometry.

35

https://mathoverflow.net/users/349

705

https://mathoverflow.net/questions/707

15

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following: > > Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$, let $p$ be a prime, at which $E$ has potentially multiplicative reduction and let $\ell$ be a prime different than $p$. Then the mod $\ell$ representation is unramified at $p$ iff $\ell$ divides the valuation of $\Delta$ at $p$. > > > This is used for instance in the proof of Fermat's last theorem. In *On modular representations of ${\rm Gal}(\overline{\mathbb Q}/\mathbb Q)$ arising from modular forms* by Ken Ribet he cites Serre's (awesome) paper *Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline{\mathbb Q}/\mathbb Q)$*, which (4.1.12) says this follows immediately from the theory of Tate curves. It is pretty easy: the Tate curve gives you a explicit description the field obtained by adjoining the $\ell$-torsion points to $\mathbb Q\_p$, and one can just check directly that the divisibility condition implies that this field (and thus the mod $\ell$ representation on the $\ell$-torsion points) is unramified at $p$. Nonetheless I'm curious to know if anyone writes this down explicitly anywhere in the literature.

https://mathoverflow.net/users/2

Reference for the `standard' Tate curve argument.

I think most people just mentally have in mind the argument you give. In my thesis I actually wrote this down semi-carefully (including the case l = p, in which case what you want to say is that E[l] is finite over Zp, where E is now the Neron model of your elliptic curve over Q\_p.) Or rather I wrote down the direction "l divides Delta => unramified" in Corollary 1.2 of [the short version of my thesis](http://www.math.wisc.edu/~ellenber/HBAVJan00.pdf). The goal of the thesis, by the way, was to extend this assertion to abelian varieties with real multiplication; the point being that it's not obvious what's supposed to play the role of Delta.

10

https://mathoverflow.net/users/431

708

https://mathoverflow.net/questions/636

6

Let Cat denote the 1-category of small categories. The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits. Does it commute with quotients by free group actions? In other words, if C is a small category and G is a group acting on C such that the action of G on the objects of C is free, does Mor(C/G) = (Mor C)/G?

https://mathoverflow.net/users/126667

Quotient of a category by a free group action

OK, I found a "high-tech" argument which is maybe more convincing. The external input to the argument is that if X -> Z <- Y is a diagram of G-sets with G acting freely on Z (and hence on X and Y) then (X xZ Y)/G = (X/G) x(Z/G) (Y/G). This is pretty clear because we can always write our original X -> Z <- Y in the form (X/G) x G -> (Z/G) x G <- (Y/G) x G. Now we use the presentation of Cat as the localization of the category of simplicial sets in which the local objects are the simplicial sets X such that Xn -> X1 xX0 ... xX0 X1 is an isomorphism for all n >= 2. By general nonsense about localizations, if C is a category (viewed as a simplicial set) with a G-action, we can compute the colimit C/G by taking the colimit levelwise and then localizing the resulting object. But the levelwise quotient is already local, by our previous observation. Since C1 represents the set of morphisms of C, we're done.

2

https://mathoverflow.net/users/126667

717

https://mathoverflow.net/questions/400

45

Around these parts, the aphorism "A gentleman never chooses a basis," has become popular. > > **Question.** Is there a gentlemanly way to prove that the natural map from $V$ to $V^{\*\*}$ is surjective if $V$ is finite-dimensional? > > > As in life, the exact standards for gentlemanliness are a bit vague. Some arguments seem to be implicitly picking a basis. I'm hoping there's an argument which is unambiguously gentlemanly.

https://mathoverflow.net/users/27

"A gentleman never chooses a basis."

Following up on Qiaochu's query, one way of distinguishing a finite-dimensional $V$ from an infinite one is that there exists a space $W$ together with maps $e: W \otimes V \to k$, $f: k \to V \otimes W$ making the usual [triangular equations](http://ncatlab.org/nlab/show/triangle+identities) hold. The data $(W, e, f)$ is uniquely determined up to canonical isomorphism, namely $W$ is canonically isomorphic to the dual of $V$; the $e$ is of course the evaluation pairing. (While it is hard to write down an explicit formula for $f: k \to V \otimes V^\*$ without referring to a basis, it is nevertheless *independent* of basis: is the same map no matter which basis you pick, and thus canonical.) By swapping $V$ and $W$ using the symmetry of the tensor, there are maps $V \otimes W \to k$, $k \to W \otimes V$ which exhibit $V$ as the dual of $W$, hence $V$ is canonically isomorphic to the dual of its dual. Just to be a tiny bit more explicit, the inverse to the double dual embedding $V \to V^{\*\*}$ would be given by $$V^{\ast\ast} \to V \otimes V^\* \otimes V^{\ast\ast} \to V$$ where the description of the maps uses the data above.

39

https://mathoverflow.net/users/2926

718

https://mathoverflow.net/questions/686

30

I subscribe to feeds from the [arXiv Front](https://arxiv.org/abs/math) for a number of subject areas, using [Google Reader](http://www.google.com/reader). This is great, but there is one problem: when a new preprint is listed in several subject categories, it gets listed in several feeds, which means I have to spend more time reading through the lists of new items, and due to my slightly dysfunctional memory, I often download the same preprint twice. Is there a way to get around this problem, by somehow merging the feeds, using a different arXiv site, or using some other clever trick? (Hope this is not too off-topic, I think a good answer could be useful to a number of mathematicians. Also, I would like to tag this "arxiv" but am not allowed to add new tags.)

https://mathoverflow.net/users/349

Handling arXiv feeds to avoid duplicates

Unless the arXiv has changed recently, articles are published daily which means that the feeds and the email are completely in step. The problem with the duplicates is that each feed is a separate request to the arXiv for information. The arXiv doesn't know that you are going to merge these results, and I've never heard of a feed reader that attempts to merge feeds to remove duplicates. However, all is not lost. The feeds that the arXiv provides are not the only way to find information. The arXiv has an API which means that you can effectively craft your own feed. For example, if you point your browser at: `http://export.arxiv.org/api/query?search_query=submittedDate:[20091014200000+TO+20091015200000]&start=0&max_results=500` then you get all the papers submitted yesterday. You can filter your search by subject. `http://export.arxiv.org/api/query?search_query=%28cat:math.AT+OR+cat:math.CT%29+AND+submittedDate:[20091014200000+TO+20091015200000]&start=0&max_results=500` Because the requests are handled all at once, there are no duplicates produced (as can be seen since Emily Riehl's paper is both math.AT and math.CT). The only catch is that you need to put the date in proper form each time, you can't put in dates such as "today" or "yesterday". Plus the timezone handling is a little weird: the arxiv publishes updates at a certain time determined by the *local* timezone, which includes daylight saving changes, but the API uses GMT/UTC. So if you want to exactly replicated the "new preprints" announcement of the arxiv then you need to do some funky timezone conversions. However, this can be done and I've done it. I use a program called [RefBase](http://www.refbase.net) for organising my references and I've modified it so that each morning it presents me with a list of what's new on the arxiv for me to scan through and decide which articles to add to my own bibliographic database. I can also scan back a few days if I've been on holiday. Buried in this extension is the code for figuring out what the date-stamp should be. I could extract it if there's any interest. Documentation on the arxiv API is at [their documentation site](http://export.arxiv.org/api_help/). The 'submittedDate' stuff isn't covered there though, that's a newer feature.

24

https://mathoverflow.net/users/45

726

https://mathoverflow.net/questions/711

8

In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our space is a *noncompact* Calabi-Yau manifold, we get a reduced Hodge diamond, to which mirror symmetry probably applies. Unfortunately, I don't know anything about mirror symmetry. Do we still get meaningful geometric information (deformations, etc.)? I'd like to know what all the subtle obstructions are to defining things in the above way.

https://mathoverflow.net/users/436

Mirror symmetry for noncompact Calabi-Yau manifolds

There is a version of mirror symmetry, called "local mirror symmetry", for certain non-compact Calabi-Yaus, for example the total space of the canonical bundle of P^2 (exercise: show this is CY). The mirror (or rather one possible mirror) of this non-compact Calabi-Yau is an affine elliptic curve in (C^\*)^2. I don't think that there is as yet a version of mirror symmetry for more general non-compact CYs, though I don't know too much about this story. In all of the papers that I've looked at on this stuff at least, the only non-compact CYs that have been considered in mirror symmetry so far are total spaces of vector bundles over compact (probably Fano) things. I guess we should probably get some sort of "Hodge diamond symmetry" in local mirror symmetry, but the story becomes more complicated. One immediate thing to notice is that, at least in the example I've given, the dimensions of the manifolds aren't the same! So things will have to be modified. The Hodge diamond symmetry in mirror symmetry for compact Calabi-Yaus should really be thought of as coming from a correspondence between certain deformations of a Calabi-Yau and certain deformations of its mirror. In the non-compact case, the deformations that we should consider will be somewhat different from the deformations that we should consider in the compact case. A somewhat recent point of view, due to Kontsevich, is that this correspondence between deformations can be gotten from homological mirror symmetry. In homological mirror symmetry for compact Calabi-Yau manifolds, we consider a derived category of coherent sheaves on one side and a Fukaya category on the other side. Then we should have an equivalence of categories, plus an equivalence of a certain structure on their Hochschild cohomologies -- in particular their Hochschild cohomologies should be equivalent at least as vector spaces. These Hochschild cohomologies should be thought as the appropriate deformation spaces (or maybe rather the tangent spaces to the appropriate deformation spaces?), and an appropriate identification of the Hochschild cohomologies (plus the extra structure that I mentioned) should give in particular the Hodge diamond symmetry. There should also be homological mirror symmetry for non-compact Calabi-Yau manifolds, but we must define the analogues of derived category and Fukaya category in this situation appropriately. Then there should be an analogous story on the Hochschild cohomologies of the categories, and an appropriate analogue of the Hodge diamond symmetry. See the paper "Hodge theoretic aspects of mirror symmetry" by Katzarkov-Kontsevich-Pantev for more details.

7

https://mathoverflow.net/users/83

738

https://mathoverflow.net/questions/723

7

Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$? What is its support?

https://mathoverflow.net/users/447

Is the Fourier transform of $\exp(-\|x\|)$ non-negative?

This Fourier transform is positive, supported everywhere, and has polynomial decay. It is the Poisson kernel evaluated at time 1, up to some rescaling. <http://en.wikipedia.org/wiki/Poisson_kernel>

8

https://mathoverflow.net/users/373

752

https://mathoverflow.net/questions/743

6

Additionally, is there any intuitive way to visualize the cardinalities that result?

https://mathoverflow.net/users/441

What do models where the CH is false look like?

I think your reading is wrong. Set theorists have studied all sorts of additional axioms, some implying CH, some being strictly weaker than CH, and many contradicting CH. My understanding is that most set theorists today, if they have any opinion on the matter, prefer to think that CH is false. In particular, Woodin has recently advanced a philosophical argument for a strong additional axiom of set theory that implies that the continuum is \aleph\_2. However, the fact of the matter is that CH has very little effect on "ordinary mathematics". You can come up with a few down-to-earth seeming combinatorial statements that are equivalent to CH, but it really just never comes up if you never deal with objects that are either very infinite or are infinite and don't have a lot of structure attached to them. For reference, it is consistent for the cardinality of the real numbers to be almost any uncountable cardinality (the only requirement is that it have uncountable [cofinality](http://en.wikipedia.org/wiki/Cofinality). I don't have time now, but if you want more references on this I can try and add some later.

14

https://mathoverflow.net/users/75

753

https://mathoverflow.net/questions/731

38

Why were algebraic geometers in the 19th Century thinking of m-Spec as the set of points of an affine variety associated to the ring whereas, sometime in the middle of the 20 Century, people started to think Spec was more appropriate as the "set of points". What are advantages of the Spec approach? Specific theorems?

https://mathoverflow.net/users/416

"Points" in algebraic geometry: Why shift from m-Spec to Spec?

The basic reason in my mind for using Spec is because **it makes the category of affine schemes equivalent to the category of commutative rings**. This means that if you get confused about what's going on geometrically (which you will), you can fall back to working with the algebra. And if you have some awesome results in commutative algebra, they automagically become results in geometry. There's another reason that Spec is more natural. First, I need to convince you that any kind of geometry should be done in **LRS**, the category of locally-ringed spaces. A locally-ringed space is a topological space with a sheaf of rings ("the sheaf of (admissible) functions on the space") such that the stalks are local rings. Why should the stalks be local rings? Because even if you generalize (or specialize) your notion of a function, you want to have the notion of a function vanishing at a point, and those functions that vanish at a point should be a very special (read: unique maximal) ideal in the stalk. Alternatively, the values of functions at points should be elements of fields; if the value is an element of some other kind of ring, then you're not really looking at a point. Suppose you believe that geometry should be done in **LRS**. Then there is a very natural functor **LRS→Ring** given by (X,OX)→OX(X). It turns out that this functor has an adjoint: our hero Spec. For any locally ringed space X and any ring A, we have Hom**LRS**(X,Spec(A))=Hom**Ring**(A,OX(X)) ... it may look a little funny because you're not used to contravariant functors being adjoints. This is another reason that spaces of the form Spec(A) (rather than mSpec(A)) are very special. *Exercise:* what if you just worked in **RS**, the category of ringed spaces? What would your special collection of spaces be? Hint: it's really boring. **Edit:** Since there doesn't seem to be much interest in my exercise, I'll just post the solution. The adjoint to the functor **RS**→**Ring** which takes a ringed space to global sections of the structure sheaf is the functor which takes a ring to the one point topological space, with structure sheaf equal to the ring.

63

https://mathoverflow.net/users/1

756

https://mathoverflow.net/questions/747

14

Could anyone point to good readable references for learning about syntomic cohomology?

https://mathoverflow.net/users/349

References for syntomic cohomology

I like the notes [here](http://math.arizona.edu/~swc/notes/files/DLSTsuji2.pdf), though I must confess I haven't read them completely. It has a nice introduction and explains the context.

4

https://mathoverflow.net/users/2

758

https://mathoverflow.net/questions/274

6

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero? What about $H^{j}\left(\mathbb{R},F\right)$ for integers $j\ge 2$ ? (Here cohomology means derived functor cohomology as in, say, Hartshorne or EGA. Anyway this cohomology coincides with Cech cohomology since $\mathbb{R}$ is paracompact.)

https://mathoverflow.net/users/450

Non-zero sheaf cohomology

The sheaf cohomology ${H}^i(X,F)$ of a (topological) manifold $X$ of dimension $n$ vanishes for $i > n$. This is a topological version of Grothendieck's vanishing theorem above. You can find this result in Kashiwara-Schapira's "[Sheaves on manifolds](http://books.google.com/books?id=qfWcUSQRsX4C&lpg=PA475&ots=Dk-Z-JgqhP&dq=kashiwara%20schapira%20sheaves%20on%20manifolds&pg=PP1#v=onepage&q=&f=false)" proposition III.3.2.2. **Reference** Masaki Kashiwara, Pierre Schapira, [Houzel, Christian] *[Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux”](http://books.google.com/books?id=qfWcUSQRsX4C&lpg=PA475&ots=Dk-Z-JgqhP&dq=kashiwara%20schapira%20sheaves%20on%20manifolds&pg=PP1#v=onepage&q=&f=false)* by Christian Houzel. (English) Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag, pp. x+512 (1990), [MR1074006](https://mathscinet.ams.org/mathscinet-getitem?mr=1074006), [Zbl 0709.18001](https://zbmath.org/0709.18001).

11

https://mathoverflow.net/users/322

770

https://mathoverflow.net/questions/760

6

More precisely, does there exist a sequence $G\_1 < G\_2 < \cdots$ of finite groups such that the irreducible representations of $G\_n$ are parameterized by the plane partitions of total size $n$?

https://mathoverflow.net/users/290

Does there exist a sequence of groups whose representation theory is described by plane partitions?

Not if you want the direct analogue of the branching rule to hold: namely, if V is the representation of Gn corresponding to a plane partition A of n, then the restriction of V to Gn-1 is the direct sum of one copy of the representation corresponding to each plane partition of n-1 contained in A. That would allow you to compute the dimension of the representation corresponding to A as the number of paths in the containment poset of plane partitions from the empty partition to A. Some computation then shows that the order of G3 would be 1+4+4+1+4+1=15, but there's only one group of order 15, the abelian one, which doesn't work. You could imagine some variations of the branching rule, though, such as "if B is obtained from A by replacing k by k-1 then the irrep corresponding to A contains k copies of the irrep corresponding to B", and maybe something like that would work.

11

https://mathoverflow.net/users/126667

779

https://mathoverflow.net/questions/406

33

There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and Classical mechanics. I think I first learned about this years ago on *This Week's Finds*. I'm wondering if anyone can give me a precise mathematical statement of this "fact". Or to the right introduction to tropical mathematics. I can do the beginning. In classical mechanics, roughly (lower down I will mention some ways what I am about to say is false), when a system transitions from one configuration to another, it takes the route that *minimizes* some "action" (this idea dates at least to Maupertuis in 1744, and Wikipedia gives ancient-Greek analogs). Thus, for a system to transition from state $A$ to state $C$ in two seconds, after one second it is in the state $B$ that minimizes the sum of the action to get from $A$ to $B$ plus the action to get from $B$ to $C$. For comparison, quantum mechanics assigns to the pair $A,B$ an "amplitude", and the amplitude to go from $A$ to $C$ in two seconds is the sum over all $B$ of the amplitude to go from $A$ to $B$ times the amplitude to go from $B$ to $C$. (This is the basic principle of Heisenberg's matrix mechanics.) Anyway, we can understand both situations within the same language by considering the a matrix, indexed by states, filled with either the actions or the amplitudes to transition. In the quantum case, the matrix multiplication is the one inherited from the usual $(\times, +)$ arithmetic on $\mathbb{C}$. In the classical case, it is the $(+, \min)$ arithmetic of the tropical ring $\mathbb{T}$. Let's be more precise. To any path through the configuration space of your system, Hamilton defines an action $\operatorname{Action}(\mathrm{path})$. The classically allowed trajectories are the critical paths of the action (rel boundary values), whereas if you believe in the path integral, the quantum amplitude is $$\int \exp\left(\frac{i}{\hbar} \operatorname{Action}(\mathrm{path}) \right) \mathrm{d}(\mathrm{path}),$$ where the integral ranges over all paths with prescribed boundary values and the measure $\mathrm{d}(\mathrm{path})$ doesn't exist (I said "if"). Here $\hbar$ is the reduced Planck constant, and the stationary-phase approximation makes it clear that as $\hbar$ goes to $0$, the integral is supported along classically allowed trajectories. Of course, the path integral doesn't exist, so I will describe one third (and more rigorous) example, this time in statistical, not quantum, mechanics. Let $X$ be the space of possible configurations of your system, and let's say that $X$ has a natural measure $\mathrm{d}x$. Let $E : X \to \mathbb{R}$ be the energy of a configuration. Then at temperature $T$, the probability that the system is in state $x$ is (unnormalized) $\exp(-(kT)^{-1} E(x))$, by which I mean if $f : X \to \mathbb{R}$ is any function, the expected value of $f$ is (ignoring convergence issues; let's say $X$ is compact, or $E$ grows quickly and $f$ does not, or...): $$\langle f \rangle\_T = \frac{\int\_X \exp(-(kT)^{-1} E(x)) f(x) \mathrm{d}x}{\int\_X \exp(-(kT)^{-1} E(x)) \mathrm{d}x}$$ Here $k$ is Boltzmann's constant. It's clear that as $T$ goes to $0$, the above integral is concentrated at the $x$ that minimize $E$. In tropical land, addition and hence integration is just minimization, so in the $T \to 0$ limit, the integral becomes some sort of "tropical" integral. Here are some of the issues that I'm having: 1. When you slowly cool a system, it doesn't necessarily settle into the state that globally minimizes the energy, just into a locally-minimal state. And good thing too, else there would not be chocolate bars. 2. The problems are worse for mechanics. Classically allowed paths are not necessarily even local minima, just critical points of the Action function, so the analogy between quantum and warm systems isn't perfect. Is there something like $\min$ that finds critical points rather than minima? 3. More generally, the path integral is very attractive, and definitely describes a "matrix", indexed by the configuration space. But for generic systems, connecting any two configurations are many classically-allowed trajectories. So whatever the classical analogue of matrix mechanics is, it is a matrix valued in sets (or sets with functions to the tropical ring), not just in tropical numbers. 4. Other than "take your equations and replace every $+$ with $\min$ and every $\times$ with $+$", I don't really know how to "tropicalize" a mathematical object.

https://mathoverflow.net/users/78

How is tropicalization like taking the classical limit?

The analogy I've worked out from pieces here and there goes like this: using the logarithm and exponential, we define for two real numbers x and y the following binary operation $x §\_h y := h .ln( e^{x/h} + e^{y/h} )$ which depends on some positive real parameter $h$. Then we observe that as $h \rightarrow 0$ the number $x §\_h y$ tends to $max(x,y)$. (Proof: assume without loss of generality that $x>y$, so $(y-x)/h <0$. But since $h. ln( e^{x/h} + e^{y/h} ) = h. ln( e^{x/h} . (1+e^{(y-x)/h} )$ as $h \rightarrow 0$ we tend to $h. ln (e^{x/h} . (1+0) ) = x = max(x,y)$. QED.) Now, in quantum mechanics the canonical commutation relations between positions and momenta operators read $[x\_u,p\_v] = i \hbar \delta\_{uv}$ and in the limit $\hbar \rightarrow 0$ those commutators thus tend to $0$, which says that we recover classical mechanics where everything commutes. And in quantum mechanics what matters are wavefunctions which are superpositions of things of the form $A.e^{iS/\hbar}$ where $A$ is some amplitude and $S$ some phase (the action of the path). Going back to $§\_h$ we can rewrite $e^{(x §\_h y)/h } = e^{x/h}+ e^{y/h}$, and so there is your analogy: the tropical mathematics operation max(,) is some kind of classical limit of the (thereby quantum) operation +.

12

https://mathoverflow.net/users/469

791

https://mathoverflow.net/questions/457

14

Let A be a non-negative integer square matrix with eigenvalues x1, x2, ... xn. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results regarding the combinatorial interpretation of these integers: * If A is the adjacency matrix of a finite directed graph G, the power symmetric functions of the eigenvalues count closed walks on A with a distinguished starting point. * Similarly, the complete homogeneous symmetric functions of the eigenvalues count non-negative integer linear combinations of aperiodic closed walks on A. * (Gessel-Viennot-Lindstrom) If Aij is the number of paths from source i to sink j on, say, a 2-D lattice where the only permissible moves are to the right and up, then the elementary symmetric functions of the eigenvalues count the number non-intersecting k-tuples of paths from the sources to the sinks. In particular det A is the number of non-intersecting n-tuples of paths. Do these results generalize to give a nice combinatorial interpretation of the value of the Schur function associated to an arbitrary partition evaluated at x1, x2, ... xn in terms of some combinatorial object attached to A? What conditions need to be placed on A so that the Schur functions are always non-negative? Feel free to either talk about the GL(n) perspective or to frame your discussion entirely in terms of tableaux.

https://mathoverflow.net/users/290

What are the Schur functions of the eigenvalues of a non-negative integer matrix counting?

This is moving far enough afield from my own understanding that I'm not sure how useful it is, but have you looked at the Ph.D. thesis of Andrius Kulikauskas? He seems to be giving some sort of combinatorial interpretation in terms of Lyndon words for the Schur functions. <http://www.selflearners.net/uploads/AndriusKulikauskasThesis.pdf>

6

https://mathoverflow.net/users/405

793

https://mathoverflow.net/questions/775

7

What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would work, but this is clearly false (take a matrix ring over a field, for instance). Benson's *Representations and Cohomology I*, on the other hand, claims that the Jacobson radical is in fact the intersection of all maximal two sided ideals. He defines the Jacobson radical as the intersection of the annihilators of simple R-modules, which are precisely the maximal two-sided ideals. Since this is the same as the intersection of the annihilators of the individual elements of the simple modules, then this is the same as the intersections of the maximal left (or right) ideals. I don't see the flaw in Benson's reasoning, but I seem to recall hearing somewhere else that the Jacobson radical is not always the intersection of the maximal two-sided ideals. Who is correct here?

https://mathoverflow.net/users/396

What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?

Certainly every maximal ideal is the annihilator of a simple R-module, but the converse isn't true. See Exercise 4.8 in Lam's "[Exercises in Classical Ring Theory](http://books.google.com/books?id=S3pZbAByfDgC&pg=PA56&lpg=PA56&dq=exercises+in+classical+ring+theory&printsec=frontcover&source=bl&ots=8Fgz07rq_Y&sig=ilAV-8aCb7lhrxw0OkbhfqBcUVs&hl=en&ei=mi3ZSvfQHYXasgO-6ZiUBg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBMQ6AEwAg#v=onepage&q=&f=false)" for an example.

7

https://mathoverflow.net/users/430

798

https://mathoverflow.net/questions/812

77

What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I *am* applying for jobs, and I need to write one of these shortly. Let us assume for the sake of argument that I *have* a teaching philosophy; I am not asking you to tell me what my teaching philosophy should be. I would like to know how those responsible for hiring view teaching statements, especially in the case of new PhD's who don't necessarily have extensive teaching experience. (I believe this is appropriate for mathoverflow because it is of interest to "a person whose primary occupation is doing mathematics", as I am.)

https://mathoverflow.net/users/143

Teaching statements for math jobs?

Having been on both sides of the issue, I might say that having considered it for some time, I really don't know! But in reality if you are looking for a position at a research university, the Dean will want to have evidence (or the non-research faculty will want to have evidence) that you care about teaching. More precisely, some subset of your peers might have a very specific teaching philosophy although they may not be able to articulate it. Those peers want to know if your teaching philosophy coincides with theirs. A few years back everyone was "hot" on the use of technology in the classroom. I don't know what that means, but suppose that it means using TI calculators, power point (the horror, the horror) or a course blog. If you have a point of view on the positive value of these things then you should say so. The problem is that each department has its own mix of bozos. I am pretty much a chalk on slate kind of guy, and when someone tells me they like clickers in large classes, I wonder do they turn around to look at their students faces. So in an ideal world you would tailor your teaching statement to the place you want to go, or to the place that you are applying. Of course, you don't want to write 200 teaching statements, so that won't work. So I am back to the original premise. They want to know that you have thought about teaching.

48

https://mathoverflow.net/users/36108

813

https://mathoverflow.net/questions/815

32

Someone recently [said](https://mathoverflow.net/questions/329/what-is-koszul-duality) "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions". I have a rough idea why this is true ("don't throw away those higher homotopies -- you might need them some time in the future"), but I would be interested to have a more informed/detailed understanding of why triangulated categories are so abominable. So, my question: Where are some good places to read about this "Down with triangulated!" philosophy? I'm interested in this because in my research I have come across some categories which, very surprisingly, have a (semi-)triangulated structure. Perhaps I should be looking for an underlying A-infinity structure.

https://mathoverflow.net/users/284

triangulated vs. dg/A-infinity

I don't really think that triangulated categories are abominable, but they certainly have their problems which are a result of having forgotten the higher homotopies. For instance, non-functoriality of mapping cones can be fixed via dg-enhancement. Another problem has to do with localization for triangulated categories. The fact that one can put a model structure on a suitable category of dg-categories and take homotopy limits for instance is a very useful thing and allows one to talk more meaningfully about gluing and working locally. One can do this in the derived category but only in quite simple situations and it requires extra structure for it to work well (e.g. the structure of a rigid tensor category compatible with the triangulation). At least one good place to read in more detail about these problems and how to fix them using dg-categories is To¨en's *Lectures on DG-categories* which can be found [here](https://perso.math.univ-toulouse.fr/btoen/files/2012/04/swisk.pdf). I haven't really said anything about the A-infinity point of view, but I don't know it so well yet - hopefully someone else can say something. But I do know that at least one of the same problems rears its head namely that of wanting to work locally. For instance in homological mirror symmetry one would like to glue Fukaya categories of complicated things together from those of easier things. To do this one would certainly need to use the A-infinity point of view before taking derived categories and if it works it should be because one can take the appropriate homotopy limit (does the model structure to do this actually exist by the way?).

21

https://mathoverflow.net/users/310

825

https://mathoverflow.net/questions/631

4

This question is based on [a blog post of Qiaochu Yuan](http://qchu.wordpress.com/2009/08/12/krafts-inequality-for-prefix-codes/). Let P be a locally finite\* graded poset with a minimal element, and w be a weight function on the elements of P. Suppose that the total weight of the elements of rank k is bounded by 1. Then is the total weight of any antichain bounded by 1, or some constant c (independent of P or w?) The answer, of course, is no, and it's not hard to construct a counterexample. So what are the minimal conditions on P and/or w needed for such a result? Note that this should specialize to several well-known theorems. Taking the poset to be a Boolean lattice and w to be 1/(n \choose k), we obtain the LYM inequality, hence the question title. Taking the poset to be the set of finite-length binary words with X \leq Y if X is a prefix of Y, and w to be 1/2^k, we get back Kraft's inequality. And finally, for arbitrary P and setting w to be the constant function 1, we get back (half of a special case of) Dilworth's theorem. A secondary question: assuming such a result exists, is there a probabilistic proof of it, similar to the probabilistic proofs of Kraft and LYM? Edit 4: Most of the counterexamples I've constructed thus far have had trees as the underlying poset (i.e., if X \leq Z and Y \leq Z, then either X \leq Y or Y \leq X). This subcase seems to simplify the analysis somewhat, so it might be worth considering only trees. In fact, here's a toy problem which itself seems rather difficult: Can we characterize the weight functions on the infinite rooted binary tree, with the weight of each graded part equal to 1, that satisfy the strong property that the weight of any antichain is at most 1? \*Edit: Actually we want something somewhat stronger than local finiteness, namely that every element is covered by finitely many elements, so that there are are only finitely many elements of any given rank. Edit^2: Of course we also want the weight function to be nonnegative, or else scary bad things can start happening. The obvious restriction on the weight function requires it to be dependent only on the rank; interestingly, this is neither necessary nor sufficient for the strong form of the conjecture (i.e. the maximal weight of any antichain \leq the maximal weight of all the elements of rank k) to hold. (Counterexamples available upon request.) I'm still searching for a counterexample in this situation to the weak form of the conjecture (Edit^3: Counterexample found.), where every rank has bounded total weight but there are arbitrarily heavy antichains.

https://mathoverflow.net/users/382

Is there a "universal LYM inequality?"

An approach to the strong form of the property, based on a probabilistic proof of the LYM inequality (apparently due to Bollobas, can be found in Tao and Vu, Ch. 7): Consider a graded poset P and let G be a compact topological group that acts on P. Construct a weight function as follows. Fix a distinguished (saturated) chain C of P, and for each X, let S(X) be the set of elements in G that take X into C. Then w(X) is the normalized Haar measure of S(X). It's clear that the total weight of every graded part is 1; now if A is an antichain of P, no automorphism of P can take two elements X, Y \in A to C simultaneously. So the sets S(X) are distinct for all elements X \in A, and the total weight is therefore at most 1. The only obvious problem with this approach is that I don't see why the sets S(X) should be measurable. However, if P is finite, no such difficulty arises, and in particular this answers Qiaochu's question (from the comments) in the affirmative, at least for finite posets. This also specializes, for finite posets, to the examples already discussed. Does every weight function on a finite poset with the strong property arise in this way?

1

https://mathoverflow.net/users/382

826

https://mathoverflow.net/questions/836

6

Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h\_Z Y)\_{hG}$ weakly equivalent to $X\_{hG} \times^h\_{Z\_{hG}} Y\_{hG}$?

https://mathoverflow.net/users/126667

Do homotopy pullbacks commute with homotopy orbits (in spaces)?

Yes. A sketch: Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration. Having done this, there is a natural long exact sequence of homotopy groups $\to \pi\_\* (U) \to \pi\_\*(U\_{hG}) \to \pi\_\*(BG) \to \dots$ and applying this to the pullback diagram you can deduce (from the 5-lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence.

7

https://mathoverflow.net/users/360

839

https://mathoverflow.net/questions/828

11

What I had in mind was something like the following: X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way. Is there a good reason why this cannot possibly be true? Here X denotes a reduced scheme of finite type of a field k (I guess people usually call this prevariety). I am mostly interested in the case where k is algebraically closed.

https://mathoverflow.net/users/438

Is there a version of the valuative criteria for separateness/properness for varieties?

If you make the statement Fix an algebraically closed base field k and let X be a scheme of finite type over k. Then X/k is proper iff for all smooth quasi-projective curves C/k and all maps f: C\c -> X then f extends uniquely to f': C -> X. it seems true to me. This should basically comes down to the fact that one can classify birational equivalence classes of curves over k in terms of the 'abstract curves' coming from all possible discrete valuations on dimension 1 function fields K/k. So using the fact that in this situation it is sufficient to check the valuative criterion on DVRs it seems like it should not be so hard to see the equivalence. The argument I had in mind is as follows: The valuative criterion implies the statement above. For the converse it is sufficient to show that any f: Spec K -> X lifts to an open subset of the curve C`_`K determined by K/k by which I mean the unique nonsingular projective curve in the birational class corresponding to K/k (the distinguished point in the complement we think of as being removed is uniquely determined by the discrete valuation we pick so that is no problem). If f just hits a closed point we can just collapse C`_`K via the structure map to Spec k and this is fine since there is nothing to lift. If not, we hit a dimension 1 point whose closure with the reduced induced structure determines some curve C' birational to the corresponding C`_`K'. The map K' -> K induces a dominant morphism g: C`_`K -> C`_`K'. We thus get a map by taking a common (up to isomorphism) open in C`_`K' and C' , taking its preimage U in C`_`K and considering U -> C' -> X which is the desired lift of f to the quasi-projective curve U. I think there is also a slicker argument using the categorical definition of finite presentation. For the same reasons this works for checking separatedness when one makes the obvious modifications to the statement. Over other bases I am not sure at the moment... I can't remember if the birational classification is still that simple (although some people implicitly mean by variety that everything is over some fixed alg. closed base). In the more general case (if your definition of variety doesn't include a finite type over a noetherian base hypothesis) where one needs non-noetherian valuation rings I think this interpretation is false - non-noetherian valuation rings can have arbitrary dimension.

4

https://mathoverflow.net/users/310

841

https://mathoverflow.net/questions/730

11

While reading a [blog post](http://sbseminar.wordpress.com/2007/12/14/the-nullstellensatz-and-partitions-of-unity/) on partitions of unity at the Secret Blogging Seminar the following question came into my mind. Let $n$ be a positive integer and let $B\_1$ and $B\_2$ be $n \times n$ matrices with integer entries. Is it true that exactly one of the following two statements is true? 1. There is a vector $v \in \mathbb{Q}^n \backslash \mathbb{Z}^n$ such that both $B\_1v$ and $B\_2v$ are in $\mathbb{Z}^n$. 2. There are matrices $A\_1$ and $A\_2$ with integer entries such that $A\_1B\_1+A\_2B\_2=I$. Here, $I$ denotes the $n \times n$ identity matrix. The case $n=1$ is Bézout's identity.

https://mathoverflow.net/users/296

An "existence contra partition of unity" statement for integer matrices?

I believe that what you say is true. I'll sketch an argument. Let f:Zn ---> Z2n be the map of free Z-modules given by the matrices B1, B2 put in column (i.e. the direct sum of the morphisms given by B1 and B2). Now we rephrase conditions (1) and (2) in a slightly more abstract way: * (1) fails to hold if, and only if there exists p:Z2n ---> Zn such that, together with f, fit in a short exact sequence 0 ---> Zn ---> Z2n ---> Zn ---> 0 (\*) Indeed, the failure of (1) means that any v in Qn such f(v) in Z2n must be integral (i.e. v in Zn). In particular, this implies that f is injective. Moreover, take w in Z2n representing a nonzero torsion element in the cokernel of f. As w represents a torsion element, Nw belongs to the image of f for some big enough positive integer N, so there is v in Zn such that f(v) = Nw. But now f(1/N v) = w, and this means, by the failure of (1), that 1/N v is integral, so w is in the image of f and the cokernel of f has no torsion. As a finitely generated torsion-free Z-module is free, we get an exact sequence like (\*) above. This argument can easily be reversed, to show the equivalence between the existence of this exact sequence and the failure of (1). * (2) holds if, and only if there exists a morphism of Z-modules r:Z2n ----> Zn such that rf = id. Let r be represented by a matrix (A1,A2). Then gf has matrix A1B1 + A2B2, and gf = id if, and only if (2) holds. Now, the proof of what you asked for is easy. (1) fails if, and only if we can form the exact sequence (\*), but such an exact sequence is always split because Z^n is projective, so we can form such exact sequence if, and only if there exists a splitting r:Z2n ----> Zn, which is exactly condition (2).

5

https://mathoverflow.net/users/322

865

https://mathoverflow.net/questions/873

13

Are there any good **short** expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.

https://mathoverflow.net/users/498

Short Introduction to Planar Algebras

My paper with Emily and Noah about the [`D_{2n}` planar algebra](http://arxiv.org/abs/0808.0764) includes our attempt at a friendly explanation. It's all about arguably the simplest non-trivial example of a *subfactor* planar algebra.

13

https://mathoverflow.net/users/3

874

https://mathoverflow.net/questions/868

27

In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation with the Artin-Schreier sequence I believe you can show that, at least, the abelianization of the absolute Galois group is terrible. What is known about the absolute Galois group of the affine line in characteristic p? In addition, can spaces which are not A^1 (or extensions of A^1 to a larger base field) occur as covers?

https://mathoverflow.net/users/360

Etale covers of the affine line

Indeed, you can get whatever genus you want even with a fixed Galois group G, so long as its order is divisible by p: this is a result of Pries: .pdf [here.](http://www.math.colostate.edu/~pries/Preprints/00DecPreprints/06genus1_05.pdf) In fact, Pries has lots of papers about exactly what can happen; looking at her papers and the ones cited therein should give you a pretty thorough picture. We don't know the Galois group of the affine line, but we do know which finite groups occur as its quotients; this is a result of Harbater from 1994 ("Abhyankar's conjecture for Galois groups over curves.") **Update**: As a commenter pointed out, Harbater proved this fact for an arbitrary affine curve; the statement for the affine line was an earlier theorem of Raynaud.

17

https://mathoverflow.net/users/431

877

https://mathoverflow.net/questions/847

23

Apologies in advance if this is obvious.

https://mathoverflow.net/users/290

Is any representation of a finite group defined over the algebraic integers?

Not a satisfying argument: We can, first of all, find a basis in which the entries lie in some algebraic number field $K$. Let $\mathcal{O}$ be the ring of integers of $K$. Then there is a locally free $\mathcal{O}$-module $M$ of rank $n$ preserved by $G$: add up all the translates of $\mathcal{O}^n$ under $G$. Now, $M$ need not itself be free, but it is isomorphic as an $\mathcal{O}$-module to the sum of various ideals of $\mathcal{O}$. Now pass to an extension $L/K$ so that every ideal class of $K$ trivializes in $L$, e.g. the Hilbert class field; then $G$ preserves a free rank $n$ module for the ring of integers of $L$. Sorry!

22

https://mathoverflow.net/users/513

882

https://mathoverflow.net/questions/827

6

Is it true that the pro-objects of an abelian category form a category with enough projectives? In general, given an abelian category A, is there a canonical way to embed it a bigger abelian category A' with enough projectives (or injectives) and such that A' is universal with respect to this property?

https://mathoverflow.net/users/344

Embedding abelian categories to have enough projectives

It seems that Pro(A) does not have enough projectives in general. In Kashiwara-Schapira's book "Categories and Sheaves" they prove (corollary 15.1.3) that Ind(k-Mod) does not have enough injectives. This means, taking opposite categories, that Pro(k-Mod^{op}) does not have enough projectives. I don't know of any universal way of adding enough projectives.

3

https://mathoverflow.net/users/322

892

https://mathoverflow.net/questions/840

3

As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces. To answer this question, one defines various properties of a space such as connectedness, compactness, the fundamental group, betti numbers etc. However, it seems that these properties can at best be used to distinguish two spaces - i.e. if X is a space with property Q, and Y is a space without property Q, then we can say with certainty that X & Y are not homeomorphic. My question is this: given two arbitrary spaces, how does one show that they are homeomporphic, without explicitly showing a homeomorphism?

https://mathoverflow.net/users/490

The core question of topology

As others have noted, it's hopeless to try to answer this question for general topological spaces. However, there are a few positive results if you assume, say, that X and Y are both simply connected closed manifolds of a given dimension. For example, Freedman showed that if X and Y are oriented and have dimension four, then to check whether they're homeomorphic you just need to compute (i) the bilinear "intersection" forms on H^2(X;Z) and H^2(Y;Z) induced by the cup product; and (ii) a Z/2-valued invariant called the Kirby-Siebenmann invariant. The invariant in (ii) obstructs the existence of a smooth structure, so if you happened to know that both X and Y were smooth manifolds (hence that their Kirby-Siebenmann invariants vanished) you'd just have to look at their intersection forms to determine whether they're homeomorphic (however a great many examples show that this wouldn't suffice to show that they're diffeomorphic). In higher dimensions, Smale's h-cobordism theorem shows that two simply connected smooth manifolds are diffeomorphic as soon as there is a cobordism between them for which the inclusion of both manifolds is a homotopy equivalence. Checking this criterion can still be subtle, but work of Wall and Barden shows that in the simply-connected 5-dimensional case it suffices to check that there's an isomorphism on second homology H2 which preserves both (i) the second Stiefel-Whitney classes, and (ii) a certain "linking form" on the torsion subgroup of H2. If you drop the simply-connected assumption, things get rather harder--indeed if n>3 then any finitely presented group is the fundamental group of a closed n-manifold (which can be constructed in a canonical way given a presentation), and Markov (son of the probabilist) showed that the impossibility of algorithmically distinguishing whether two presentations yield the same group translates to the impossibility of algorithmically classifying manifolds. Even assuming you already knew the fundamental groups were isomorphic, there are still complications beyond what happens in the simply-connected case, but these can sometimes be overcome with the s-cobordism theorem. In a somewhat different direction, in dimension 3 one can represent manifolds by link diagrams, and Kirby showed that two such manifolds are diffeomorphic (which in dimension 3 is equivalent to homeomorphic) iff you can get from one diagram to the other by a sequence of moves of a certain kind. (see [Kirby calculus](http://en.wikipedia.org/wiki/Kirby_calculus) in Wikipedia; similar statements exist in dimension 4). I suppose that one could argue that this isn't an example of what you were looking for, since if one felt like it one could extract diffeomorphisms from the moves in a fairly explicit way, and one can't (AFAIK) just directly extract some invariants from the diagrams which completely determine whether the moves exist.

14

https://mathoverflow.net/users/424

894

https://mathoverflow.net/questions/903

44

I've been doing functional programming, primarily in OCaml, for a couple years now, and have recently ventured into the land of monads. I'm able to work them now, and understand how to use them, but I'm interested in understanding more about their mathematical foundations. These foundations are usually presented as coming from category theory. So we get explanations such as the following: > > A monad is a monoid in the category of endofunctors. > > > Now, my goal (partially) is to understand what that means. Can anyone suggest a gentle introduction to category theory, particularly one aimed at programmers already familiar with a functional language such as ML or Haskell, with references for further reading? Resources not necessarily aimed at programmers but accessible to readers with a background in discrete math and first-order logic would be quite acceptable as well.

https://mathoverflow.net/users/550

Resources for learning practical category theory

Online resources: * [The Catsters channel](http://www.youtube.com/user/TheCatsters) * [MATH198 course notes](http://haskell.org/haskellwiki/User:Michiexile/MATH198) - examples in Haskell * [Rydehard, Burstall: Computional Category Theory](http://www.cs.man.ac.uk/%7Edavid/categories/book/) - examples in ML (free reprint of a book) * [MAGIC course](http://www.cs.manchester.ac.uk/%7Ehsimmons/MAGIC-CATS/magic-cats.html) * [Barr, Wells: Category theory for computing science](http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf) (TAC TR22 is a free reprint of the book) * [Jaap van Oosten: basic category theory](http://www.itu.dk/%7Ebirkedal/teaching/category-theory-Fall-2001/basiccat.ps.gz) * [Tom Leinster](http://www.maths.gla.ac.uk/%7Etl/ct/) * [Eugenia Cheng](http://cheng.staff.shef.ac.uk/catnotes/categorynotes-cheng.pdf) * [Steve Awodey](http://www.andrew.cmu.edu/course/80-413-713/notes/) - very similar to the book mentioned by Quadrescence * [Daniele Turi](http://www.dcs.ed.ac.uk/home/dt/CT/) * [Thomas Streicher](http://www.mathematik.tu-darmstadt.de/%7Estreicher/CTCL.pdf) * [Abstract and concrete categories: the joy of cats](http://katmat.math.uni-bremen.de/acc/) - might be considered too verbose, but it's full of examples; slightly newer (?) version as [TAC TR17](http://tac.mta.ca/tac/reprints/articles/17/tr17.pdf) * [Spivak: Category Theory for Scientists](http://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013/textbook/MIT18_S996S13_textbook.pdf) - free textbook of a 2013 MIT OpenCourseWare; an updated (and non-free) version was published by MIT Press in 2014. * [Emily Riehl: Category Theory in Context](https://math.jhu.edu/%7Eeriehl/context.pdf) Books (not free): * Benjamin Pierce: Basic category theory for computer scientists, MIT Press 1991; a slight expansion/update of the earlier (and free) [CMU-CS-88-203 report](http://repository.cmu.edu/cgi/viewcontent.cgi?article=2846&context=compsci) * MacLane - solid mathematical foundations, but hardly any references to computing * [Martin Brandenburg](https://mathoverflow.net/users/2841/martin-brandenburg) - Einführung in die Kategorientheorie (in german) Category theory in Haskell: * [Wikibooks introductory text](http://en.wikibooks.org/wiki/Haskell/Category_theory) * [sigfpe's blog](http://blog.sigfpe.com) has a lot of category theory articles - (di)natural transformations, monads, Yoneda lemma... * [Comonad.Reader](http://comonad.com/reader/) * [The Monad.Reader](http://www.haskell.org/sitewiki/images/8/85/TMR-Issue13.pdf) - check "Calculating monads with category theory" * [Bartosz Milewski - Category Theory for Programmers](https://www.goodreads.com/book/show/33618151-category-theory-for-programmers) [Another list](http://www.cs.le.ac.uk/people/akurz/books.html)

46

https://mathoverflow.net/users/158

906

https://mathoverflow.net/questions/908

15

I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the search for larger and larger prime numbers such as in the [GIMPS project](http://www.mersenne.org/). It would seem to me since that since it already proven that there are an infinite number of primes, I am not quite sure why working to finding bigger and bigger really matters!? Is this a "*Climbing Mt Everest because it is there*" kind of thing, or is the search and finding bigger results somehow furthering mathematics in some kind of way?

https://mathoverflow.net/users/538

Why the search for ever larger primes?

Well the M in GIMPS stands for Mersenne, and it hasn't been proven that there are infinitely many Mersenne primes. But it's widely believed to be true--in fact there is a conjectural [estimate](http://en.wikipedia.org/wiki/Lenstra%E2%80%93Pomerance%E2%80%93Wagstaff_conjecture#Lenstra.E2.80.93Pomerance.E2.80.93Wagstaff_conjecture) of their frequency. I think the search for Mersenne primes is mostly a "because it is there" thing, but it could provide numerical evidence for or against this conjecture. GIMPS is probably more interesting as an experiment in massively distributed computation.

11

https://mathoverflow.net/users/126667

914

https://mathoverflow.net/questions/924

1

In the chapter 6.4 on normal and self-adjoint operators, there is an example of an infinite dimensional inner product space H that has a normal operator but that has no eigenvectors. The space is the set of functions f\_n(t) = e^(int) , t in [0,2pi] with inner product = 1/2pi \* integral\_0\_2pi(e^(-int) e^(imt))dt The operator T(f\_n) = f\_(n+1) My question is what is the basis for the validity of the following statement. any vector f in the inner product space can be represented as f = sum\_i=n\_to\_i=m(a\_i \* f\_i), a\_m <> 0 They use this fact to prove that the operator as no eigenvectors. But it seems to imply that any member of an infinite dimensional inner product space can be represented with a finite number of basis elements, since n and m are finite. Are there no elements of an infinite dimensional inner product space that require an infinite number of terms to represent?

https://mathoverflow.net/users/570

Friedberg, Insel, and Spence Linear Algebra example

The space being defined is the **span** of the functions fn, and in the definition of the span we only allow finite sums of the basis vectors. Edit: I should also mention that the notion of infinite sum in an inner product space doesn't make sense unless the space is also complete with respect to the induced norm, i.e. is a Hilbert space. In the nice situation a Hilbert space has an "orthonormal basis," which is not a basis in the linear algebra sense but in the sense that the span of the basis is dense.

4

https://mathoverflow.net/users/290

925

https://mathoverflow.net/questions/849

4

For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$

https://mathoverflow.net/users/492

Conjugation in SU(2)

It's not hard to explicitly construct G using the quaternions, assuming P is not \pm Q, and I think this is worth working out in detail because I really like this picture of SU(2). Identify SU(2) with the unit quaternions by the isomorphism ``` [a b ] [-\bar{b} \bar{a}] --> a+bj ``` and since the trace of such a matrix is 2\*Re(a), the traceless matrices in SU(2) correspond exactly to the purely imaginary unit quaternions. Now if we consider R^3 as the space of all imaginary quaternions, we can describe the SU(2) action on it geometrically: Let v=cos(t)+sin(t)w, where w is purely imaginary and |w|=1. Then the conjugation map x -> vxv^{-1} is a rotation of the plane orthogonal to w -- you can easily check that it's in SO(3), and that it fixes w because vw=wv. It's also not hard to check that the angle of this rotation is 2t. Consider P and Q as imaginary quaternions, hence also as vectors in R^3. Then PQ = -(P.Q) + PxQ, so let G=Im(PQ)/|Im(PQ)|. Now G is a unit vector orthogonal to both P and Q, and conjugation by G is the same as rotating the plane through P and Q by \pi, so GPG^{-1}=-P and GQG^{-1}=-Q as desired. Finally, we just go back from the unit quaternions to SU(2): up to scale, G was supposed to be the imaginary part of the quaternion PQ, so the matrix G is some constant times the traceless part of the matrix PQ.

4

https://mathoverflow.net/users/428

942

https://mathoverflow.net/questions/929

5

Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction thrilled me and I'm still curious about how that developed and if now a general theory of configurations as continuation of classical geometry exists. Do you know something about it? copy from the article: "Classical projective geometry was a beautiful field in mathematics. It died, in our opinion, not because it ran out of theorems to prove, but because it lacked organizing principles by which to select theorems that were important. Also, it was isolated from the rest of mathematics. Much of what we do may be regarded as direct continuation of nineteenth century synthetic geometry. In fact, we hope the new motivation of studying C-complexes will provide projective geometry with one organizational principle, and with one relation tying it to "mainstream" mathematics. We note … representable matroids, arrangements of hyperplanes, and motivic cohomology. A large part of this paper's exposition is motivated by this dream of continuing classical projective geometry." Edit: [*Mnev's theorem*](http://inc.web.ihes.fr/prepub/PREPRINTS/M02/M02-31.pdf), that every scheme over Z "is" a moduli space for point-configurations in the plane, makes me ask about applications and if versions for other number rings exist?

https://mathoverflow.net/users/451

a general theory of configurations?

We model theorists have been studying such things for the past couple of decades, so I know something about this. Suppose that (S, cl) is a matroid -- i.e. a set S endowed with a closure operator satisfying a couple of natural axioms; canonical examples are where S is a vector space and cl(X) = Span(X), or when S a "projective space" over some field (i.e. the set of all 1-dimensional subspaces of a vector space, and cl is the closure operator induced by linear span). In addition to the standard matroid axioms, let's assume: 1. cl(emptyset) = emptyset; 2. cl({a}) = {a} for any a in S; 3. (nontriviality) There is a subset X of S such that cl(X) is not the union of {cl(a) : a is in X}; 4. (modularity) For any finite-dimensional closed subsets X and Y of S, dim(X union Y) = dim(X) + dim(Y) - dim(X intersect Y); 5. dim(S) is at least 4, and any 2-dimensional subset contains at least 3 elements. Then it turns out that (S,cl) is isomorphic to a projective space over some skew field. I believe this was first proved in Emil Artin's book *Geometric Algebra* (1957). In the context of model theory, there has been a lot of research recently on matroids that arise in the models of certain theories (only we call them "pregeometries") and how properties of these matroids are related to definable group actions and vector spaces, in analogy to Artin's result above. See, for instance, Hrushovski's "stable group configuration theorem," which says that there is an infinite definable group whenever you have a certain finite configuration in a model of your theory. <http://en.wikipedia.org/wiki/Pregeometry_(model_theory)>

4

https://mathoverflow.net/users/93

949

https://mathoverflow.net/questions/947

22

I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as: Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away. Assume a (admittedly miraculous) input of the list of the exact latitude/longitude of every person on Earth at a particular time. Also take as provided a function $d(p\_1, p\_2)$ that returns the distance on the surface of the earth between $p\_1$ and $p\_2$ - I know this is not trivial, but it's "just spherical geometry" and not the important (to me) part of the question. What's the most efficient way to find the loneliest person? Certainly one solution is to calculate $d(\ldots)$ for every pair of people on the globe, then sort every person's list of distances in ascending order, take the first item from every list and sort those in descending order and take the largest. But that involves $n(n-1)$ invocations of $d(\ldots)$, $n$ sorts of $n-1$ items and one last sort of $n$ items. Last I checked, $n$ in this case is somewhere north of six billion, right? So we can do better?

https://mathoverflow.net/users/587

How does one find the "loneliest person on the planet"?

The paper *Vaidya, Pravin M.*, [**An $O(n \log n)$ algorithm for the all-nearest-neighbors problem**](https://doi.org/10.1007/BF02187718), Discrete Comput. Geom. 4, No. 2, 101-115 (1989), [ZBL0663.68058](https://zbmath.org/?q=an:0663.68058) gives an $O(n \log n)$ algorithm for the "all-nearest-neighbors" problem: given a set of points $S$, find all the values $m(p)$ where $p$ is a point of $S$ and $m(p)$ is the minimum distance from $p$ to a point of $S \setminus \{p\}$. Then the "loneliest point" is the point $p$ which maximizes $m(p)$. So your problem can be solved in $O(n \log n)$ time, which is pretty good. (In case it's not clear, I'm applying their algorithm to the set of points viewed as living inside $\mathbb{R}^3$, using the fact that there's an order-preserving relationship between distance along the sphere and straight-line distance in $\mathbb{R}^3$.)

29

https://mathoverflow.net/users/126667

955

https://mathoverflow.net/questions/953

52

The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched cover over a set of codimension 3, since I guess, my intuition is all about 1-D loops, and not spheres.

https://mathoverflow.net/users/353

Analogue to covering space for higher homotopy groups?

There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi\_n$ for $n \geq 2$. We could next ask for a $2$-connected cover $X''$ of $X'$: a space $X''$ with $\pi\_kX = 0$ for $k \leq 2$ and a map $X'' \to X'$ which is an isomorphism on $\pi\_n$ for $n \geq 3$. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension $1$ - it will be a $K(\pi\_2X, 1)$. (For the universal cover the fiber was the discrete space $\pi\_1X = K(\pi\_1X, 0)$.) An example is the Hopf fibration $K(\mathbb{Z}, 1) = S^1 \to S^3 \to S^2$. Geometrically it's harder to see what's going on with the $2$-connected cover than with the universal cover, because fibrations with fiber of the form $K(G, 1)$ are harder to describe than fibrations with discrete fibers (covering spaces).

62

https://mathoverflow.net/users/126667

957

https://mathoverflow.net/questions/951

4

I'm aware of the great body of work on [Legendrian knot](http://en.wikipedia.org/wiki/Legendrian_knot) theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops based at a point in a contact manifold? Can that be made into a "Legendrian fundamental group" somehow? I've heard that h-principles are somehow involved, but I'm not sure what the punchline is.

https://mathoverflow.net/users/353

Legendrian homotopy of curves in a contact structure?

In general, the (parametric) h-principle for Legendrian immersions implies that Legendrian immersions f:L->(M,\xi) are classified up to homotopy (through Legendrian immersions) by the following bundle-theoretic invariant: Choosing a compatible almost complex structure on \xi allows one to complexify the differential of f to an isomorphism d\_C f: TL\otimes C -> f\*\xi, and the relevant invariant is the homotopy class of this isomorphism of complex vector bundles (of course this is independent of the almost complex structure since the space of compatible almost complex structures is contractible). The above holds in any contact manifold (M,\xi) of arbitrary dimension. Of course when M is 3-dimensional and L is S^1, f^\*\xi is the unique complex line bundle over S^1, automorphisms of which are parametrized up to homotopy by pi\_1(U(1))=Z. So (given that the h-principle also implies that any loop in a 3-manifold is homotopic to a Legendrian loop) it appears to always be the case that the "Legendrian fundamental group" surjects onto the standard fundamental group, with kernel Z. When M=R^3 this invariant is equivalent to the rotation number that Steven mentioned. There's a proof of the relevant h-principle in the book by Eliashberg and Mishachev. The above discussion is partly based on Section 3.3 of arXiv:0210124 by Ekholm-Etnyre-Sullivan.

4

https://mathoverflow.net/users/424

963

https://mathoverflow.net/questions/983

16

So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based. I was wondering if anyone had a suggestion where I should go from here if I'm interested in learning about similar fields. If would be nice if I could relate it back to programming, but not necessarily a prerequisite. Thanks.

https://mathoverflow.net/users/621

What is lambda calculus related to?

To get you started, you could try this: <http://math.ucr.edu/home/baez/week240.html> It has a whole lot of references, some of which you might want to follow up. In particular, some of them describe the close relations between functional programming, lambda calculus and cartesian closed categories. One reference that looks particularly good is the set of lecture notes by Peter Selinger, which are written from a somewhat computer sciencey perspective.

18

https://mathoverflow.net/users/586

987

https://mathoverflow.net/questions/968

14

In the answers to [Qiaochu's post](https://mathoverflow.net/questions/847/is-any-representation-of-a-finite-group-defined-over-the-algebraic-integers) on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend on whether you require your representation to be free or just projective. If your representation V is defined over a number field $K,$ it contains a projective submodule $V'$ for the integers $O$ of $K$ such that $V'\otimes\_O K=V,$ but it's not at all clear if this can be chosen to be free. Luckily, there is a very nice theory of algebraic number theory that says that any projective module over the ring of integers of a number field becomes free when you extend scalars to the Hilbert class field. So, since all representations of finite groups are defined over cyclotomic fields (in fact, you just need the roots of unity for the orders of elements in G), every representation has as an integral basis in the Hilbert class field of a cyclotomic field. Which is.....? Edit: while the class numbers are interesting that's not the question I asked. I want to actually what the Hilbert class field is, or something about it. For example, is it cyclotomic (seems unlikely, but cyclotomic fields are nice...)?

https://mathoverflow.net/users/66

What is the Hilbert class field of a cyclotomic field?

Giving an "explicit" description of the Hilbert class field of a number field K (or, more generally, all abelian extensions of K) is Hilbert's 12th problem, and has only been solved for Q and for imaginary quadratic fields. The Hilbert class field H of Q(zeta) will only be contained in a cyclotomic field if H = Q(zeta) itself --- since one can explicitly compute that any abelian extension of Q properly containing Q(zeta) is ramified at some place.

14

https://mathoverflow.net/users/nan

999

https://mathoverflow.net/questions/915

76

The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}\_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for $2$. So in these examples characteristic $2$ is a messy special case. On the other hand, certain types of combinatorial questions can be reduced to linear algebra over $\mathbb{F}\_2,$ and this relationship doesn't seem to generalize to other finite fields. So in this example characteristic $2$ is a nice special case. Is anything deep going on here? (I have a vague idea here about additive inverses and Fourier analysis over $\mathbb{Z}/2\mathbb{Z}$, but I'll wait to see what other people say.)

https://mathoverflow.net/users/290

Is there a high-concept explanation for why characteristic 2 is special?

I think there are two phenomena at work, and often one can separate behaviors based on whether they are "caused by''one or the other (or both). One phenomenon is the smallness of $2$, i.e., the expression $p-1$ shows up when describing many characteristic $p$ and $p$-adic structures, and the qualitative properties of these structures will change a lot depending on whether $p-1$ is one or greater than one. For example: * Adding a primitive $p^\text{th}$ root of unity $z$ to ${\bf Q}\_p$ yields a totally ramified field extension of degree $p-1$. The valuation of $1-z$ is $1/(p-1)$ times the valuation of $p$. This is a long way of saying that $-1$ lies in ${\bf Q}\_2$. * The group of units in the prime field of a characteristic $p$ field has order $p-1$. This is the difference between triviality and nontriviality. * As you mentioned, some combinatorial questions can be phrased in Boolean language and attacked with linear algebra. The other phenomenon is the evenness of $2$. Standard examples: * Negation has a nontrivial fixed point. This gives one way to explain why there are $4$ square roots of $1 \pmod {2^n}$ (for $n$ large), but only $2$ in the $2$-adic limit. If you combine this with smallness, you find that negation does nothing, and this adds a lot of subtlety to the study of algebraic groups (or generally, vector spaces with forms). * The Hasse invariant is a weight $p-1$ modular form, and odd weight forms behave differently from even weight forms, especially in terms of lifting to characteristic zero, level 1. This is a bit related to David's mention of abelian varieties — I've heard that some Albanese "varieties" in characteristic $2$ are non-reduced.

63

https://mathoverflow.net/users/121

1003

https://mathoverflow.net/questions/994

9

The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it satisfies the following criterion: > > For any valuation ring R (with K=Frac(R)) and any morphisms Spec(R)→Y and Spec(K)→X making the following square commute > > > > ``` > > Spec(K) ---> X > | | > | | f > v v > Spec(R) ---> Y > > ``` > > there exists at most one (resp. exactly one) morphism Spec(R)→X filling in the diagram. > > > But if Y is locally noetherian and f:X→Y is of finite type, then this condition only needs to be verified for **discrete** valuation rings. Does anybody know of an example where it is not sufficient to use DVRs? In other words, does there exist a morphism of schemes f:X→Y which *is not* separated (resp. proper), but *does* satisfy the valuative criterion for DVRs? *Reference:* [EGA II](http://www.numdam.org/numdam-bin/item?id=PMIHES_1961__8__5_0), Proposition 7.2.3 and Theorem 7.3.8

https://mathoverflow.net/users/1

Example where you *need* non-DVRs in the valuative criteria

You can probably just take Y to be the spectrum of a valuation ring A which is not a DVR, for example the integral closure of C[[t]] in an algebraic closure of C((t)). In this case any homomorphism from A to a DVR R has to factor through the quotient field or the residue field of A. For an explcit example, Let X be two copies of Spec(A) glued along the complement of the closed point and let X --> Y be the map which is the identity on both copies. The fibres of this map are both proper so it satisfies the valuative criterion using only DVRs by the observation above but the map itself fails to be proper.

14

https://mathoverflow.net/users/519

1042

https://mathoverflow.net/questions/889

8

Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor powers of $L$ is a polynomial in $k$. If L is very ample, this is just the Hilbert polynomial, and this can be proven by an induction argument twisting the short exact sequence $0 \to \mathcal O(-1) \to \mathcal O \to K \to 0$. More generally, if $L$ (or $L^\*$) is an ideal sheaf, the same argument should work. Why does the result still hold for arbitrary $L$? Edit: I'd be particularly interested in an elementary proof that does not involve proving an entire Riemann-Roch theorem--Mumford is using this result to *prove* Riemann-Roch for abelian varieties!

https://mathoverflow.net/users/75

Why is the Euler characteristic of powers of a line bundle a polynomial in the power?

OK, here is another way to see it more in line with what you had in mind I think. Write your $L$ as $\mathcal O(D)$ for some divisor $D$ on $X$. Set $J\_1$ to be the ideal sheaf defined by $\mathcal O(-D) \cap \mathcal O\_X$ and $J\_2$ to be the ideal sheaf defined by $\mathcal O(D) \cap \mathcal O\_X$ (intersections taken inside of $K\_X$). Let $Y\_i$ be the closed subschemes of $X$ defined by these ideal sheaves (they have dimension smaller than that of $X$). Then we have the exact sequences $$0 \to J\_1(kD) \to \mathcal O(kD) \to \mathcal O\_{Y\_1}(kD) \to 0$$ $$0 \to J\_2((k-1)D) \to \mathcal O((k-1)D) \to \mathcal O\_{Y\_2}((k-1)D) \to 0$$ The two left hand terms are equal by construction. Then by the induction hypothesis, and chasing the Euler characteristics, $\chi(kD) - \chi((k-1)D)$ is a numerical polynomial. This implies that that $\chi(kD)$ itself is a numerical polynomial (Section 1.7 of Harshorne's Algebraic Geometry). (Here I swept something under the rug, because the subschemes $Y\_i$ may not be as nice as $X$ was. But they are at least proper, and we should show that the result we want is that for a proper variety $W$, $\chi(kD)$ is polynomial for a divisor $D$. Then reduce this to the case where $W$ is reduced by looking at the inclusion of $W\_\mathrm{red}$ into $W$. Then further reduce to the case where $W$ is integral.)

5

https://mathoverflow.net/users/397

1044

https://mathoverflow.net/questions/1039

6

Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the direct summands of the pushforward f\*ℚX[dim X]. My question is as follows: What more general statements are there that enable us to explicitly calculate the direct summands of the pushforward? I'm thinking especially of the case where f:X→Y is as above (so in particular semismall), but we replace the constant sheaf ℚX with an arbitrary perverse sheaf (of geometric origin).

https://mathoverflow.net/users/425

Explicit Direct Summands in the Decomposition Theorem

I'm a little confused about your question. If you just want to know what the summands are there's nothing special about f\*ℚX[dim X]. The only way I know of understanding that sheaf is a general algorithm for understanding all semi-simple perverse sheaves. The only fact you use is that a simple perverse sheaf is concentrated in the highest degree allowed by perversity on the largest stratum in its support, and strictly below this degree on all smaller strata (do I need some hypothesis for this? This tends to be the sort of thing I forget). So, if I have a semi-simple perverse sheaf F, all I have to is look at the restriction of F to each stratum. This will be a complex, whose cohomology in each term is a local system. Perversity includes an upper bound on the degrees that this cohomology can be non-zero. I take the local system in the highest degree allowed by perversity on each stratum, and take the IC sheaves of all those local systems. It happens that in the case of f\*ℚX[dim X], this local system has a geometric interpretation (it's the highest degree cohomology of the fiber allowed by semi-smallness), but that's the only thing that's special about this case.

2

https://mathoverflow.net/users/66

1045

https://mathoverflow.net/questions/1024

2

Let G be the [gamma function](http://en.wikipedia.org/wiki/Gamma_function), and b be a constant in (-2,inf). Let H(n, i) = G(i+1+b) \* G(n-i+1+b) / [G(i+1) \* G(n-i+1)] for integers n > i > 0. Let S(n) = \sum\_{i=1}^{i=n-1} H(n, i). Let x\_ n = H(n,1) / S(n). Note x\_ 2 = 1, x\_ 3 = 1/2 for all b. I am convinced that as n -> inf, x\_ n -> 0 for b >= -1, and x\_ n -> (-b-1)/2 for -2 < b < -1. I can prove the b >= -1 case, but not the other, except when b=-3/2. Can anyone help with a proof? I have found recursive relationships between x\_ n and S(n): S(n+1) = (1/(n+1)) [n + 2 + 2b + 2(n+b)x\_ n/n] S(n) x\_ {n+1} = (n+b)/n x\_ n S(n) / S(n+1) = (n+b)(n+1)x\_ n / [n(n+2+2b) + 2(n+b)x\_ n] which may be of use. One way to deal with the b >= -1 case is use the latter to relate 1/x\_ {n+1} and 1/x\_ n and show this tend to infinity. For background, see section 4 of Probability Distributions on Cladograms (1996) by David Aldous In Random Discrete Structures (its available free online) Graham

https://mathoverflow.net/users/650

Limit of sequence involving gamma functions

Using Mathematica and using reflection formulae for Gamma one finds: x[n,b] = (b+1) n/(n+b) G[n+b+1]/G[n+2b+2] / ( G[b+1]/G[2b+2] - 2 G[n+b+1]/G[n+2b+2] ) Now, observe that for b<-1 the quotients G[n+b+1]/G[n+2b+2] tend to infinity as n->oo (this follows from Stirling's approximation). Accordingly, for such b, x[n,b] -> (b+1) / (-2) which is what you predicted. I don't think that b>-2 is needed. To prove the above formula by hand (or to see why a computer can do this), you may want to have a look at the WZ method (the book A=B by Petkovsek, Wilf and Zeilberger is a wonderful and freely available introduction).

3

https://mathoverflow.net/users/359

1049

https://mathoverflow.net/questions/1048

46

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do $$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. Why do we need to square and then square-root the numbers?

https://mathoverflow.net/users/668

Why is it so cool to square numbers (in terms of finding the standard deviation)?

One answer is mathematical convenience. The theory is much simpler using powers of two rather than other powers. There are justifications. Squaring makes small numbers smaller and makes big numbers bigger. You could argue that a useful measure of dispersion should be forgiving of small errors but weigh larger errors more heavily, so it makes sense to square the deviations from the mean.

-4

https://mathoverflow.net/users/136

1057

https://mathoverflow.net/questions/530

18

More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex representation? I know that e ≤ 3. For various special values of n we can attain the lower bound e ≥ 2. For example, if n+1 = q is a power of a prime then the group of affine linear transformations ax + b on Fq is doubly transitive, so the linearization of this group action decomposes into the trivial representation and an irreducible representation. Edit: Oops. Reading Noah's comment, I probably should've justified the upper bound. I hope this example is correct: let zeta be an nth root of unity and consider the group generated by the diagonal matrix (zeta, zeta, ... zeta), the diagonal matrix (1, zeta, zeta^2, ... zeta^{n-1}), and a cyclic permutation matrix of order n. It can be verified that this is a group of order n^3 and that its elements span M\_n(C) , so the given representation is irreducible.

https://mathoverflow.net/users/290

How small can a group with an n-dimensional irreducible complex representation be?

I [posted](https://mathoverflow.net/questions/834/arithmetic-progressions-without-small-primes) a question about primes in arithmetic progressions and was told my understanding is wrong. Although proving this would be harder than the generalized Riemmann hypothesis, the expectation of experts is that, for any n, there is a prime q which is 1 mod n and is O(n^{1+ \epsilon}). In that case, Z/n acting on Z/q would achieve e = 2+\epsilon. It seems like the best result we are likely to prove is that the only way to beat e=3 is to find a small prime power which is 1 mod n.

6

https://mathoverflow.net/users/297

1066

https://mathoverflow.net/questions/1075

49

Let G be a finite group of order n. Must every automorphism of G have order less than n? (David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil don't!)

https://mathoverflow.net/users/126667

Order of an automorphism of a finite group

Yes every automorphism has order bounded by |G|-1, provided G is not the trivial group. A reference is M V Horoševskiĭ 1974 Math. USSR Sb. 22 584-594 which can be found at <http://www.iop.org/EJ/abstract/0025-5734/22/4/A08/> It is even shown that the upper bound is reached only for elementary abelian groups.

49

https://mathoverflow.net/users/310

1077

https://mathoverflow.net/questions/1095

5

Sorry if the terminology's wrong, I don't know differential topology. Also, this is more of a brain-teaser than a bona fide research question, but it's hopefully a "real mathematician"-level brain-teaser. So the crossing number inequality gives a lower bound for how many intersections you have to have if you draw a graph with enough edges in the plane. The thing is, the crossing number inequality counts intersections "with multiplicity." It's pretty easy to see that if you don't count intersections with multiplicity, you can draw any finite graph in the plane with only one "crossing point." However, while there's an easy explicit description of such a drawing, the edges aren't smooth (or even first-differentiable!) So my question is in two parts: 1. Is there a drawing of K\_n in the plane such that there is exactly one point where edges can intersect, and all edges are smooth embeddings of the (open) unit interval in R^2? 2. Is there an explicit description of such a drawing? (E.g., can you write the edges as real algebraic curves?) (Note to moderators: you might want to tag this as "recreational" or "brain-teaser" or something of that sort; I don't have 250 reputation and so can't. :-/)

https://mathoverflow.net/users/382

Smooth immersion(?) of graphs into the plane

If a graph can be embedded into the plane with smooth edges and one point of crossing, then it can be embedded smoothly into the projective plane without crossings, by blowing up the crossing point (and conversely). But not every graph can be embedded into the projective plane, so not every graph has a smooth embedding of the type you describe.

6

https://mathoverflow.net/users/440

1098

https://mathoverflow.net/questions/1058

92

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological spaces of finite dimension, and well-ordered sets. However, it fails in other categories: topological spaces, groups, rings, fields, graphs, posets, etc. Can we caracterize Cantor-Bernsteiness in terms of other categorical properties? [Edit: Corrected misspelling of Bernstein]

https://mathoverflow.net/users/416

When does Cantor-Bernstein hold?

Whenever the objects in your category can be classified by a bounded collection of cardinal invariants, then you should expect to have the Schroeder-Bernstein property. For example, vector spaces (over some fixed field $K$) or algebraically closed fields (of some fixed characteristic) can each be classified by a single cardinal invariant: the dimension of the vector space, or the transcendence degree of the field. More interesting example: countable abelian torsion groups. Suppose A and B are two such groups, $A$ is a direct summand of $B$, and vice-versa; are they isomorphic? By Ulm's Theorem, $A$ and $B$ are determined up to isomorphism by countable sequences of cardinal numbers -- namely, the number of summands of $\mathbb{Z}\_p^\infty$ and the "Ulm invariants," which are dimensions of some vector spaces associated with $A$ and $B$. All of these invariants behave nicely with respect to direct sum decompositions, so it follows that $A$ and $B$ are isomorphic. (See Kaplansky's *Infinite Abelian Groups* for a very nice, and elementary, proof of all this.) If you like model theory, I could tell you a lot about when the categories of models of a complete theory have the Schroeder-Bernstein property (under elementary embeddings). If not, at least I can tell you this: 1. Categories of structures with "definable" partial orderings with infinite chains (e.g. real-closed fields, atomless Boolean algebras) will NOT have the S-B property. Again, I need some model theory to make this statement precise... 2. Let $C$ be a first-order axiomatizable class of structures (in a countable language) which is "categorical in $2^{\aleph\_0}$" -- i.e. any two structures in $C$ of size continuum are isomorphic. Then $C$ has the S-B property with respect to elementary embeddings. (This generalizes the cases of vector spaces and algebraically closed fields.) Addendum: A completely different way that a category $C$ might be Schroeder-Bernstein is if every object is "surjunctive" (i.e. any injective self-morphism of an object is necessarily surjective). This covers Justin's example of the category of well-orderings.

79

https://mathoverflow.net/users/93

1101

https://mathoverflow.net/questions/1076

9

The pentagon and hexagon axioms in the definition of a symmetric monoidal category are one example that I was thinking of here; the axioms of a weak 2-category are another. I understand that it can be checked laboriously that these few coherence axioms are sufficient to show, e.g. in the first case, that all coherence conditions we want on associativity and commutativity to hold do, but this is rather tedious. Is there some other motivation for the choice of coherence axioms?

https://mathoverflow.net/users/344

Motivation for coherence axioms

I agree with Viritrilbia: there is in general no "nice" or canonical choice of coherence axioms. By "in general" I mean for an arbitrary 2-dimensional theory, such as the theory of weak 2-categories, monoidal categories, braided monoidal categories with duals, etc. This is true for the same reason that there is in general no "nice" or canonical presentation of a given group. Nevertheless, there are some famous patterns in the coherence axioms for certain commonly encountered theories. The associahedra, or Stasheff polytopes, give the higher associativity coherence axioms. (There is one such polytope of each dimension. The pentagon is the associahedron of dimension 2.) The hexagon in the definition of symmetric monoidal category is also part of a general pattern. Topologists often sweep units under the carpet. In the definition of monoidal category, the pentagon is not the only coherence axiom: there is also the triangle, which has to do with the unit coherence isomorphisms $X \otimes I \to X$. People talk much less about the triangle, even though it's a crucial part of the definition. I don't know of a general pattern that it fits into. Incidentally, when the definition of monoidal category was first given (by Mac Lane?), there were one or two extra coherence axioms that were later shown to be redundant. This is an indication of how non-obvious the choice of axioms is.

10

https://mathoverflow.net/users/586

1104

https://mathoverflow.net/questions/1124

13

In Bonn, we've been have a discussion on the topic in the title: > > Suppose that $A$ and $B$ are classes and that there are injections from $A$ to $B$ and from $B$ to $A$. Does it follow that there is a bijection between $A$ and $B$? > > > Example: Let $A$ the class of sets of cardinality one and let $B$ be the class of sets of cardinality two. There is an injection $A\to B$ sending $a$ to $\{a,\varnothing\}$, $B\to A$ sending b to $\{\{b\}\}$. Does it follow that there is a bijection between $A$ and $B$?

https://mathoverflow.net/users/296

Does Cantor-Bernstein hold for classes?

Ignoring set-theoretic technicalities of formulating the question properly, I see no reason that the usual proof of Schroder-Bernstein wouldn't work. (Set-theoretic technicalities: In the standard language of set theory, you can't quantify over classes, so you can't quite state this. However, you can prove a metatheorem saying that whenever you exhibit two such injections, you can prove there is also a bijection. Alternatively, you could work in [set theory with classes](http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory), in which the statement can be made properly and you ought to be able to prove it just like ordinary Schroder-Bernstein. Alternatively, it is a trivial corollary of the "global" axiom of choice (which implies, in particular, that all proper classes have the same size), though this is kind of applying a sledgehammer.)

7

https://mathoverflow.net/users/75

1127

https://mathoverflow.net/questions/1129

3

Sorry if this is obvious. I'd like to understand why the map WC(E/Q) -> H^1(Gal(**Q**/Q), E(**Q**)) is bijective. Thanks.

https://mathoverflow.net/users/436

Weil-Châtelet group

This should be in chapter 10 of Silverman's AEC (sorry if I'm wrong about this!).

4

https://mathoverflow.net/users/2

1130

https://mathoverflow.net/questions/1108

9

I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she realized she wasn't going to make it. Here is my question, loosely phrased: Given that we want to make it across the intersection before the light is up, what should be our position function? To make things precise, we could specify some initial conditions, put a cap on the car's acceleration/deceleration (or its speed or its jerk or whatever), fix a probability distribution on when the light will change, and assign some point values to our happiness if we make the light vs. don't make the light vs. end up entering the intersection after the light has changed and get a ticket for it. And then of course we could throw in the extra curveball of a yellow light warning you that it's about to turn red... A similar question arises if you're approaching a red light but you think it might turn green soon. Ideally you'd like to enter the intersection at the highest possible speed just after the light has turned green, but then again you don't want to enter when it's still red. I'm sure a computer could solve such problems easily, but it seems like there should be some better way to think about this than just asking a machine to do it for me. For the first question, it seems like the answer will just be either "hit the gas and go for it" or "cruise to a stop and don't plan on going through", depending on the parameters. (Of course, there might be something in the "go just fast enough that you can slow down and not enter on a red" plan if the cost of a ticket is high enough.) On the other hand, the second question seems to admit much more interplay between probability and differential equations. The real issue here is that I know almost nothing about either of these two fields. Any ideas?

https://mathoverflow.net/users/303

easy(?) probability/diff eq. question

I think there's potentially an interesting set of problems here - I thought about it myself some time ago, and I'm [not the only one](http://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/#comment-33878) (if you're too lazy to follow the link, this is Tim Gowers posing exactly this question on Terry Tao's blog). I would classify them as "Operations Research" rather than probability or DE; the latter could be tools to be used here, but it's unlikely that there are significant theoretical challenges here once the problem is clearly posed. Of course, one of the challenges here is posing the right problem. For example, it's not hard to see that if you're at rest at a certain known distance from the light, and your acceleration is limited to some value A, and you know when the light is about to turn green, then your best strategy is to stand still until the correct (easily calculated) time, and then slam the gas pedal and speed so as to make it to the light just in time. In other instances such as if your initial speed is too high to make it before the light turns green, you may be better off breaking first and then accelerating. In general, it is typical of such problems that the strategy is "extremal" in the sense that the action at each time is either full deceleration or full acceleration; it may be hard to find a problem where the strategy is more subtle. Still, determining the optimal timings etc. could be an interesting challenge. I did not yet spend any time trying to find a good definition of the problem, but it's a worthwhile thing to think about.

6

https://mathoverflow.net/users/25

1134

https://mathoverflow.net/questions/1081

9

It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes: "The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category."

https://mathoverflow.net/users/689

What is an example of a topological space that is not homotopy equivalent to a CW-complex?

My favorite example of a space which is not homotopy equivalent to a CW complex is the Long Line. All it's homotopy groups vanish (exercise 1) but the long line is not contractible (exercise 2). It's too long!

10

https://mathoverflow.net/users/184

1137

https://mathoverflow.net/questions/1142

6

**Edit:** Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not? Question as originally stated: Suppose X is a complete nonsingular curve (smooth proper integral scheme of dimension 1 over C) and D ∈ DivX. I have heard that very ampleness of D is determined entirely by its degree and the genus of X. How can this be done explicitly? Thanks!

https://mathoverflow.net/users/84526

Is very ampleness of a divisor on a curve determined entirely by degree and genus?

Let $C\_1$ be a hyperelliptic curve of genus $g \geq 3$ (example: $y^7 = x^2 + 1$ for $g = 3$), and $C\_2$ be a non-hyperelliptic curve of the same genus $g$ (for example, the Klein quartic with $g = 3$ again: I'll use it in the form $y^7 = x^2(x-1)$). Then let $K\_1$, $K\_2$ be the canonical divisors of $C\_1$, $C\_2$ respectively (In the example above: $K\_1 = 4[\infty\_1]$ and $K\_2 = 4[\infty\_2]$, where $\infty\_1, \infty\_2$ are the "points at infinity" on the curves above: note that the projective curves corresponding to the equations above both have singularities at infinity, but in both cases these singularities can be resolved to give a single point at infinity on each curve.) Then $K\_1$ is not very ample but $K\_2$ is: this is a standard application of Riemann-Roch (see also Hartshorne, Chapter IV, Proposition 5.2.) The reason that $K\_1$ is not very ample is that the map $C\_1 \rightarrow P^2$ associated to $K\_1$ is a double cover of a rational curve (a conic, in fact) in $P^2$, rather than an embedding.

11

https://mathoverflow.net/users/422

1158

https://mathoverflow.net/questions/1159

32

A lot of times I see theorems stated for local rings, but usually they are also true for "graded local rings", i.e., graded rings with a unique homogeneous maximal ideal (like the polynomial ring). For example, the Hilbert syzygy theorem, the Auslander-Buchsbaum formula, statements related to local cohomology, etc. But it's not entirely clear to me how tight this analogy is. I certainly don't expect all statements about local rings to extend to graded local rings, so I'd like to know about some "pitfalls" in case I ever decide to make an "oh yes, this obviously extends" fallacy. What are some examples of statements which are true for local rings whose graded analogues are not necessarily true? Or another related question: what kind of intuition should I have when I want to conclude that statements have graded versions? There is a notion of "generalized local ring" due to Goto and Watanabe which includes graded local rings and local rings: a positively graded ring that is finitely generated as an algebra over its zeroth degree part, and its zeroth degree part is a local ring, so one possibility is just to see if this weaker definition is enough to prove the statement. Of course the trouble comes when the proofs cite other sources, and become unmanageable to trace back to first principles.

https://mathoverflow.net/users/321

Graded local rings versus local rings

One small thing I know of which changes is that if one has a Z-graded-commutative noetherian ring (where Z is the integers) Matlis' classification of indecomposable injective modules goes through but with one small hiccup. Every indecomposable injective is isomorphic to E(R/p)[n] for some unique homogeneous prime ideal p but the integer shift n is not necessarily unique although under the hypotheses I think you are interested in one probably gets uniqueness. I can't think of an example where this really causes much of a problem though. Having thought about this some more I think that non-negative integer graded-local noetherian rings, in particular those generated in degree 1 such that the maximal homogeneous ideal is also maximal if one forgets the grading, are incredibly well behaved and the analogy with local rings is very good. In fact, there is even a version of Nakayama's lemma for such rings (maybe one needs a little more) which is stronger than the usual one in the sense that one can drop the finiteness condition on the module. There are also no problems with graded versions of prime avoidance etc... in general. I'd recommend section 1.5 of Cohen-Macaulay Rings by Bruns and Herzog where they prove that a bunch of standard facts still go through and one can see what does and doesn't change in the proofs. As I mentioned in the comment I think one has to be most careful when considering rings graded by things like monoids which aren't as nice as the non-negative integers. In particular, if the grading is not positive (i.e. some elements of the monoid are invertible) and/or if the monoid is not cancellative at the identity (i.e. a+b = a does not imply b is the identity). I think in the non-cancellative case one can construct a counterexample to Nakayama's lemma but I am not 100% sure on this.

8

https://mathoverflow.net/users/310

1163

https://mathoverflow.net/questions/1151

58

In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come into play? It is very confusing and awkward to me that why taking injective stuff into consideration would allow you to "extend" a left exact functor.

https://mathoverflow.net/users/nan

Sheaf cohomology and injective resolutions

Since everybody else is throwing derived categories at you, let me take another approach and give a more lowbrow explanation of how you might have come up with the idea of using injectives. I'll take for granted that you want to associate to each object (sheaf) $F$ a bunch of abelian groups $H^i(F)$ with $H^0(F)=\Gamma(F)$, and that you want a short exact sequence of objects to yield a long exact sequence in cohomology. I also want one more assumption, which I hope you find reasonable: if $F$ is an object such that for any short exact sequence $0\to F\to G\to H\to 0$ the sequence $0\to \Gamma(F)\to \Gamma(G)\to \Gamma(H)\to 0$ is exact, then $H^{i}(F)=0$ for $i>0$. This roughly says that $H^{i}$ is zero unless it's forced to be non-zero by a long exact sequence (you might be able to run this argument only using this for $i=1$, but I'm not sure). Note that this implies that injective objects have trivial $H^{i}$ since any short exact sequence with $F$ injective splits. Now suppose I come across an object $F$ that I'd like to compute the cohomology of. I already know that $H^{0}(F)=\Gamma(F)$, but how can I compute any higher cohomology groups? I can embed $F$ into an injective object $I^{0}$, giving me the exact sequence $0\to F\to I^{0}\to K^{1}\to 0$. The long exact sequence in cohomology gives me the exact sequence $$0\to \Gamma(F)\to \Gamma(I^{0})\to \Gamma(K^{1})\to H^{1}(F)\to 0 = H^1(I^{0})$$ That's pretty good; it tells us that $H^{1}(F)= \Gamma(K^{1})/\mathrm{im}(\Gamma(I^{0}))$, so we've computed $H^{1}(F)$ using only global sections of some other sheaves. We'll come back to this, but let's make some other observations first. The other thing you learn from the long exact sequence associated to the short exact sequence $0\to F\to I^{0}\to K^{1}\to 0$ is that for $i>0$, you have $$H^{i}(I^{0}) = 0\to H^{i}(K^{1})\to H^{i+1}(F)\to 0 = H^{i+1}(I^{0})$$ This is great! It tells you that $H^{i+1}(F)=H^{i}(K^{1})$. So if you've already figured out how to compute $i$-th cohomology groups, you can compute $(i+1)$-th cohomology groups! So we can proceed by induction to calculate all the cohomology groups of $F$. Concretely, to compute $H^{2}(F)$, you'd have to compute $H^{1}(K^{1})$. How do you do that? You choose an embedding into an injective object $I^{1}$ and consider the long exact sequence associated to the short exact sequence $0\to K^{1}\to I^{1}\to K^{2}\to 0$ and repeat the argument in the third paragraph. Notice that when you proceed inductively, you construct the injective resolution $$0\to F\to I^{0}\to I^{1}\to I^{2}\to\cdots$$ so that the cokernel of the map $I^{i-1}\to I^{i}$ (which is equal to the kernel of the map $I^{i}\to I^{i+1}$) is $K^{i}$. If you like, you can define $K^{0}=F$. Now by induction you get that $$H^{i}(F) = H^{i-1}(K^{1}) = H^{i-2}(K^{2}) = \cdots = H^{1}(K^{i-1}) = \Gamma(K^{i})/\mathrm{im}(\Gamma(I^{i-1})).$$ Since $\Gamma$ is left exact and the sequence $0\to K^{i}\to I^{i}\to I^{i+1}$ is exact, you have that $\Gamma(K^{i})$ is equal to the kernel of the map $\Gamma(I^{i})\to \Gamma(I^{i+1})$. That is, we've shown that $$H^{i}(F) = \ker[\Gamma(I^{i})\to \Gamma(I^{i+1})]/\mathrm{im}[\Gamma(I^{i-1})\to \Gamma(I^{i})].$$ Whew! That was kind of long, but we've shown that if you make a few reasonable assumptions, some easy observations, and then follow your nose, you come up with injective resolutions as a way to compute cohomology.

92

https://mathoverflow.net/users/1

1165

https://mathoverflow.net/questions/1102

16

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular construction if G is finitely presented, etc. What I'm wondering about, is there a notion of a smooth classifying space? That is, when can a classifying space for a group be given a smooth structure?

https://mathoverflow.net/users/343

Smooth classifying spaces?

The answer to this does depend highly on the category in which you are prepared to work. If by "smooth structure" you mean "when is BG a **finite dimensional** manifold" then the answer is, as Andy says, "not many". However if you are prepared to admit that there are more things that deserve the name "smooth" than just finite dimensional manifolds, then the answer ranges from "a few" to somewhere near "all". To illustrate this with examples, the classifying space of ℤ is, of course, S1 whilst the classifying space of ℤ/2 is ℝℙ∞. Both are manifolds, but only the first is finite dimensional. Here are some more details for the "somewhere near all". Take any topological model for BG. Then consider all continuous maps ℝ → BG. These correspond to G-bundles over ℝ. Amongst those will be certain bundles which deserve the name "smooth" bundles. By taking the corresponding curves, one determines a family of curves ℝ → BG which should be called "smooth". Using this one can define a Frölicher space structure on BG. (It is possible that you will get more smooth bundles than you bargained for this way. If that's a problem, you could work in the category of diffeological spaces but then you'd need to use all the ℝns). In the middle, one can consider infinite dimensional manifolds. Then as your group is discrete it would be enough to ensure that you have a properly discontinuous action on an infinite sphere (there's a question somewhere around here about that being contractible). Some would say that your sphere "ought" to be the sphere in some Hilbert space. Failing that, if you have a faithful action on a Hilbert space (or more generally Banach space) with one or two topological conditions then you can quotient the general linear group by your group. Indeed, if your group is discrete then take the obvious action on ℓ2(G) (square summable sequences indexed by G). A good example, but which is about as far from your situation for discrete groups as possible, is that of diffeomorphisms on a manifold. The classifying space of this group is the space of embeddings of that manifold in some suitable infinite dimensional space. For more on the categories behind all this, see the nlab entries starting with [generalised smooth spaces](http://ncatlab.org/nlab/show/generalized+smooth+space) and the references therein. Also, anything by Kriegl, Michor, or Frolicher in the literature is worth a look.

13

https://mathoverflow.net/users/45

1177

https://mathoverflow.net/questions/1166

2

More precisely, is every concrete category C isomorphic to a category C' of small categories such that the morphisms between two elements of C are precisely the functors between their images in C'? At some point I started adopting this point of view as a philosophy without ever bothering to actually verify it.

https://mathoverflow.net/users/290

Can the objects of every concrete category themselves be realized as small categories?

No. There is no full subcategory of Cat equivalent to • ⇉ •, because for any two objects C and D of Cat, either both Hom(C, D) and Hom(D, C) are nonempty, or one of C and D is the empty category and then one of Hom(C, D) and Hom(D, C) is empty and the other is the category •.

4

https://mathoverflow.net/users/126667

1178

https://mathoverflow.net/questions/1184

23

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and then ask how far they can be generalized. * A good place to start is [Hadwiger's theorem](http://en.wikipedia.org/wiki/Hadwiger%27s_theorem). Define a "Hadwiger measure" *m* on **R**n to be a thing that assigns (possibly negative) real numbers to (nice?) subsets of **R**n in such a way that the assignment is invariant under rigid transformations (i.e. isometries) and satisfies the "inclusion-exclusion" principle that *m*(\_A\_ ∪ *B*) = *m*(\_A\_) + *m*(\_B\_) - *m*(\_A\_ ∩ *B*); Hadwidger measures are also required to satisfy some analytic properties. Then Hadwidger proves that the space of measures on **R**n is precisely (*n*+1)-dimensional, and has a basis *mi* with *mi*([0,1]i) = 1 and *mi*(λ *A*) = λi *mi*(*A*), where λ *A* is the set rescaled by a factor of λ in every direction. In particular, *m*0 of a finite set counts the number of points, and agrees with Euler characteristic for compact regions; the function *m*0 is the "combinatorial Euler characteristic". It is not homotopy-invariant: *m*0([0,1]) = 1 whereas *m*0(**R**) = -1. It is multiplicative. Incidentally, Hadwiger's paper is in German and so I cannot read it. Apparently all this material is in Rota's book "Introduction to Geometric Probability", but I have been away from a library and haven't read it yet. Thus I don't know the precise statement of "nice". * Schanuel in MR1173024 various "geometric categories". Namely, say that a subset of **R**n is a "polyhedron" if it is the positive locus finitely many affine maps to **R**; close the collection of polyhedra under union, intersection, and complement, and thus recover the notion of "polyhedral set" (so that a polyhedral set is actually a pair (*n*,\_S\_) where *S* is a subset of **R**n satisfying certain properties). Then morphism of polyhedral sets is a set-theoretic function whose graph (as a subset of **R**n x **R**m) is polyhedral. Then it's straightforward to check that a morphism is an isomorphism if it is a set-theoretic bijection — morphisms allow gluing and cutting. Or replace the word "affine" with "polynomial" and thus recover the notion of "semi-algebraic set". Or restrict your attention to bounded polyhedral sets. Anyway, each of these geometric categories has well-behaved product and coproduct, and so a "Burnside Rig" (ring without negation) whose elements are isomorphism classes of objects. Schanuel computes each of these Burnside rigs, and shows that the universal cancelative quotient of each is the integers; this map to **Z** is the combinatorial Euler characteristic. * Apparently there are also more analytic definitions. Schanuel in MR842922 (wonderful but only trying to develop intuition and motivation) suggests that each of the Hadwiger measures can be defined in terms of curvatures and whatnot, but the formulas he gives only make sense for compact manifolds (with boundaries, corners...). Chen (MR1215324) describes the combinatorial Euler characteristic with the following fun integral: let *f*: **R** → **R** be continuous except for finitely many jump and/or removable discontinuities, and define ∫Eulerf = Σ*x*∈**R** [ f(x) - (1/2) (f(x+) + f(x-)) ]; then try to compute Euler integrals of characteristic functions. The problem is that he then defines the multi-dimensional version via the Fubini theorem, but suggests that his integrals depend on a choice of basis. * The definition of combinatorial Euler characteristic is great for "finite polyhedral complexes", I think. By a "finite polyhedral complexes" I mean glue together finitely many polyhedra, but you're allowed to leave some faces open, so that unlike a CW complex not every cell must have complex closure. Then you can calculate Euler characteristic with the usual formula: (number of cells of even dimension) - (number of cells of odd dimension). I think this is a topological (but not homotopy!) invariant. Anyway, so first, are there references I've missed? Second and more importantly, all the references consider only subsets of Euclidean space (well, Schanuel briefly mentions the Burnside rig of varieties/**C**, but only computes a quotient). Why? Why isn't there an intrinsic topological description, or perhaps manifold-theoretic description? In particular, a "measure-theoretic" version that does not rely on embeddings in Euclidean space would be great, as it would presumably give "measures" against which we could integrate smooth functions. Any ideas?

https://mathoverflow.net/users/78

Is there a topological description of combinatorial Euler characteristic?

the best approach to the geometric euler characteristic comes from the theory of o-minimal structures. the best reference in this area is the book "tame topology and o-minimal structures" by lou van den dries. requires very little background to understand. in brief: an o-minimal structure is collection of boolean algebras of subsets of $R^n$ which satisfies a short list of axioms. (the name comes from model theory, but you don't need to know any model theory to understand the results) examples of o-minimal structures include the semialgebraic sets, the globally subanalytic sets, and (if you tweak the definitions a bit) the piecewise-linear sets. elements of an o-minimal structure are "tame" or "definable" sets. mappings between tame sets are tame iff their graph is a tame set. basic relevant results: every tame set has a well-defined euler characteristic. two tame sets are "definably homeomorphic" (there is a tame bijection between them --- not necessarily continuous!) iff they have the same dimension and euler characteristic. (yes, i wrote iff - this is the first surprise in this subject) one can so this for more general manifolds as well. concerning integration with respect to euler characteristic: 1) in the o-minimal framework, one can integrate all constructible functions, as noted by viro and schapira in the 1980s, based on works of macpherson and kashiwara in the 1970s. these results follow from sheaf theory. though more difficult than the combinatorial approach, all these proofs are "natural" and don't rely on "luck". 2) if you want to integrate non-constructible (e.g., smooth) integrands, the theory of chen (really due to rota) will fail -- that integral vanishes on all continuous integrands. 3) baryshnikov and ghrist have extensions of the integral to definable integrands (see 2009 arxiv paper). there are two such extensions, and they are dual. there are deep connections with morse theory, but the integral operators are unfortunately non-linear, and the fubini theorem does not hold in full generality.

15

https://mathoverflow.net/users/15245

1192

https://mathoverflow.net/questions/1162

56

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of time discussing the topology and algebra, but very little time on the analysis. Question : is there a "fun" source for reading about the appropriate parts of analysis?

https://mathoverflow.net/users/317

Atiyah-Singer index theorem

I found Booss, Bleecker: "Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics" ([review](http://projecteuclid.org/euclid.bams/1183553481)) very beautifull and had read it just for fun. It is a very nice piece of exposition, motivates everything and demands from the reader only very little preknowledge.

29

https://mathoverflow.net/users/451

1193

https://mathoverflow.net/questions/1186

3

I'm trying to compute the singular cohomology of SO(4), just as practice for using spectral sequences. I got H0=**Z**, H1=0, H2=**Z**/2**Z**, H3=**Z**⊕**Z**, H4=0, H5=**Z**/2**Z**, and H6=**Z**. Are these correct? I'm not sure if I'm reading it right, but these calculations seem to disagree with [this](http://www.math.cornell.edu/~hatcher/SO/SO(n).pdf) pretty cool little note on the cohomology ring of SO(n) (check out the crazy pictures at the bottom!). Also, in the spirit of "teaching a man to fish", does anyone know of some place where people have collected all these sorts of calculations (and possibly also homology and homotopy calculations)? Lastly, how can I determine the ring structure on H\*(SO(4)) from these calculations? Supposedly the isomorphism of whatever your usual cohomology is (de Rham, singular, whatever) with the cohomology of the double complex respects the ring structure, but is it really just as easy as saying that the product of a (p,q)-element with an (r,s)-element lives in (p+r,q+s) and is the thing you'd expect it to be? ADDENDUM: Since it's likely that they're incorrect, I'll lay out my process here and hopefully someone with some spare time on their hands can tell me where I went wrong. (I apologize in advance for trying to describe the spectral sequence of a double complex without any diagrams! I tried, but apparently tables don't work on Math Overflow just yet.) I'm working out of Bott & Tu. I'm using the standard fibration of SO(4) over S3 with fiber SO(3). They have Leray's theorem (15.11, at least in my very old edition) giving that, since the base is simply-connected, E2p,q=Hp(S3,Hq(SO(3);**Z**)). We know that H0(SO(3))=H3(SO(3))=**Z**, H2(SO(3))=**Z**/2**Z**, and Hn(SO(3))=0 otherwise. By the universal coefficient theorem, the singular cohomology of S3 with coefficients in an abelian group G is just G in dimensions 0 and 3, and 0 elsewhere. So I've got E2 only nonzero in columns 0 and 3, where it's **Z**, 0, **Z**/2**Z**, **Z**, 0, 0, 0... This is in fact E∞, since the only potentially nonzero map from here on out is from the (0,2) entry to the (3,0) entry, but this has to be a homomorphism from **Z**/2**Z** to **Z**, which is necessarily zero. Summing along the diagonals yields the results I gave above.

https://mathoverflow.net/users/303

singular cohomology of SO(4)

Your calculation is correct. In Hatcher's description of integral cohomology mod torsion the last generator has to be interpreted as an extra generator in addition to the previous ones. The integral cohomology ring of the limit is in general not isomorphic to the E\_{\infty} ring, but the E\_{\infty} ring is the associated graded to a filtration on the limit. You might want to look at the Leray-Hirsch theorem which tells you about part of the cup product structure whenever the fundamental group of the base acts trivially and there are no differentials. (And you can use that the induced maps H^*(B)\to H^*(E)\to H^\*(F) in cohomology are ring isomorphisms.) In your particular case it turns out that there is only one possibility: the E\_{\infty} and the limit are isomorphic as rings.

4

https://mathoverflow.net/users/86

1200

https://mathoverflow.net/questions/1123

13

Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}\_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}\_m$ as objects in the abelian category of sheaves of abelian groups on the fppf site of $S$, and take $\mathrm{Ext}$'s between them. We know that $\mathrm{Ext}^1(A,\mathbf{G}\_m)$ is the dual abelian scheme; but what is $\mathrm{Ext}^i(A,\mathbf{G}\_m)$ for $i>1$? Here is an argument why these higher $\mathrm{Ext}$'s contain no important information: the dual abelian scheme captures already all the data of $A/S$, since applying $\mathrm{Ext}^1(\cdot, \mathbf{G}\_m)$ one more time recovers $A$; therefore the higher $\mathrm{Ext}$'s cannot hold any more information. Still, we should know explicitly what they are.

https://mathoverflow.net/users/307

What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?

It seems that here you are talking about the local exts, i.e. about the sheaves $\underline{Ext}^{i}(A,G\_m)$ on $S$. The question is rather tricky actually. The problem is that before we try to answer it, we should first specify what we mean by $\underline{Ext}^{i}(A,G\_m)$. We could mean exts in the category of sheaves of abelian groups in the flat topology on $S$, or we could mean exts in the category of commutative group schemes. The latter is not an abelian category so you have to do something before you can define exts. It is possible to do this though. The standard lore is to use Yoneda exts. This is carried out in detail in the LNM 15 book by Oort. Among other things Oort checks that if $S$ is the spectrum of an algebraically closed field, then ext sheaves are all zero for $i \geq 2$. Over general base schemes the situation is more delicate. First of all there are examples of Larry Breen showing that the ext sheaves in the category of sheaves of abelian groups are strictly larger than the ext sheaves in the category of commutative group schemes. In his thesis Breen, Lawrence Extensions of abelian sheaves and Eilenberg-MacLane algebras. Invent. Math. 9 1969/1970 15--44. Breen also showed that over a regular noetherian base schemes the global ext groups (in either category) are torsion if $i \geq 2$. Later in Breen, Lawrence Un théorème d'annulation pour certains $E{\rm xt}\sp{i}$ de faisceaux abéliens. Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 339--352. he strengthened his result to show that the higher exts sheaves are always zero for $1 < i < 2p-1$, where $p$ is a prime which is smaller than the (positive) residue characteristic of any closed point in $S$.

11

https://mathoverflow.net/users/439

1220

https://mathoverflow.net/questions/1238

4

This is a pretty basic question but I have been stuck on it for a while. Given an abstract simplicial complex X and a subcomplex A, why does \* suffice to show that the map |A|->|X| induced by inclusion is a homotopy equivalence: * Let g: (|K|,|L|) -> (|X|,|A|) be a continuous map, where K is a **finite** simplicial complex and L a subcomplex of K. Any such g is homotopic rel |L| to a map sending |K| into |A|. Here |.| denotes the geometric realization. I'm trying to understand the very first step of the proof of Proposition 2.2 of J-C. Hausmann's paper "[On the Vietoris-Rips complexes and a cohomology theory for metric spaces](http://www.ams.org/mathscinet-getitem?mr=1368659)".

https://mathoverflow.net/users/353

proving that an inclusion map from a subcomplex is a homotopy equivalence

By taking K = a simplex and L = its boundary you can show that |A| -> |X| is an isomorphism on all homotopy groups (do surjectivity and injectivity separately). Then apply Whitehead's theorem.

3

https://mathoverflow.net/users/126667

1244

https://mathoverflow.net/questions/1237

72

The automorphism group of the symmetric group $S\_n$ is $S\_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S\_6$ with the (cyclic) group of order $2$. (For this famous outer automorphism, see for instance wikipedia or Baez's thoughts on the number $6$.) On the other hand, $S\_2$ is the automorphism group of $Z\_3$, $Z\_4$ and $Z\_6$ (and only those groups among finite groups). Hence my question: is $S\_6$ the automorphism group of a group? of a finite group?

https://mathoverflow.net/users/336

Is ${\rm S}_6$ the automorphism group of a group?

${\rm S}\_6$ is not the automorphism group of a finite group. See H.K. Iyer, *On solving the equation Aut(X) = G*, Rocky Mountain J. Math. 9 (1979), no. 4, 653--670, available online [here](http://dx.doi.org/10.1216/RMJ-1979-9-4-653). This paper proves that for any finite group $G$, there are finitely many finite groups $X$ with ${\rm Aut}(X) = G$, and it explicitly solves the equation for some specific values of $G$. In particular, Theorem 4.4 gives the complete solution for $G$ a symmetric group, and when $n = 6$ there are no such $X$.

89

https://mathoverflow.net/users/428

1245

https://mathoverflow.net/questions/605

13

One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I\_n$, type $II\_1$, and type $III$ factors are algebraically simple (any 2-sided ideal must contain a projection. All projections are comparable in a factor, so you can show 1 is in the ideal). Ideals in $B(H)$ ($\dim(H)=\infty$, $H$ separable) have been studied extensively. What about ideals in $II\_\infty$ factors? One might think, since every $II\_\infty$ factor $M$ can be written as $N\overline{\otimes} B(H)$ for $N$ a $II\_1$ factor, if $I\subset B(H)$ is an ideal, then $N\otimes I$ is a 2-sided ideal. This is false. One needs to take the ideal generated by $N\otimes I$. What does that mean from a von Neumann algebra viewpoint? Is it the same as taking the norm closure? We can also describe some ideals in terms of the trace. One has the equivalent of the Hilbert-Schmidt operators: $$I\_2=\{x\in M | tr(x^\ast x)<\infty\}$$ and the trace class operators: $$I\_1=\{x\in M | tr(|x|)<\infty\}=I\_2^\ast I\_2 =\left\{\sum^n\_{i=1} x\_i^\ast y\_i | x\_i, y\_i\in I\_2\right\}.$$ What is the relation of $I\_j$ to $N\otimes L^j(H)$ for $j=1,2$ (where $L^2(H)$ is the Hilbert-Schmidt operators and $L^1(H)$ is the trace class operators in $B(H)$)? Is $I\_j$ the norm closure of $N\otimes L^j(H)$?

https://mathoverflow.net/users/351

Ideals in Factors

Blackadar in his textbook on operator algebras gives a complete classification of norm-closed ideals in factors. See Proposition III.1.7.11.

12

https://mathoverflow.net/users/402

1258

https://mathoverflow.net/questions/1206

9

Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let $$S(\alpha) = \sum\_{n \leq N} \Lambda(n) e(\alpha n).$$ Now, using, say, Lemma 7.15 in Iwaniec-Kowalski (or the same result in Montgomery), we get $$\sum\_{q \leq q\_0} \sum\_{a \pmod{q}: \gcd(a,q)=1} \lvert S(a/q)\rvert^2 \leq \frac{(N + Q^2) N \log N}{\sum\_{\substack{q\leq Q \text{ squarefree} \\ \gcd(q,P(q\_0))=1}} \phi(q)^{-1}},$$ where $Q$ is arbitrary and $P(z):=\prod\_{p \leq z} p$. In practice, we would choose $Q$ slightly smaller than $\sqrt{N}$, and obtain $$\sum\_{q \leq q\_0} \sum\_{\substack{a \pmod{q} \\ \gcd(a,q)=1}} \lvert S(a/q) \rvert^2 \leq (1+\epsilon) 2 e^\gamma N^2 \log q\_0,$$ where gamma is Euler's constant $0.577\cdots$ and $\epsilon$ is very small. Now, the 2 in the bound $\leq (1+\epsilon) 2 e^\gamma N^2$ is due to the parity problem, and thus should be next to impossible to remove (except for very small $q\_0$). However, the factor of $e^\gamma$ clearly has no right to exist. The true asymptotic should be simply $N^2 \log q\_0$. Can we remove that nasty $e^\gamma$? That is, can you prove a bound of type $$\sum\_{q \leq q\_0} \sum\_{\substack{a \pmod{q} \\ \gcd(a,q)=1}} \lvert S(a/q)\rvert^2 \leq (1+\epsilon) 2 N^2 \log q\_0 ?$$ Harald

https://mathoverflow.net/users/398

The large sieve for primes

Heh, I think I know why you are interested in this question, Harald, as Ben and I thought about essentially the same problem for what I suspect to be the same reason :-) Anyway, we were able to get rid of the e^gamma factor. One way to proceed is to work not with the exponential sums, but rather the inner product of Lambda with Dirichlet characters. Using a Selberg sieve majorant (the same one used to prove, say, the Brun-Titschmarsh inequality) and the TT^\* method, one gets a nice l^2 bound on these inner products (losing only the 2 from the parity problem, or not even that if one deletes the principal character), and then one can do some Fourier analysis on finite groups to pass from Dirichlet characters back to exponentials. It helps a little bit to smooth the sum, but it's not too essential here. (See also my paper with Ben on the restriction theory of the Selberg sieve for some related results.) Our argument isn't written up (or checked, for that matter) yet, but we can talk more if you want to know more details.

10

https://mathoverflow.net/users/766

1276

https://mathoverflow.net/questions/1051

20

Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would be least surprised if the answer were 3. Notice that the 2 subsets of integers such that highest power of 5 that divides them is a) even b) odd manage to split most primitive triples, plus all the multiples of those. Notice also that Schur proved the positive integers cannot be split into any finite number of sum-free subsets (i.e. no finite partition can split all power-of-1 Fermat triples), while Fermat's theorem proves that all power-of-n (n>2) triples can be split by the trivial partition into 1 set. **Edit:** Since this turns out to be a known open problem, we're adding the tag [open-problem] and converting this question to community wiki. The idea is to have a separate answer for each possible approach to solving this problem. If you have some additional insight or a reference to contribute to an answer, you only need 100 rep to do so. We're still figuring out exactly how to handle open problems on MO. The discussion is happening on [this tea.mathoverflow.net thread](http://mathoverflow.tqft.net/discussion/8/are-open-questions-acceptable-how-should-they-be-treated).

https://mathoverflow.net/users/2480

Splitting Pythagorean triples

This problem appears in Croot and Lev's 2007 "Open Problems in Additive Combinatorics" (<http://people.math.gatech.edu/~ecroot/E2S-01-11.pdf> ), where it is attributed to Erdos and Graham (the latter of whom offers $250 for its solution). Other references may include Cooper and Poirel's "Notes on the Pythagorean Triple System" (<http://www.math.sc.edu/~cooper/pth.pdf> ), which mentions that even the case of whether 2 colors is enough is open. They also exhibit a 2-coloring of the integers from 1 to 1344 without any monochromatic Pythagorean triples. UPDATE (5/4/16): A [new preprint of Heule, Kullmann, and Marek](http://arxiv.org/abs/1605.00723) (to appear in the SAT 2016 conference) answers the question for $2$ subsets in the negative: If $\{1,2,\dots,7825\}$ is split into two subsets, one subset must contain a pythagorean triple. The $7825$ here is tight. The proof is heavily computer-assisted, and the key ideas seem to be to cast the problem in the language of satisfiability, and set up a divide-and-conquer method so that the (enormous) resulting instance is handleable by state-of-the-art satisfiability solvers.

18

https://mathoverflow.net/users/405

1279

https://mathoverflow.net/questions/854

2

I am programming an algorithm where I have broken up the surface of a sphere into grid points (for simplicity I have the grid lines are parallel and perpendicular to the meridians). Given a point A on the sphere, I would like to be able to efficiently take any grid "square" and determine the point B in the square with the least spherical coordinate distance AB. * Near the poles, the "squares" degenerate into "triangles". * As noted by Trithematician, this is a special case of to finding the shortest distance from a point to an arc on a sphere. For the longitudinal lines, these are arcs of great circles, but for the latitudinal they are not. * TonyK provides a method below that solves the longitudinal case. * Minimising the distance in 3d also minimises the distance on the sphere I actually asked this question here: <https://stackoverflow.com/questions/1463606/closest-grid-square-to-a-point-in-spherical-coordinates> and decided to use an approximation instead, but I am still curious as to whether there is an exact solution. I thought that since this is really more of a maths question than a programming question that I would repost it here.

https://mathoverflow.net/users/494

Closest grid square to a point in spherical coordinates

The tricky part is to find the nearest point on a meridian to a given point P. Let's fix the meridian M at φ=0, to keep the algebra simple. Suppose P has spherical coordinates (θ, φ), with 0 ≤ θ ≤ π and -π < φ ≤ π. Let C be the great circle through P which is perpendicular to M; then we are looking for an intersection of C and M. (There are two such intersections; in the end we will simply choose the one that is on the same side of the equator as P). The normal of C meets the sphere on the meridian M, at A = (ρ, 0), say. P is on C, so OP is perpendicular to OA, where O is the centre of the sphere. In cartesian coordinates, with r = 1 for simplicity: A = (sin ρ, 0, cos ρ) P = (cos φ sin θ, sin φ sin θ, cos θ) The scalar product is A.P = sin ρ cos φ sin θ + cos ρ cos θ This must be zero, giving tan ρ = -1/(cos φ tan θ) The required point, Q say, is on M and perpendicular to A, so if Q = (σ, 0) in spherical coordinates, we have tan σ = -1/tan ρ = cos φ tan θ So Q = (arctan (cos φ tan θ), 0) If this Q is on the wrong side of the equator, take Q = (π - arctan (cos φ tan θ), π)

2

https://mathoverflow.net/users/767

1281

https://mathoverflow.net/questions/1283

10

I've seen the following lower bound for the complementary error function ([erfc](http://en.wikipedia.org/wiki/Erfc)) but I haven't been able to prove it. Does anyone know how to establish the following? $$erfc(x) > \frac{ x \exp(-x^2) }{ \pi(1 + 2x^2) }$$

https://mathoverflow.net/users/136

erfc lower bound

Durrett, *Probability: Theory and Examples*, 3rd edition, p. 6 gives $$(x^{-1} - x^{-3}) e^{-x^2/2} \le \int\_x^\infty e^{-y^2/2} \: dy $$ The proof Durrett gives is from the observation that $$ \int\_x^\infty (1-3y^{-4}) e^{-y^2/2} \: dy = \left( x^{-1} + x^{-3} \right) e^{-x^2/2} $$ which I suspect can be found by integration by parts, although I haven't written it out; in any case, differentiate it to check. After this, some changes of variables give $$ \left( {1 \over z} - {1 \over 2z^3} \right) e^{-z^2}/\sqrt{\pi} \le erfc(z). $$ Finally, $z/(1+2z^2) < 1/z-1/(2z^3)$ for $z > 2^{-1/4}$, giving your bound for $z > 2^{-1/4}$ if $\pi$ is replaced with $\sqrt{\pi}$. Obviously this is a hack trying to get your proposed bound in the form of the bound I already knew, but hopefully it helps.

7

https://mathoverflow.net/users/143

1290

https://mathoverflow.net/questions/1267

3

How to classify **K3 surfaces** over an arbitrary field *k*?

https://mathoverflow.net/users/65

K3 over fields other than C?

The "standard" definition of a K3 surface is field independent (unless you are a physicist): $p\_g=1, q=0$, and trivial canonical class. Some results: * Mumford and Bombieri showed that you get (just as in the complex case) a 19 dimensional family of K3 surfaces for any degree (the 19 dimensional thingy is a deformation theory argument which is completely algebraic). * Deligne showed that all the K3 surfaces in finite characteristics are reductions mod p. What you obviously don't get is the fact that all these spaces sit together in a nice 20 dimensional complex ball. I also don't know if you can carry over any of the recent Kodaira dimension computation of these moduli (which are very analytic in nature). Reference: Complex algebraic surfaces (Beauville): Chapter VIII and Appendix A.

8

https://mathoverflow.net/users/404

1299

https://mathoverflow.net/questions/1291

127

Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be useful for other people too. I'm interested in learning modern Grothendieck-style algebraic geometry in depth. I have some familiarity with classical varieties, schemes, and sheaf cohomology (via Hartshorne and a fair portion of EGA I) but would like to get into some of the fancy modern things like stacks, étale cohomology, intersection theory, moduli spaces, etc. However, there is a vast amount of material to understand before one gets there, and there seems to be a big jump between each pair of sources. Bourbaki apparently didn't get anywhere near algebraic geometry. So, does anyone have any suggestions on how to tackle such a broad subject, references to read (including motivation, preferably!), or advice on which order the material should ultimately be learned--including the prerequisites? Is there ultimately an "algebraic geometry sucks" phase for every aspiring algebraic geometer, as Harrison suggested on these forums for pure algebra, that only (enormous) persistence can overcome?

https://mathoverflow.net/users/344

A learning roadmap for algebraic geometry

FGA Explained. Articles by a bunch of people, most of them free online. You have Vistoli explaining what a Stack is, with Descent Theory, Nitsure constructing the Hilbert and Quot schemes, with interesting special cases examined by Fantechi and Goettsche, Illusie doing formal geometry and Kleiman talking about the Picard scheme. For intersection theory, I second Fulton's book. And for more on the Hilbert scheme (and Chow varieties, for that matter) I rather like the first chapter of Kollar's "Rational Curves on Algebraic Varieties", though he references a couple of theorems in Mumfords "Curves on Surfaces" to do the construction. And on the "algebraic geometry sucks" part, I never hit it, but then I've been just grabbing things piecemeal for awhile and not worrying too much about getting a proper, thorough grounding in any bit of technical stuff until I really need it, and when I do anything, I always just fall back to focus on varieties over C to make sure I know what's going on. EDIT: Forgot to mention, Gelfand, Kapranov, Zelevinsky "Discriminants, resultants and multidimensional determinants" covers a lot of ground, fairly concretely, including Chow varieties and some toric stuff, if I recall right (don't have it in front of me)

41

https://mathoverflow.net/users/622

1305

https://mathoverflow.net/questions/1294

36

Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long chains of conditional probabilities) of deriving the probability 'p' that the minimum distance between any pair of these points is larger than some value 'k' -i.e. if the line was an array, there would be more than 'k' slots/positions between any two point. Well that, or an expression for the mean minimum distance (MMD) for a pair of points in the set - referring to the smallest distance between any two points that can be found, not the mean minimum/shortest distance between all possible pairs of points. I was unable to find an answer to this question after a literature search, so I was hoping someone here might have an answer or point me in the right direction with a reference. This is for recreational purposes, but maybe someone will find it interesting. If not, apologies for the spam.

https://mathoverflow.net/users/774

Mean minimum distance for N random points on a one-dimensional line

This can answered without any complicated maths. It can be related to the following: Imagine you have $N$ marked cards in a pack of $m$ cards and shuffle them randomly. What is the probability that they are all at least distance $d$ apart? Consider dealing the cards out, one by one, from the top of the pack. Every time you deal a marked card from the top of the deck, you then deal $d$ cards from the bottom (or just deal out the remainder if there's less than $d$ of them). Once all the cards are dealt out, they are still completely random. The dealt out cards will have distance at least d between all the marked cards if (and only if) none of the marked cards were originally in the bottom $(N-1)d$. The probability that the marked cards are all distance d apart is the same as the probability that none are in the bottom $(N-1)d$. The points uniformly distributed on a line segment is just the same (considering the limit as $m$$\rightarrow∞$). The probability that they are all at least a distance $d$ apart is the same as the probability that none are in the left section of length $(N-1)d$. This has probability $(1-\frac{(N-1)d}{L})^N$. Integrating over $0$$\le$$d$$\le$$\frac{L}{(N-1)}$ gives the expected minimum distance of $\frac{L}{(N^2-1)}$.

60

https://mathoverflow.net/users/1004

1308

https://mathoverflow.net/questions/316

6

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{vol}(M)\ast A^n + o(\mathrm{something}) $$ (Perhaps one could consider first the case of a Kähler manifold? The Laplacian is particularly simple there.)

https://mathoverflow.net/users/65

Eigenvalues of Laplacian

The most natural way is to study the short-time asymptotics of the heat or wave kernel on M. For example, you can use the heat kernel $p\_t(x,y) = \sum\_i e^{-\lambda\_i t} f\_i(x) \overline{f\_i(y)}$ where $f\_i$ are the eigenfunctions with eigenvalues $\lambda\_i$. This is a fundamental solution to the heat equation. When $t$ is small then you can construct a good approximation to $p\_t$ near any particular $x$ by hand, using Fourier analysis in local co-ordinates. The end result is that that $p\_t(x,x) \approx C t^{-n/2}$. Now integrate this estimate $dx$, noting that $\int\_M p\_t(x,x)dx$ basically counts eigenvalues with $\lambda\_i \leq 1/t$.

7

https://mathoverflow.net/users/327

1315

https://mathoverflow.net/questions/385

35

There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for example <http://arxiv.org/abs/math/0507284> I've seen this idea attributed to big names like Quillen, Drinfeld, and Deligne -- so it must be true, right? ;-) An example of this philosophy is the deformation theory of a compact complex manifold: It is "controlled" by the Kodaira-Spencer dg Lie algebra: holomorphic vector fields tensor Dolbeault complex, with differential induced by del-bar on the Dolbeault complex, and Lie bracket induced by Lie bracket on the vector fields (I think also take wedge product on the Dolbeault side). I seem to recall that there is a general theorem which justifies this philosophy, but I don't remember the details, or where I heard about it. The statement of the theorem should be something like: > > Let k be a field of characteristic zero. Given a functor F: (Local Artin k-algebras) -> (Sets) satisfying some natural conditions that a "deformation functor" should satisfy, then there exists a dg Lie algebra L such that F is isomorphic to the deformation functor of L, which is the functor that takes an algebra A and returns the set of Maurer-Cartan solutions (dx + [x,x] = 0) in (L^1 tensor mA) modulo the gauge action of (L^0 tensor mA), where mA denotes the maximal ideal of A. > > > Furthermore, I think such an L should be unique up to quasi-isomorphism. Does anyone know a reference for something along these lines? Any other nice examples of cases where this philosophy holds would also be appreciated.

https://mathoverflow.net/users/83

Deformation theory and differential graded Lie algebras

I hope to write more on this later, but for now let me make some general assertions: there are general theorems to this effect and give two references: arXiv:math/9812034, DG coalgebras as formal stacks, by Vladimir Hinich, and the survey article arXiv:math/0604504, Higher and derived stacks: a global overview, by Bertrand Toen (look at the very end to where Hinich's theorem and its generalizations are discussed). The basic assertion if you'd like is the Koszul duality of the commutative and Lie operads in characteristic zero. In its simplest form it's a version of Lie's theorem: to any Lie algebra we can assign a formal group, and to every formal group we can assign a Lie algebra, and this gives an equivalence of categories. The general construction is the same: we replace Lie algebras by their homotopical analog, Loo algebras or dg Lie algebras (the two notions are equivalent --- both Lie algebras in a stable oo,1 category). We can associate to such an object the space of solutions of the Maurer-Cartan equations -- this is basically the classifying space of its formal group (ie formal group shifted by 1). Conversely from any formal derived stack we can calculate its shifted tangent complex (or perhaps better to say, the Lie algebra of its loop space). These are equivalences of oo-categories if you set everything up correctly. This is a form of Quillen's rational homotopy theory - we're passing from a simply connected space to the Lie algebra of its loop space (the Whitehead algebra of homotopy groups of X with a shift) and back. So basically this "philosophy", with a modern understanding is just calculus or Lie theory: you can differentiate and exponentiate, and they are equivalences between commutative and Lie theories (note we're saying this geometrically, which means replacing commutative algebras by their opposite, ie appropriate spaces -- in this case formal stacks). Since any deformation/formal moduli problem, properly formulated, gives rise to a formal derived stack, it is gotten (again in characteristic zero) by exponentiating a Lie algebra. Sorry to be so sketchy, might try to expand later, but look in Toen's article for more (though I think it's formulated there as an open question, and I think it's not so open anymore). Once you see things this way you can generalize them also in various ways -- for example, replacing commutative geometry by noncommutative geometry, you replace Lie algebras by associative algebras (see arXiv:math/0605095 by Lunts and Orlov for this philosophy) or pass to geometry over any operad with an augmentation and its dual...

38

https://mathoverflow.net/users/582

1320

https://mathoverflow.net/questions/1269

8

Are **supersingular primes** and **supersingular elliptic curves** related? (this was essentially a subquestion in [my earlier question](https://mathoverflow.net/questions/1249/ways-to-characterize-supersingular-primes), but still looks sufficiently different to me to deserve a separate post)

https://mathoverflow.net/users/65

What does "supersingular" mean?

Let F be the finite field with p elements. A supersingular elliptic curve is an elliptic curve E/F with the property that the endomorphism ring (ring of homomorphisms from E to E) of E over the algebraic closure of F\_p is has rank 4 as a Z-module. It is a theorem that End(E/F) has rank 2 or rank 4 (and that in the former case, End(E/F) is isomorphic to an order in an imaginary quadratic number ring whereas in the latter case, End(E/F) is isomorphic to an order in a quaternion algebra over Q). Hence, a supersingular elliptic curve over F\_p can be thought of as an elliptic curve over F with a "big" endomorphism ring. It is a theorem that another characterization supersingular elliptic curves is that E/F is supersingular is if and only if the number of points on E/F is exactly p + 1 (edit: for p > 3, see Voloch's comment below). I believe that the etymology of the term "supersingular" is as follows: if you start with an elliptic curve E/Q over Q, for all but finitely many primes p, reduction (mod p) gives an elliptic curve E/F. For a generic E/Q (specifically, one without "complex multiplication") then the set of primes such that reduction (mod p) turns E/Q into a supersingular E/F has asymptotic density 0. Such primes are called "supersingular primes for E/Q" - supersingular refers to "really unusual." The reductions for such primes are then called supersingular elliptic curves over F. I'm pretty sure that every elliptic curve over F is a (mod p) reduction of an elliptic curve over Q so that all supersingular elliptic curves arise in this way. I'll remark (following Silverman) that a supersingular elliptic curve over F is not "singular" in the sense of algebraic geometry - by definition all elliptic curves are nonsingular. I do not know much about supersingular primes in the context of monstrous moonshine. According to Wikipedia, a supersingular prime is a prime that divides the order of the monster group; and there are 15 such primes. Given E/Q, by a theorem of Elkies there will be infinitely many super singular primes p for E/Q. So it's difficult to imagine how the list of 15 supersingular (with respect to moonshine) primes could emerge from the notion of "supersingular elliptic curve." I imagine that the etymology of the term "supersingular" in the context of moonshine is again that that supersingular primes are special - but that they special in a completely different way from the supersingular primes for a elliptic curve over Q.

10

https://mathoverflow.net/users/683

1324

https://mathoverflow.net/questions/1325

2

The mathedu mailing list has a recent longish thread at <http://www.nabble.com/Why-do-we-do-proofs--to25809591.html> which discussed among other things whether we should teach triangles as labeled or unlabeled to high school students (this is a vast oversimplification of the thread). I have long been concerned with how we think (informally and formally) about mathematical objects, so naturally I started to consider how we think about triangles. Consider circles. Most informal and formal descriptions involve an embedding into R^2, but they *can* be characterized as manifolds (even as Riemannian manifolds) of dimension 1 with specific properties, independent of any embedding. This sort of thing has turned out to be a major way to think about all sorts of spaces. So can we describe triangles in a similar way? Unfortunately, manifolds are far removed from my usual mathematical work (category theory). What I *think* I understand is that there can be *piecewise* linear manifolds, even Riemannian ones. So perhaps we can say a triangle is a piecewise linear manifold of dimension 1 with certain properties. Now, I want to define a triangle so that it comes complete with information about the lengths of its sides and what the three angles are. Riemannian manifolds have a way to specify length and angles, and I can believe you can make the sides have specific lengths. But the angles? It seems to me that the tangent spaces (like those on a circle) result in all angles being 0 or pi, except at the corners where they don't exist. But I may not understand the situation correctly. So my question is: Is there a known methodology that allows triangles to be characterized independent of embeddings in such a way that incorporates information about side lengths and angles?

https://mathoverflow.net/users/342

Characterizing triangles unembeddedly

I'm not sure what you have in mind for circles--the only invariant of a connected compact Riemannian 1-manifold is its length; they have only extrinsic curvature (defined for a manifold embedded in Euclidean space) not intrinsic curvature. You'd have to consider Riemannian manifolds with some extra structure--perhaps just a function which is supposed to be the curvature of an embedding into R^2, and integrates to 2π. Then you could represent triangles similarly, with a curvature "function" which is the sum of three delta functions, one at each vertex. Of course the side lengths of a triangle determine its angles, so you might produce a "triangle" which cannot be embedded in R^2! Interestingly for the case of 2-manifolds the situation is nicer. There is a theorem that a Riemannian 2-manifold of positive curvature homeomorphic to the sphere has a unique (up to rigid motion) isometric immersion in R^3 as a convex surface. There is an analogous statement for polyhedra but I am not sure of the precise statement. I think these results are due to Pogorelov and Alexandrov.

4

https://mathoverflow.net/users/126667

1341

https://mathoverflow.net/questions/1329

14

So, when doing LaTeX, it is absolutely necessary for ones sanity to using a preview program which updates automatically every time you compile. Of course, any previewer designed for DVIs will do this, but as far I can tell, Adobe Acrobat not only does not automatically update, but will not let you change the PDF with it open. On a Mac one can get around this by using Skim, and on \*nix by using xpdf, but what should one do on Windows?

https://mathoverflow.net/users/66

Automatically updating PDF reader for Windows

Use [SumatraPDF](http://blog.kowalczyk.info/software/sumatrapdf/index.html). It is a lightweight pdf viewer which updates automatically. It also allows syncing with [TeXnicCenter](http://transact.dl.sourceforge.net/project/texniccenter/Tutorials/How_to_Sumatra_EN_%282009.09.23%29.pdf) and [WinEdt](http://robjhyndman.com/researchtips/synchronizing-winedt-and-pdf-files/).

18

https://mathoverflow.net/users/261

1347

https://mathoverflow.net/questions/1312

13

Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather than some other compactification? In any case, what other compactifications of the moduli of curves have been studied? Similarly, what other compactifications of the moduli of maps have been studied? Do any of these other compactifications lead to an interesting "Gromov-Witten theory"?

https://mathoverflow.net/users/83

Gromov-Witten theory and compactifications of the moduli of curves

I can give individual answers to a lot of your questions, but I can't answer any of them completely, nor can I fit all these answers together into a coherent whole. For string theory, there does seem to be something special about the Deligne-Mumford compactification. Morally, what's going on is this: string theorists are allowing cylinder-shaped submanifolds of their Riemann surfaces to become infinitely long. The only finite energy fields on such infinitely long submanifolds are constant, so you can replace the long cylinder with a node. (Likewise, morally, if you allow vertex operators at two marked points to come together, you should take their operator product. This is what bubbling when marked points collide does for you.) Somewhat more technically: The first step in (bosonic) string theory is to compute the partition function of the nonlinear sigma model as a function on the space of metrics on your worldsheet. This function on metrics descends to a section of some line bundle on the moduli stack of complex structures on the worldsheet. When you can compute it at all, you can show that this section has exactly the right pole structure it needs to be a section of the 13th power of the canonical bundle tensored with the 2nd power of the dual of the line bundle corresponding to the boundary divisor of the Deligne-Mumford compactification. (There's an old Physics Report by Phil Nelson that explains this pretty well, although not with anything you'd call a proof. Should also credit Belavin & Knizhik, who did the initial calculations.) There's a somewhat more modern perspective on this (Zwiebach, Sullivan, Costello,...) that says that the generating function of string theory correlation functions for smooth Riemann surfaces satisfies a certain equation (a "quantum master equation"), which gives instructions for how to extend the theory to nodal Riemann surfaces. Different master equations give different recipes for extending to the boundary, if I understand your advisor correctly. People have played around with other compactifications. There are a lot of different compactifications of the stack of smooth marked curves. People have already mentioned a few of them. David Smyth has some cool results which classify the "stable modular" compactifications of the stack of curves (<http://arxiv.org/abs/0902.3690> ). For compactifications of the moduli of maps, the only one that comes immediatley to mind is Losev, Nekrasov, and Shatashvili's "freckled instanton compactification", in which IIRC, you allow zeros and poles to collide and cancel each other out.

12

https://mathoverflow.net/users/35508

1357

https://mathoverflow.net/questions/1365

21

I want to say that a [group object](http://en.wikipedia.org/wiki/Group_object) in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One problem with this idea is that this diagram $D$ as a category on its own doesn't have enough structure to make the object labelled $``G\times G"$ really the product of $G$ with itself in $D$. > > Is there a category $U$ with a group object $G$ in it such that every group object in every other category $C$ is the image of $G$ under a product-preserving functor $F:U\rightarrow C$, unique up to natural isomorphism? > > > (It's okay with me if "product-preserving" or "up to natural isomorphism" are replaced by some other appropriate qualifiers, like "limit preserving"...)

https://mathoverflow.net/users/84526

Is there a "universal group object"? (answered: yes!)

Yes, the category U is the opposite of the full subcategory of Grp on the free groups on 0, 1, 2, ... generators. This is an instance of Lawvere's theory of "theories". See [this nLab entry](http://ncatlab.org/nlab/show/Lawvere+theory) for a discussion (of this example in fact).

32

https://mathoverflow.net/users/126667

1370

https://mathoverflow.net/questions/1363

11

Let's say that I want to prove that a language is not regular. The only general technique I know for doing this is the so-called "pumping lemma", which says that if $L$ is a regular language, then there exists some $n>0$ with the following property. If $w$ is a word in $L$ of length at least $n$, then we can write $w=xyz$ (here $x$, $y$, and $z$ are subwords) such that $y$ is nontrivial and $xy^{k}z$ is an element of $L$ for all $k>0$. This lemma basically reflects the trivial fact that in any directed graph, there is some $n$ such that any path of length at least n contains a loop. Question: are there any other general techniques for proving that a language is not regular?

https://mathoverflow.net/users/317

Regular languages and the pumping lemma

For necessary and sufficient conditions for a language to be regular (sometimes useful in proving nonregularity when simpler tricks like the pumping lemma fail) see the [Myhill–Nerode theorem](http://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem).

18

https://mathoverflow.net/users/440

1375

https://mathoverflow.net/questions/1367

29

Note: This comes up as a byproduct of Qiaochu's question ["What are examples of good toy models in mathematics?"](https://mathoverflow.net/questions/1354/what-are-examples-of-good-toy-models-in-mathematics) There seems to be a general philosophy that problems over function fields are easier to deal with than those over number fields. Can someone actually elaborate on this analogy between number fields and function fields? I'm not sure where I can find information about this. Ring of integers being Dedekind domains, finite residue field, RH over function fields easier to deal with, anything else? Being quite ignorant about this analogy, I am actually not even convinced that why working over function fields "should" give insights about questions about number fields.

https://mathoverflow.net/users/nan

Global fields: What exactly is the analogy between number fields and function fields?

There's a really nice table in section 2.6 of [these](http://www-math.mit.edu/~poonen/papers/curves.pdf) notes from a seminar that Bjorn Poonen ran at Berkeley a few years ago.

44

https://mathoverflow.net/users/2

1376

https://mathoverflow.net/questions/1373

6

Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where doing the multiplication out is too tedious for me to have any chance at accuracy in the final answer, and want an efficient way to tell a computer to do it (things that will just multiply pairs, and then you input the next set of pairs don't count).

https://mathoverflow.net/users/622

Is there a software package that does Schubert Calculus computations?

There's a Littlewood-Richardson calculator here: <http://math.rutgers.edu/~asbuch/lrcalc/> I usually use the "SchurRings" package in Macaulay 2 ( <http://www.math.uiuc.edu/Macaulay2/> ) though. No particular reason why, just that Macaulay 2 is something I am used to using. It's very easy to use, here's an example (it doesn't print correctly on this page) where the 4 in the first command means use 4 variables (so we're working in Gr(4, infinity)). i1 : S = schurRing(s,4) o1 = S o1 : SchurRing i2 : s\_{2,2} \* s\_{3,1} o2 = s + s + s + s + s + s + s ``` 5,3 5,2,1 4,3,1 4,2,2 4,2,1,1 3,3,2 3,2,2,1 ``` o2 : S

5

https://mathoverflow.net/users/321

1379

https://mathoverflow.net/questions/1383

3

I feel a little embarrassed to be asking this question here, since I think it should be much easier than I'm making it, but here goes: Given a finite poset P, does there necessarily exist some chain that intersects every maximal antichain? (Note: By maximal antichain, I mean that there's no antichain strictly containing our antichain.) The answer seems to be "no" for infinite posets, but I can't find either a reference or a proof when it comes to finite posets. Sorry if this is an undergrad-homework-level problem...

https://mathoverflow.net/users/382

Chains intersecting antichains in finite posets

No. Consider the poset of subsets of {x,y,z} under inclusion. The maximal chain Ø, {x}, {x,y}, {x, y, z} does not intersect every maximal antichain: it misses the maximal antichain {y}, {x,z}. By symmetry every other maximal chain also misses some maximal antichain.

5

https://mathoverflow.net/users/440

1386

https://mathoverflow.net/questions/1346

9

I know that the individual cohomology groups are representable in the homotopy category of spaces by the Eilenberg-MacLane spaces. Is it also true that the entire cohomology ring is representable? If so, is there a geometric interpretation of the cup product as an operation on the representing space?

https://mathoverflow.net/users/788

Representablity of Cohomology Ring

The total cohomology of spaces should be thought of as a **graded ring**, or even more precisely as a **graded E\* algebra** where E\* is the cohomology of a point. It is representable in the sense that there is a *graded E\* algebra object* in hTop representing it. Let's unpack that a little. First, you have to understand about *group objects* and so forth. The full story is in [Lawvere theories](http://ncatlab.org/nlab/show/Lawvere+theory), but the basic idea is that a, say, group object in a category 𝒞 is an object, say C, of 𝒞 together with all the structure needed to ensure that the contravariant hom-functor 𝒞(-,C) lifts from a functor to Set to a functor to Group. Providing 𝒞 has enough products of C, it's simple to write down a few morphisms that C must have, together with some diagrams that must commute. Lawvere theories are the correct way of thinking about these things in general, but in a specific instance it can be instructive to just write everything down. Now, that works fine for groups, abelian groups, modules, rings, and so forth, but cohomology isn't any of those things. Cohomology is a *graded* ring. So we need graded objects in our category. To talk about graded objects we need a grading set, say Z. This doesn't have to have any structure whatsoever. A Z-graded object in 𝒞 is just a functor Z → 𝒞 where Z is viewed as a discrete category. So it's a family of objects in 𝒞, indexed by the elements (objects) of Z. A Z-graded object in 𝒞 represents a functor from 𝒞 into SetZ, the category of Z-graded sets. While we can send a Z-graded set to a set either by its coproduct or its product, we shouldn't do so. We should keep the labelling. That's because we now want to mix these two things and talk of **graded rings**, or more generally *graded* theories, also called *many-sorted* theories. In a single-sorted theory, such as groups or rings, the standard set-based (or more generally 𝒞 based) groups or rings consist of a set (object) together with certain functions (morphisms), called *operations*, from certain n-fold products of that set (object) to itself. In a *graded* theory, the operations go from products of components of the graded set (object) to other components. Thus in a graded abelian group, say A, we have operations A(z) × A(z) → A(z) but not A(z) × A(z') → A(z''). That is, we can only add terms in the same component. So back to cohomology. Cohomology is a representable functor into the category of graded E\* algebras, where the grading set is ℤ. So for each integer n we have a space En in hTop and for each operation of a graded E\* algebra we have an operation En × Em → El. In particular, multiplication corresponds to a map (technically, a homotopy class of maps as we're in hTop) En × Em → En+m. For ordinary cohomology, these spaces are the Eilenberg-Maclane spaces. Several remarks are in order here. Sometimes, the operations in a graded object come in families. Then it can be severely tempting to put the graded pieces together into one single thing, either by taking the coproduct or the product. However, this **destroys information**. Just because the primary operations come in families doesn't mean that *all* of the operations come in families. For example, the primary structure of a graded E\* algebra come in suitable families and it can be tempting to put the pieces of the cohomology together. (One has to choose what method to use: product or coproduct. Product generally works better because in a product one doesn't have to worry about things remaining finite, which is good, because you have no control over this.) But cohomology theories don't just have their primary structure operations, they also have a whole raft of other operations. The main division of these two is into **stable** and **unstable** operations. By putting your theory together into a single object, you are effectively saying that you will only consider **stable** operations and will ignore **unstable** ones. While this is basically okay for cohomology, for K-theory it is a disaster because *we don't know what the stable ones are!*. Secondly, to correct something Chris said, the spaces En do keep track of the suspension isomorphisms. The fact that there are suspension isomorphisms in cohomology theories is encoded in the fact that that there is an equivalence ΩEn ≃ En-1. Also, the Mayer-Vietoris maps are part of the fact that cohomology is representable (you get MV from the long exact sequence for pairs, and that's needed for representability). Of course, knowing that ΩEn ≃ En-1 allows you to construct a spectrum from the spaces En, even an Ω-spectrum, and those operations that come in families, namely the stable ones, become morphisms of spectra. Once you have that, you can define the associated homology theory and lots of other wonderful things. But the point is that there's a lot that you can do before going to spectra and, up to a point, the language of spectra is merely a way of keeping track of the grading. Finally, this probably isn't geometric enough for the cup product in ordinary cohomology, but there you're asking a difficult question. To give a truly geometric interpretation, you would first need to find good geometric models of **all** the K(π,n)s. While this can be done for a few, I don't know of a good family for all of them ("good" in the sense of "easy to think about" rather than anything technical). One can start out fairly well, say with π = ℤ, with ℤ, S1, ℂℙ∞, but then it gets a bit sticky. Another model for K(ℤ,2) is the projective unitary group on a Hilbert space and this acts on the general linear group for the space of Hilbert-Schmidt operators on that same Hilbert space so you could make K(ℤ,3) as the quotient of GL(H⊗H)/ℙ(H) and this is essentially the start of gerbe theory, but, as I'm sure you'll agree, it's hard to look at that and say "Aha! Now I understand K(ℤ,3)." in quite the same way as one can look at ℂℙ∞ and understand the connection between H2(X;ℤ) and line bundles.

9

https://mathoverflow.net/users/45

1392

https://mathoverflow.net/questions/1387

9

Does anybody know good references to learn about Lie superalgebras? I started with Howe's "Remarks on classical invariant theory", which contains a study of osp(m,2n), and now I am reading Kac's '77 Advances paper. I wonder if there are other helpful sources. I am especially interested in getting a feel for the representation theory.

https://mathoverflow.net/users/316

References for Lie superalgebras

Have you seen the survey by Frappat-Sciarrino-Sorba, "Dictionary on Lie Superalgebras" listed [here](http://ncatlab.org/nlab/show/super+Lie+algebra)? When you have collected more references, please feel encouraged to add them to that list there...

3

https://mathoverflow.net/users/381

1400

https://mathoverflow.net/questions/1359

20

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules Qcoh(U) and Qcoh(X)? --- To begin, it might be helpul to simply assume that U ⊆ X is an open subscheme, and consider some properties of the resulting functors. I can think of one interesting functor: there is an exact functor Qcoh(X) → Qcoh(U) given by restriction of sheaves. I assume this is a special case of some more general construction (direct or inverse image functor?) that probably has an adjoint on some side. Can we continue to list enough functors and properties of these functors to the point where we have determined precisely when the above map f is an isomorphism onto an open subscheme?

https://mathoverflow.net/users/778

Functorial characterization of open subschemes?

The abelian category of quasicoherent sheaves on a schemes determine the scheme. This is an old result of Gabriel ("[des categories abeliennes](http://www.numdam.org/item?id=BSMF_1962__90__323_0)" 1962), proved in full generality by [Rosenberg](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948). This means that, $\operatorname{QCoh}(X)$ does not only tell you the open subschemes of $X$ but also gives you the structure sheaf! I've known this result for some time but I had never looked at it in detail until today. I'll sketch what I have just learned hoping not to make big mistakes... An abelian subcategory $B$ of an abelian category $A$ is said to be a **thick subcategory** if it is full and for any exact sequence in $A$ $$0\to M'\to M \to M''\to 0,$$ $M$ belongs to $B$ if and only if $M'$ and $M''$ do. If $B$ is a thick subcategory of $A$ there is a well defined localization $A/B$, which is again an abelian category. $A/B$ has the same objects as $A$ and a morphism $f:M\to N$ in $A/B$ is an isomorphism if and only if $\ker f$ and $\operatorname{coker} f$ belong to $B$. Let $T\colon A\to A/B$ be the localization functor. Then $B$ is said to be a **localizing subcategory** if $B$ is thick and $T$ has a right adjoint. The condition of being localizing can be rephrased only in terms of $A$ and $B$. see Gabriel's thesis above (proposition 4 in chapter III). Finally, if $M$ is an object of $A$, we denote by $\langle M\rangle$; the smallest localizing subcategory containing $M$. Now let $X$ be a scheme, $j\colon U \to X$ an open embedding and $i\colon Y\to X$ its closed complement. Then there are a bunch of adjunctions between the categories of quasicoherent sheaves of $U,X,Y$: $i\_\*\colon \operatorname{QCoh}(Y)\to \operatorname{QCoh}(X)$ has a left adjoint $i^\*\colon \operatorname{QCoh}(X)\to \operatorname{QCoh}(Y)$ and a right adjoint $i\_!\colon \operatorname{QCoh}(X) \to \operatorname{QCoh}(Y)$. On the other hand, the functor $j^\*\colon \operatorname{QCoh}(X)\to \operatorname{QCoh}(U)$ has a left adjoint $j\_!\colon \operatorname{QCoh}(U)\to \operatorname{QCoh}(X)$ and a right adjoint $j\_\*\colon \operatorname{QCoh}(U)\to \operatorname{QCoh}(X)$. This is sometimes called a recollement. Let's assume that $X$ is Noetherian and let $A = \operatorname{QCoh}(X)$. We have an exact sequence of abelian categories $$0 \to \operatorname{QCoh}(Y) \to A \to \operatorname{QCoh}(U) \to 0$$ in the sense that the category $\operatorname{QCoh}(Y)$ happens to be a localizing subcategory of $A$ and its quotient is identified with $\operatorname{QCoh}(U)$. The first map in the exact sequence is $i\_\*$ and the second $j^\*$. Moreover, I think that $\operatorname{QCoh}(Y)$ is the smallest localizing subcategory of $\operatorname{QCoh}(X)$ containing $i\_\*O\_Y$. Gabriel proves that there are no more such localizing subcategories, that is closed subschemes of $X$ correspond exactly to localizing subcategories $\langle M\rangle$ generated by a single coherent sheaf (i.e. Noetherian object in $A$). Moreover, irreducible closed subsets (the points in the underlying topological space of $X$) are given by localizing subcategories $\langle I\rangle$ for $I$ an indecomposable injective. We have described the points of $X$ and its closed sets in terms of only the category $A$, so we can recover the underlying topological space of $X$ from $A$. In particular, an open subscheme $U$ of $X$ gives a complementary closed subscheme $Y$, which is in correspondence with a localizing subcategory $\langle M\rangle$ and, moreover, $\operatorname{QCoh}(U) = A/\langle M\rangle$. So, responding to the queston above, for any $f\colon U\to X$, $U$ is an open subscheme if and only if the kernel of $f^\*\colon \operatorname{QCoh}(X) \to \operatorname{QCoh}(U)$ is a localizing subcategory of the form $\langle M\rangle$ for a coherent sheaf $M$. Regarding the structure sheaf $O\_X$ there is an isomorphism between $O\_X(U)$ and the ring of endomorphisms of the identity functor on $\operatorname{QCoh}(U)$ (which happens to be $A/\operatorname{QCoh}(Y)$), so the structure sheaf can be recovered only in terms of the category $A$. Finally, just say that there are other results in the spirit of reconstructing a scheme from some category of sheaves on it. This is the starting point for using such categories of sheaves as a definition of noncommutative scheme. There is more information on [this entry](http://ncatlab.org/nlab/show/noncommutative+algebraic+geometry) in nlab.

22

https://mathoverflow.net/users/322

1417

https://mathoverflow.net/questions/1438

28

This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?

https://mathoverflow.net/users/788

Why is homology not (co)representable?

Corepresentable functors preserve products; homology does not. One replacement is the following. Let X be a CW-complex with basepoint. Then the spaces {K(Z,n)} represent reduced integral homology in the sense that for sufficiently large n, the reduced homology Hk(X) coincides with the homotopy groups of the smash product: ``` pin+k(X ^ K(Z,n)) = [Sn+k, X ^ K(Z,n)] ``` This is some kind of "stabilization", and it factors through taking the n-fold suspension of X. Taking suspensions makes wedges more and more closely related to products. This doesn't make homology representable, but provides some alternative description that's more workable than simply an abstract functor.

24

https://mathoverflow.net/users/360

1444

https://mathoverflow.net/questions/1443

3

I am given a graph defined by vertexes and edges. I have to obtain all the cycle bases in a network. **No coordinates will be given for the nodes.** Here's a [sketch](http://deluxecourse.com/network.png) that illustrates my point. **Note that inside a cycle it must not contain any edge**

https://mathoverflow.net/users/807

Algorithm to find all the cycle bases in a graph

Maybe what you want is a cycle basis? That is, a set of cycles such that any other cycle can be found by adding and subtracting combinations of cycles in the basis. One can find a cycle basis easily for any graph by finding a spanning tree and then, for each edge that's not in the tree, reporting the cycle formed by that edge together with the tree path connecting its endpoints. In a plane-embedded graph, the set of interior faces forms a cycle basis, matching what the sketch describes. Finding the shortest cycle basis is more complicated but still known in polynomial time; see e.g. [Kavitha et al, ICALP 2004](http://dx.doi.org/10.1007/b99859).

11

https://mathoverflow.net/users/440

1466

https://mathoverflow.net/questions/1420

111

For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.

https://mathoverflow.net/users/66

What's the "best" proof of quadratic reciprocity?

I think by far the simplest easiest to remember elementary proof of QR is due to Rousseau ([On the quadratic reciprocity law](https://stacky.net/files/115/RousseauQR.pdf "G. Rousseau: On the quadratic reciprocity law")). All it uses is the Chinese remainder theorem and Euler's formula $a^{(p-1)/2}\equiv (\frac{a}{p}) \mod p$. The mathscinet review does a very good job of outlining the proof. I'll try to explain how I remember it here (but the lack of formatting is really rough for this argument). Here's the outline. Consider $(\mathbb{Z}/p)^\times \times (\mathbb{Z}/q)^\times = (\mathbb{Z}/pq)^\times$. We want to split that group in "half", that is consider a subset such that exactly one of $x$ and $-x$ is in it. There are three obvious ways to do that. For each of these we take the product of all the elements in that "half." The resulting three numbers are equal up to an overall sign. Calculating that sign on the $(\mathbb{Z}/p)^\times$ part and the $(\mathbb{Z}/q)^\times$ part give you the two sides of QR. In more detail. First let me describe the three "obvious" halves: 1. Take the first half of $(\mathbb{Z}/p)^\times$ and all of the other factor. 2. Take all of the first factor and the first half of $(\mathbb{Z}/q)^\times$. 3. Take the first half of $(\mathbb{Z}/pq)^\times$. The three products are then (letting $P = (p-1)/2$ and $Q=(q-1)/2$): 1. $(P!^{q-1}, (q-1)!^P)$. 2. $((p-1)!^Q, Q!^{p-1})$. 3. $\left(\dfrac{(p-1)!^Q P!}{q^P P!},\dfrac{(q-1)!^P Q!}{p^Q Q!}\right)$. All of these are equal to each other up to overall signs. Looking at the second component it's clear that the sign relating 1 and 3 is $\left(\frac{p}{q}\right)$. Similarly, the sign relating 2 and 3 is $\left(\frac{q}{p}\right)$. So the sign relating 1 and 2 is $\left(\frac{p}{q}\right) \left(\frac{q}{p}\right)$. But to get from 1 to 2 we just changed the signs of $\frac{p-1}{2} \frac{q-1}{2}$ elements. QED

135

https://mathoverflow.net/users/22

1472

https://mathoverflow.net/questions/1440

16

Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a fully faithful, exact embedding into the module category of some ring. Therefore, I'd like a theorem like this: Any small triangulated category is a fully faithful, triangulated subcategory of the unbounded derived category of modules on some ring. My guess is that this fails to be true, for similar reasons to a triangulated category not always being the derived category of its core. Is there a simple example of this? -and- Is there a set of properties that do imply the above theorem?

https://mathoverflow.net/users/750

Freyd-Mitchell for triangulated categories?

There are some things like what you ask for but as Tyler points out one needs restrictions on the categories one can consider. Any algebraic triangulated category which is well generated is equivalent to a localization of the derived category of a small DG-category - this is a theorem of Porta (ref is M. Porta, The Popescu-Gabriel theorem for triangulated categories. arXiv:0706.4458). Here algebraic (in the sense of Keller) means that the category is equivalent as a triangulated category to the stable category of a Frobenius category (Schwede has a paper on this as well giving conditions in terms of Koszul type objects). An answer (maybe closer to what you ask) is the following. If one has a Grothendieck abelian category then Gabriel-Popescu tells you it comes from a torsion theory on some category of modules. It turns out this lifts to the level of derived categories so one can view the derived category of a Grothendieck abelian category as a localization of the derived category of R-modules for some R (in particular it comes with a fully faithful embedding into the derived category of R-modules).

9

https://mathoverflow.net/users/310

1478

https://mathoverflow.net/questions/1465

19

Polynomials in $\mathbb Z[t]$ are categorified by considering Euler characteristics of complexes of finite-dimensional graded vector spaces. Now, given a rational function that has a power series expansion with integer coefficients, it seems natural to consider complexes of (locally finite-dimensional) graded vector spaces. Are there nice examples of this in nature?

https://mathoverflow.net/users/813

Can we categorify the equation (1 - t)(1 + t + t^2 + ...) = 1?

Yes, the particular equation you wrote is categorified by the free resolution of k as module over k[x] by the complex $k[x] \overset{x}\longrightarrow k[x]$ given by multiplication by x. It also appears in the numerical criterion for Koszulity of k[x] (see the paper of [Beilinson, Ginzburg and Soergel](http://mathaware.org/jams/1996-9-02/S0894-0347-96-00192-0/S0894-0347-96-00192-0.pdf)).

22

https://mathoverflow.net/users/66

1481

https://mathoverflow.net/questions/1479

5

I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes: I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh coefficients of the "Majority" function on 2n+1 binary inputs. Long story short, that boiled down to finding the asymptotics of the following quantity: L[n] := sum{k=0..n} F[n,k], where F[n,k] := (2n+1)!! / [(2k+1) k! (n-k)!]. Here is the set of steps that a naive person such as me always follows in such a scenario: Step 1. Type it into Maple. In this case, Maple reports back L[n] = ( (2n+1) (2n choose n) / 2^n ) hypergeom([1/2, -n], [3/2], -1). I don't know exactly what this hypergeom is, nor how to evaluate its asymptotics. I can't seem to get Maple to tell me (with asympt()) either. Step 2. Evaluate the first few terms (1, 4, 14, 48, 166, 584, 2092, etc.) and type it into OEIS. It's clearly sequence A082590. There are many definitions given for this sequence; One definition is (basically) the above hypergeometric formula. Another is that it is 2^n sum{k=0..n} 2^(-k) (2k choose k). Given this, it's pretty easy to deduce from Stirling that the rough asymptotics is Theta(2^(2n) / sqrt(n)). Actually, I originally didn't notice this simple definition on the OEIS page. Instead, the simplest thing I noticed was the title definition, the [x^n] coefficient of 1/((1-2x) sqrt(1-4x)). Since that was the simplest thing I initially noticed, I wondered how to find the asymptotics of that. I was sure many people would know that, but mathoverflow didn't exist at the time. So... Step 3. Email Doron Zeilberger out of the blue, asking for help. He very kindly responded: "You could use the contour integral... or my favorite way would be to use my Maple packages: read AsyRec: read EKHAD: Asy(AZd( 1/(1-2\*x)/(1-4\*x)^(1/2)/(x^(n+1)),x,n,N)[1],n,N,2); which produces 2^(2\*n)*(1/n)^(1/2)*(1+3/8/n+121/128/n^2)." Great! There's the precise answer! But I don't know much about how Zeilberger's packages work, don't know what the contour integral is, and anyway, it's all roundabout story. If you were writing this in a paper, what would be the shortest way to go from the definition of L\_n to the fact L[n] = (2^(2n) / sqrt(n)) (1+o(1))?

https://mathoverflow.net/users/658

Asymptotics of a hypergeometric series/Taylor series coefficient.

Okay, you want the asymptotics of [z^n] 1/((1-2z) sqrt(1-4z)). In a neighborhood of z = 1/4, you have 1/((1-2z) sqrt(1-4z)) = 2/sqrt(1-4z) \* (1+o(1)) Then a theorem of Flajolet and Odlyzko (see Flajolet and Sedgewick, *Analytic Combinatorics*, Cor. VI.1; the book is available for free download from [Flajolet's web site](http://algo.inria.fr/flajolet/Publications/books.html) tells you that [z^n] 1/((1-2z) sqrt(1-4z)) = (1+o(1)) \* 2 [z^n] (1-4z)^{-1/2}. This is the well-known generating function for the sequence of central binomial coefficients C(2n, n). So [z^n] (1-4z)^{-1/2} = 4^n / sqrt(Pi\*n) \* (1+o(1)). Therefore [z^n] 1/((1-2z) sqrt(1-4z)) = 4^n/sqrt(n) \* 2/sqrt(Pi) \* (1+o(1)). For further development of the asymptotic series, read Chapter 6 of Flajolet and Sedgewick.

4

https://mathoverflow.net/users/143

1486

https://mathoverflow.net/questions/1489

71

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup generated?

https://mathoverflow.net/users/828

Cohomology and fundamental classes

Rene Thom answered this in section II of "Quelques propriétés globales des variétés différentiables." Every class $x$ in $H\_r(X; \mathbb Z)$ has some integral multiple $nx$ which is the fundamental class of a submanifold, so the homology is at least rationally generated by these fundamental classes. Section II.11 works out some specific cases: for example, every homology class of a manifold of dimension at most 8 is realizable this way, but this is not true for higher dimensional manifolds and the answer in general has to do with Steenrod operations.

71

https://mathoverflow.net/users/428

1495

https://mathoverflow.net/questions/834

16

The following question came up in the discussion at [How small can a group with an n-dimensional irreducible complex representation be?](https://mathoverflow.net/questions/530/how-small-can-a-group-with-an-n-dimensional-irreducible-complex-representation-be) : Is it known that there are infinitely many primes p for which the least prime q which is 1 mod p is > c p^2 (for some positive constant c, independent of p)? Wikipedia's article on primes in arithmetic progressions says that the expected bound for the least prime is p^{2 + \epsilon}, given various strengthenings of the Riemann Hypothesis, but it doesn't say much about lower bounds. By the way, for the applications in the linked post, it would be even better if the same lower bound applied to finding a prime power q which is 1 mod p.

https://mathoverflow.net/users/297

Arithmetic progressions without small primes

It is firmly expected that for every \epsilon > 0 each aritmetic progression with difference q and terms coprime with q will contain a prime <<{\epsilon} q^{1 + \epsilon}. This is a direct consequence of a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the Elliott-Halberstam conjecture. But the upper bound $\ll {\epsilon} q^{1 + \epsilon}$ for the least prime in an arithmetic progression was conjectured by S. Chowla (in his book "The Riemann Hypothesis and Hilbert's tenth problem"), and probably by others independently of him, years before Montgomery made his conjecture, and presumably on the basis of the same kind of heuristic argument that moonface advances. In fact, I think that even an upper bound as strong as qlog(q)^2 has been conjectured, though I won't swear to that. But it is definitely known that qlog(q) won't work. The Montgomery conjecture seems reasonable, because it rests on the assumption that there is square root cancellation in a certain sum with D-characters as coefficients in an explicit formula - this ties in with moonface's comment about the proof that GRH implies L \leq 2 being "lossy". On the other hand, one cannot expect to get q^{1 + \epsilon} out of the Elliott-Halberstam conjecture in any direct way, because that is an averaging kind of statement. You would not expect to get L \leq 2 out of the Bombieri-A. I. Vinogradov theorem either, for that is an averaged version of GRH. The point is that information is needed about every single one of the arithmetic progressions individually.

13

https://mathoverflow.net/users/3304

1498

https://mathoverflow.net/questions/1492

10

Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a complex orbifold) to a projective algebraic variety.

https://mathoverflow.net/users/683

Moduli spaces of complex curves as algebraic varieties

The classical (pre Deligne-Mumford) approach is to map $\mathcal{M}\_g$ into $\mathcal{A}\_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}\_g$ is quasi-projective is to define it as the Siegel upper half space $\mathcal{H}\_g$ ($g$ by $g$ complex matrices with positive define imaginary part), modulu $\mathrm{SP}(2g,\mathbb{Z})$; We define the level cover $\mathcal{A}\_g(m)\to\mathcal{A}\_g$, which is the moduli of $\mathcal{A}\_g$ plus torsion points, which is the quotient of $\mathcal{H}\_g$ by $\Gamma(m)$ (the matrixes in $\mathrm{SP}(2g,\mathbb{Z})$ which are trivial modulo $n$), and send $\mathcal{A}\_g(m)$ to some projective space using polynomials in the theta constants. Reference (for both Torreli and the embedding of $\mathcal{A}\_g$ above) : Mumfords curves and their Jacobians (now bundled together with the red book in LNM 1358) Lecture IV. In Lecture II (same place) Mumford sketches two more "coordinate oriented" methods * Tracking the Weierstrass point of curves. * Tracking invariants of the Chow form of the canonical curve (the Chow form is the equation for pairs of hyperplanes such that $H\_1 \cap H\_1 \cap$ canonical-curve is not $0$). He also says these are the only "coordinate oriented" methods he knows of.

12

https://mathoverflow.net/users/404

1500

https://mathoverflow.net/questions/1504

34

More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its integral closure A' (in its field of fractions), what's the geometric relationship between V(A) and V(A')? If you can, phrase your answer in terms of complex affine varieties.

https://mathoverflow.net/users/290

What is the geometric meaning of integral closure?

The property you are interested in is known as being *normal*. For affine varieties, the definition of normal is just that the coordinate ring is integrally closed, and the operation on varieties that corresponds to taking the integral closure of the coordinate ring is known as *normalization* (a general variety is said to be normal if it is locally isomorphic to a normal affine variety.) So far I've just restated your question; but there are a number of things known. For this we need the notion of smoothness: for varieties over $\mathbb{C}$, this should be equivalent to being a smooth manifold (the general definition is a bit technical). 1. Any smooth variety is normal. 2. The set of singular points of a normal variety has codimension $\geq 2$. **Corollary:** For curves, normal $\iff$ smooth. Shafarevich's *Basic Algebraic Geometry* vol. 1 is a good reference for this from the varieties point of view (and deals with smoothness more rigorously). As regards the relationship between the varieties corresponding to $A$ and $A'$: I mostly just have intuition for curves, so I'll stick to talking about them. For curves, $A'$ is a version of $A$ with the singularities "resolved": more specifically, $V(A')$ is a smooth variety equipped with a surjective morphism of varieties from $A' \to A$ which is an isomorphism away from the preimages of the singular points of $A$. (This should be true in higher dimensions too I think: it's definitely true if one is talking about schemes, but I think it's also true that for affine varieties the map $A \to A'$ induces a map $V(A') \to V(A)$.) The two basic examples to keep in mind here are the cuspidal cubic $C\_1: y^2 = x^3$ and the nodal cubic $C\_2: y^2 = x^3 + x^2$. In the case $C\_1$: the coordinate ring $\mathbb{C}[x, y]/(y^2 - x^3)$ has integral closure isomorphic to $\mathbb{C}[t]$, and the map of varieties here is the map from the affine line to $C\_1$ given by $t \mapsto (t^2, t^3)$. In this case the map is a bijection as sets (but not an isomorphism of affine varieties! because the inverse map cannot be expressed as a polynomial map), and the "cusp" of $C\_1$ that is visible at the point $(0,0)$ is no longer evident. In the case $C\_2$: the coordinate ring also has integral closure isomorphic to $\mathbb{C}[t]$: this time the map is a bit more complicated, but it's $t \mapsto (t^2 -1 , t(t^2-1))$. How did I find that? In this case, looking at the curve one sees that it has a self-intersection at the origin. This means that there should be two distinct points in the normalization that have been sent to the same point in $C\_2$. Another way of stating that is that because the curve appears to have two tangent lines at the origin, there really should be two different points there, one on each tangent line. How to tell them apart? Well, as one approaches the origin from one direction, the ratio $y/x$ tends to $1$ in the limit, whereas if one approaches it from the other direction, the ratio $y/x$ tends to $-1$: so at one of our two points, $y/x=1$, and at the other one, $y/x = -1$. Since $y/x$ is well-defined everywhere else on the curve, this suggests that we want $t = y/x$ to belong to our coordinate ring at the origin. Indeed, $t^2 = x +1$, so $t$ is integral, and we can solve for $x$ and $y$ in terms of $t$ to get the original answer. So in this case we have a surjective map from the affine line to a self-intersecting curve which is injective everywhere except at the preimage of the singular point at the origin.

29

https://mathoverflow.net/users/422

1512