microlinear space

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#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#

* table of contents
{:toc}

## Idea

In traditional [[differential geometry]] a [[smooth manifold]] may be thought of as a "locally linear space": a space that is locally isomorphic to a
[[vector space]] $\simeq \mathbb{R}^n$.

In the broader context of [[synthetic differential geometry]]  there may exist [[spaces]] --- in a [[smooth topos]] $\mathcal{T}$ with line object $R$ ---
considerably more general than manifolds. While for all of them  there is a notion of [[tangent bundle]] $T X : = X^D$ (sometimes called a _[[synthetic tangent bundle]]_, with $D$ the [[infinitesimal space|infinitesimal interval]]), not all such tangent bundles
necessarily have $R$-linear fibers!

A _microlinear space_  is essentially an object $X$ in a [[smooth topos]],  such that its [[tangent bundle]] does have $R$-linear fibers.

In fact the definition is a bit stronger than that, but the main point in practice of microlinearity is that the linearity of the fibers of the tangent bundle allows to apply most of the familiar constructions in [[differential geometry]] to these spaces.

## Definition

+-- {: .num_defn}
###### Definition
**(microlinear space)**

Let $\mathcal{T}$ be a [[smooth topos]] with line object $R$.  An object $X \in \mathcal{T}$ is a __microlinear space__ if for each [[diagram]] $\Delta : J \to \mathcal{T}$ of [[infinitesimal spaces]] in $\mathcal{T}$ and for each [[colimit|cocone]] $\Delta \to \Delta_c$ under it such that homming into $R$ produces a [[limit]] diagram , $R^\Delta_c \simeq \lim_{j \in J} R^{\Delta_j}$, also homming into $X$ produces a limit diagram: $X^\Delta_c \simeq \lim_{j \in J} X^{\Delta_j}$.

=--

The main point of this definition is the following property.

+-- {: .num_prop}
###### Proposition
**(fiberwise linearity of tangent bundle)**

For every microlinear space $X$, the [[tangent bundle]] $X^D \to X$ has a natural fiberwise $R$-[[module]]-structure.

=--

+-- {: .proof}
###### Construction and Proof

We describe first the addition of tangent vectors, then the $R$-action on them and then prove that this is a [[module]]-structure.

* **Addition** With $D = \{\epsilon \in R| \epsilon^2 = 0 \}$ the
  [[infinitesimal space|infinitesimal interval]] and
  $D(2) = \{(\epsilon_1, \epsilon_2) \in R \times R | \epsilon_i^2 = 0\}$
  we have a cocone

  $$
    \array{
      D(2) &\leftarrow& D
      \\
      \uparrow && \uparrow
      \\
      D &\leftarrow& {*}
    }
  $$

  such that

  $$
    \array{
      R^{D(2)} &\to & R^D
      \\
      \downarrow && \downarrow
      \\
      R^D &\to& R
    }
  $$

  is a [[limit]] cone, by the [[Kock-Lawvere axiom]] satisfied in the [[smooth topos]] $\mathcal{T}$. Since $X$ is microlinear,
  also the canonical map

  $$
    r \colon X^{D(2)} \to X^D \times_X X^D
  $$

  is an [[isomorphism]]. With $\Id \times Id : D \to D(2)$ the diagonal
  map, we then define the fiberwise addition $X^D \times_X X^D \to X^D$ in
  the [[tangent bundle]] $X^D$
  to be given by the map

  $$
    + : X^D \times_X X^D \stackrel{r^{-1}}{\to} X^{D(2)} \stackrel{X^{Id \times Id}}{\to}
    X^D
    \,.
  $$

  On elements, this sends two elements $v_1, v_2 \in X^D$ in the same fiber to the element $v_1 + v_2$ of $X^D$ given by the map $(v_1 + v_2) : d \mapsto r^{-1}(v_1,v_2)(d,d)$.

*  **Multiplication** $ \cdot : R \times X^D \to X^D$ is defined componentwise by

   $(\alpha \cdot v) : d \mapsto v (\alpha \cdot d)$.

One checks that this is indeed unital, associative and distributive. ...

=--

## Examples

A large class of examples is implied by the following proposition.

+-- {: .num_prop}
###### Proposition
**(closedness of the collection of microlinear spaces)**

In every [[smooth topos]] $(\mathcal{T},R)$ we have
the following.

1. The standard line $R$ is microlinear.

1. The collection of microlinear spaces is closed under [[limits]] in $\mathcal{T}$:

   for $X = \lim_i X_i$ a [[limit]] of microlinear spaces $X_i$, also $X$ is microlinear.

1. Mapping spaces into microlinear spaces are microlinear: for $X$ any microlinear space and $\Sigma$ any space, also the [[internal hom]] $X^\Sigma$ is microlinear.

=--

+-- {: .proof}
###### Proof

This is obvious from the standard properties
of [[limits]] and the fact that the [[internal hom]]-functor
$(-)^Y : \mathcal{T} \to \mathcal{T}$ preserves limits.
(See [[limits and colimits by example]] if you don't find it obvious.)

1. by definition

1. Let $\Delta$ be the tip of a cocone $\Delta_j$ of infinitesimal spaces such
   that $\lim_j R^{\Delta_j} = R^\Delta$. Then

   $$
     \begin{aligned}
       X^{\Delta}
        &= (\lim_i X_i)^\Delta
        \\
        &\simeq \lim_i X_i^{\Delta}
        \\
        & \simeq \lim_i \lim_j X_i^{\Delta_j}
        \\
        & \simeq \lim_j \lim_i X_i^{\Delta_j}
        \\
        & \simeq \lim_j X^{\Delta_j}
      \end{aligned}
   $$

1. with $\Delta$ as above we have (writing $[A,B]$ for the [[internal hom]] otherwise
   equivalently denoted $B^A$)

   $$
     \begin{aligned}
       [\Delta, [\Sigma, X]] & \simeq [\Delta \times \Sigma, X]
       \\
       & \simeq [\Sigma, [\Delta, X]]
       \\
       & \simeq [\Sigma, \lim_j [\Delta_j, X]]
       \\
       & \simeq \lim_j [\Sigma, [\Delta_j, X]]
       \\
       & \simeq  \lim_j [\Delta_j, [\Sigma, X]]
     \end{aligned}
   $$

=--

+-- {: .num_prop}
###### Proposition
**(microlinear loci)**

Let $\mathcal{F}$, $\mathcal{G}, \mathcal{Z}, \mathcal{B}$ be the [[smooth toposes]] of the same name that are discussed in detail in [[Models for Smooth Infinitesimal Analysis|MSIA, capter III]]. These are constructed there as [[category of sheaves|categories of sheaves]] on a [[subcategory]] of the category $\mathbb{L} = (C^\infty Ring^{fin})$ of [[smooth loci]].

All [[representable functor|representable objects]] in these [[smooth toposes]] are microlinear.

=--

+-- {: .proof}
###### Proof

For $\mathcal{F}$ and $\mathcal{G}$ this is the statement of [[Models for Smooth Infinitesimal Analysis|MSIA, chapter V, section 7.1]].

For $\mathcal{Z}$ and $\mathcal{B}$ the argument is similarly easy:

These are categories of sheaves on the full category $\mathbb{L} = (C^\infty Ring^{fin})^{op}$.  The line object $R$ is representable in each case, $R = \ell C^\infty(\mathbb{R})$.  Every object in $\mathbb{L}$ is a limit (not necessarily finite) over copies of $R$ in $\mathbb{L}$. Accordingly, every object $\ell A$ of $\mathbb{L}$ satisfies the microlinearity axioms in $\mathbb{L}$ in that for each cocone $\Delta \to \Delta_c : J \to \mathbb{L}$ of [[infinitesimal objects]] such that $R^{\Delta_c} \simeq \lim_{j \in J} R^{\Delta_j}$ we also have $(\ell A)^{\Delta_c} \simeq \lim_{j \in J} (\ell A)^{\Delta_j}$.  Now, the [[Yoneda embedding]] $Y : \mathbb{L} \to PSh(C)$ preserves limits and exponentials.  Since the [[Grothendieck topology]] in question is [[subcanonical coverage|subcanonical]], $Y((\ell A)^{\Delta_j})$ is in $Sh(C)$ and hence is the exponential object $Y(\ell A)^{Y \Delta_j}$ there.  Finally, the _finite_ limit over $J$ is preserved by the reflection $PSh(C) \to Sh(C)$ (sheafification, which acts trivially on our representables), so $Y(\ell A)^{\Delta_c} \simeq \lim_{j \in J}  Y(\ell A)^{Y(\Delta_j)}$ and hence all $Y(\ell A)$ are microlinear in $\mathcal{Z}$ and $\mathcal{B}$.

=--

## References

The notion of microlinear space in the above fashion is due to

* F. Bergeron, (1980)

and was studied further under the name _strong infinitesimal linearity_

* [[Anders Kock]], R. Lavendhomme, _Strong infinitesimal linearity, with applications to string difference and affine connections_, Cahiers de Top. 25 (1984)

This is similar to but stronger than the earlier "condition (E)" given in

* Demazure (1970)

which apparently was also called "infinitesimal linearity" (without the "strong").

Spaces satisfying this condition were called _infinitesimally linear spaces_, for instance in

* [[Anders Kock]] (1981) .

The later re-typing of that book

* [[Anders Kock]], _Synthetic Differential Geometry_ (2006)  ([pdf](http://home.imf.au.dk/kock/sdg99.pdf))

contains in its appendix D the definition of microlinearity as above.

A comprehensive discussion of microlinearity is in chapter V, section 1 of

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_

[[!redirects microlinear spaces]]

[[!redirects microlinearity]]


microlocal analysis

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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In the original sense, _microlocal analysis_ (e.g. [Strohmaier 09](#Strohmaier09)) is the study of the [[functional analysis]] of [[generalized functions]]/[[distributions]] with attention paid not just to their [[singular support]], i.e. to the points around which they are singular as generalized functions, but also the directions of [[propagation of singularities theorem|propagation of their singularities]] at each singular point, which is the set of [[covectors]] known as their [[wave front set]]. This extra directional ("microlocal") information governs the basic operations on distributions, notably the [[pullback of distributions]] and the [[product of distributions]].

Since the wave front set is the set of (co-)directions along which, locally, the [[Fourier transform of distributions]] is not rapidly decreasing (the set of "[[UV divergences]]" in applications to [[perturbative quantum field theory]]), much of microlocal analysis is concerned with constructions related to Fourier transformation, such as the discussion of [[pseudodifferential operators]].

Microlocal analysis in this sense was introduced by ([Sato 70](#Sato70)),  soon  followed  by  ([Hörmander 71](#Hoermander71), [Hörmander 83](#Hoermander83)) who  both  introduced the notion of [[wave front set]].
This microlocal point of  view was then extended to [[sheaf  theory]] by [Kashiwara-Schapira 82](#KashiwaraSchapira82), [Kashiwara-Schapira 85](#KashiwaraSchapira85)) who  introduced  the  notion  of _[[microsupport]]_  of  sheaves giving rise to _[[microlocal sheaf theory]]_ (see [Schapira 17](#Schapira17)). Beware that some authors still use the term "microlocal analysis" for this sheaf theoretic concept.

## Overview

To accommodate an intuitive notion of a "function of differential operators" there is a simple trick used: consider the [[Fourier transform]]. Then the [[differential operators]] become polynomials. This correspondence of operators and their [[symbol of a differential operator|symbols]] may, with some analytic care, be extended to define generalizations of differential operators by suitably extending a notion of symbols. Thus the [[pseudodifferential operator]]s of Kohn and Nirenberg appeared in 1965 with soon following  revolution in harmonic analysis and analysis in [[PDE]]. This includes a  further generalization, the Fourier integral operators of [[Lars Hörmander]] and [[Victor Maslov|V. Maslov]]. A part of harmonic analysis involving geometric aspects in the cotangent bundles of such methods is called **microlocal analysis**. The geometric aspects include the support, wavefront set, characteristics...of distributions, pseudodifferential operators and their symbols. There are more technical definitions (involving [[wavefront set]]s, supports and filtrations on the algebras of symbols) of various "microlocal" properties of symbols: [[microlocalization]], microhypoellipticity, microparametrix etc.). In addition to the analytic microlocalization there is a formal microlocalization; and a version of filtered localization theory in noncommutative algebra, so called [[algebraic microlocalization]], which is however not used in operator theory. While local aspect of a differential operator is about its behaviour around a point in coordinate space, the microlocal aspect is about a point in the cotangent bundle, hence it also localizes around the fixed covector direction, hence "micro".

This is clearly related to the general study of oscillating integrals, including the stationary phase method and WKB-method (and generalizations) in particular. These kind of approximations and related estimates are of importance to the study of the propagation of singularities of differential equations, wave fronts, eikonal equations, and so on.

As oscillating integrals are involved in the analysis of various [[Green functions]] like the [[heat kernel]] there is also a connection to [[index theorems]] for [[elliptic differential operators]], see [Hörmander 83](#Hoermander83).

## In perturbative quantum field theory

In [[perturbative quantum field theory]] microlocal analysis is used to define [[Wick algebras]] of [[quantum observables]] on [[free fields]]: The product on these algebras is a [[Moyal star-product]] induced from the [[Peierls-Poisson bracket]], whose [[integral kernel]] is the [[causal propagator]] on the given [[globally hyperbolic spacetime]]. But the [[wave front set]] of this propagator is such that its [[UV-divergences]] in general collide with those of [[local functionals]] ([here](Wick+algebra#CompactlySupportedPolynomialLocalDensities)). This is fixed by modifying the causal propagator to a [[Hadamard propagator]]. The resulting change of the algebra structure is known as [[normal ordering]] of quantum fields. It yields the properly defined _[[Wick algebras]]_ of free quantum fields.

As the name suggests, normal ordering was originally an operation implementd simply by re-arranging the order of [[Fourier transform|Fourier]] modes of quantum fields. The desire to generalize this procedure from [[Minkowski spacetime]] to general [[globally hyperbolic Lorentzian manifolds]] was what required use of tools from microlocal analysis. See at _[[locally covariant perturbative AQFT]]_ and at _[[S-matrix]]_ for more.

## Related concepts

* [[wave front set]], [[microsupport]]

* [[microlocal sheaf theory]], [[microlocal category]]

* [[propagation of singularities theorem]]

* [[microformal morphism]]

## References

Microlocal analysis of [[distributions]] in terms of [[wave front sets]] was introduced in

* {#Sato70} [[Mikio Sato]], _Regularity of hyperfunctions solutions of partial differential equations 2 (1970), 785–794.

* {#Hoermander71} [[Lars Hörmander]], _Fourier integral operators_ I. Acta Math. 127 (1971)

* {#Hoermander83} [[Lars Hörmander]], _The analysis of linear partial differential operators_, in 4 vols.: _I. Distribution theory and Fourier analysis_, _II. Differential operators with constant coefficients_, _III. Pseudo-differential operators_, _IV. Fourier integral operators_, Grundlehren der mathematischen Wissenschaften 256, Springer 1983, 1990

Survey is in

* {#Strohmaier09} Alexander Strohmaier, chapter _Microlocal analysis_ ([web](https://link.springer.com/chapter/10.1007%2F978-3-642-02780-2_4)) in [[Christian Bär]], [[Klaus Fredenhagen]] _Quantum Field Theory on Curved Spacetime_, 2009 ([web](https://link.springer.com/book/10.1007
/978-3-642-02780-2))

* A. Kaneko, _[Microlocal analysis](http://eom.springer.de/M/m063760.htm)_, Springer Online Enc. Of Math.

Comprehensive lecture notes are in

* {#Melrose03} [[Richard Melrose]], _Introduction to microlocal analysis_, 2003 ([pdf](http://www-math.mit.edu/~rbm/iml90.pdf))

See also

* C. Bardos, L. Boutet de Monvel, _From atomic hypothesis to microlocal analysis_ (lecture notes) [pdf](www.math.jussieu.fr/~boutet/Micro_equations.pdf)

* A. Grigis, J. Sjöstrand, _Microlocal analysis for differential operators: an introduction_,   Cambridge U.P. 1994.

* [[Hans Duistermaat]], _Fourier integral operators_, Progress in Mathematics, Birkhäuser 1995.

* [[Victor Guillemin]], [[Masaki Kashiwara]], Takahiro Kawai, _Seminar on micro-local analysis_, Ann. of Math. Studies 93 (1979), [googlebooks](https://books.google.com/books?isbn=0691082324)
* [[Victor Guillemin]], [[Shlomo Sternberg]], _Some problems in integral geometry and some related problems in microlocal analysis_, Amer. J. Math. __101__ (1979), 915--955 [jstor](https://www.jstor.org/stable/2373923)

* Yu. V. Egorov,  _Microlocal analysis_, Ю. В. Егоров, _Микролокальный анализ_, Дифференциальные уравнения с частными производными – 4, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 33, ВИНИТИ, М., 1988, 5-–156, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=intf&paperid=116&what=fullt&option_lang=rus), [MR93e:35002](http://www.ams.org/mathscinet-getitem?mr=1175403), Eng. translation in Partial Differential Equations IV (Egorov, Shubin eds.), Springer 1993 [doi](https://dx.doi.org/10.1007/978-3-662-09207-1)

* Yu. Safarov, _Distributions, Fourier transforms and microlocal analysis_ (course online notes), [pdf](https://nms.kcl.ac.uk/yuri.safarov/Lectures/LTCC2014a/pdos-2014a1.pdf)

Application in [[perturbative quantum field theory]]:

* Christian Gérard, *Microlocal analysis of quantum fields on curved spacetimes* ([arXiv:1901.10175](https://arxiv.org/abs/1901.10175))

Some categories related to microlocal analysis (see also [[microlocal category]])

* [[Dmitry Tamarkin]], _Microlocal category_, [arXiv:1511.08961](https://arxiv.org/abs/1511.08961)

> Given a compact symplectic manifold whose symplectic form has integer periods, one associates to it a dg category based on the microlocal analysis according to Kashiwara-Schapira.

* [[Boris Tsygan]], _A microlocal category associated to a symplectic manifold_, In: M. Hitrik, D. Tamarkin, B. Tsygan, S. Zelditch (eds), Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proc. in Math. & Stat. __269__ [doi](https://doi.org/10.1007/978-3-030-01588-6_4)
* [[Jun Zhang]], _Quantitative Tamarkin theory_, book, CRM Short Courses, [arXiv:1807.09878](https://arxiv.org/abs/1807.09878)
[doi](https://doi.org/10.1007/978-3-030-37888-2)

> ...we try to explain how standard symplectic techniques, for instance, generating function, capacities, symplectic homology, etc., are elegantly packaged in the language of sheaves as well as related intriguing sheaf operators. In addition, many concepts developed in Tamarkin category theory are natural generalizations of [[persistent homology]] theory...

category: analysis, geometry


microlocal category

Related notions: [[microlocal analysis]]

* Dmitry Tamarkin, _Microlocal category for a closed symplectic manifold_, talk at IAS [youtube](https://www.youtube.com/watch?v=GxdwyLt-i-k)
* Boris Tsygan, _A microlocal category associated to a symplectic manifold_, talk [youtube](https://www.youtube.com/watch?v=tQc3_j0qEIs)
* [[Dmitry Tamarkin]], _Microlocal category_, [arXiv:1511.08961](https://arxiv.org/abs/1511.08961)

> Given a compact symplectic manifold whose symplectic form has integer periods, one associates to it a dg category based on the microlocal analysis according to Kashiwara-Schapira.

* [[Boris Tsygan]], _A microlocal category associated to a symplectic manifold_, In: M. Hitrik, D. Tamarkin, B. Tsygan, S. Zelditch (eds), Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proc. in Math. & Stat. __269__ [doi](https://doi.org/10.1007/978-3-030-01588-6_4)
* [[Jun Zhang]], _Quantitative Tamarkin theory_, book, CRM Short Courses, [arXiv:1807.09878](https://arxiv.org/abs/1807.09878)
[doi](https://doi.org/10.1007/978-3-030-37888-2)

> ...we try to explain how standard symplectic techniques, for instance, generating function, capacities, symplectic homology, etc., are elegantly packaged in the language of sheaves as well as related intriguing sheaf operators. In addition, many concepts developed in Tamarkin category theory are natural generalizations of [[persistent homology]] theory...

* Vivek Shende, _Microlocal category for Weinstein manifolds via h-principle_, [arXiv:1707.07663](https://arxiv.org/abs/1707.07663)

* Sheng-Fu Chiu, _Non-squeezing property of contact balls_, [doi](https://doi.org/10.1215/00127094-3715517) [arXiv:1405.1178](https://arxiv.org/abs/1405.1178)

> In this paper we solve a contact non-squeezing conjecture proposed by Eliashberg, Kim and Polterovich.
Let BR be the open ball of radius $R$ in $\mathbb{R}^{2n}$ and let $\mathbb{R}^{2n}\times S^1$ be the prequantization space equipped with the standard contact structure. Following [[Dmitry Tamarkin|Tamarkin]]’s idea, we apply microlocal category methods to prove that if $R$ and $r$ satisfy $1\leq \pi r^2 \lt \pi R^2$, then it is impossible to squeeze the contact ball $B_R\times S^1$ into $B_r\times S^1$ via compactly supported contact isotopies.

category: geometry, analysis


microlocal defect functional

Microlocal defect functionals -- microlocal defect measures (H-measures), H-distributions,...

* Luc Tartar, _$H$-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations_, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 193–230 [MR1069518 (91h:35042)](http://www.ams.org/mathscinet-getitem?mr=1069518), [doi](http://dx.doi.org/10.1017/S0308210500020606); _$H$-measures and applications_, ICM 1992 lecture, [pdf](http://www.mathunion.org/ICM/ICM1990.2/Main/icm1990.2.1215.1224.ocr.pdf); _On mathematical tools for studying partial differential equations of continuum physics: $H$-measures and Young measures_, Developments in partial differential equations and applications to mathematical physics (Ferrara, 1991) Plenum, New York, 1992, pp. 201–217 [MR1213932 (94c:35031)](http://www.ams.org/mathscinet-getitem?mr=1213932); _Compensated compactness and applications to partial differential equations_,
Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Pitman 1979, pp. 136–212, MR81m:35014
* Patrick Gérard, _Microlocal defect measures_, Communications in Partial Differential Equations __16__:11 (1991) 1761-1794 [doi](http://dx.doi.org/10.1080/03605309108820822) [MR 1135919 (92k:35027)](http://www.ams.org/mathscinet-getitem?mr=1135919)

category: analysis


microlocal formulation of index theory

[[!redirects Microlocal formulation of index theory]]

#Contents#
* table of contents
{:toc}

## Idea

One may give an abstract definition of an [[elliptic complex]] over a [[complex analytic space]]: an elliptic pair is the datum of an $\mathbb{R}$-[[constructible sheaf]] $F$ and a complex $\mathcal{M}$ of $\mathcal{D}$-[[D-module|modules]], such that the intersection
$$char(\mathcal{M})\cap SS(F)=T^*_M M\subset T^*M$$
of the characteristic variety of $\mathcal{M}$ with the microsupport of $F$ is contained in the zero section of the cotangent bundle.

To such an elliptic pair, one may associate a pair of trace kernels $K_{\mathcal{M}}$ and $K_F$ such that the trace kernel $K_{\mathcal{M}}\otimes K_F$ is well defined (because of the microsupport condition). The index of the elliptic pair is given by the Hochshild class $eu(K_{\mathcal{M}}\otimes K_F)$, and the index theorem essentially says that this class may be computed as the product of the classes of the two kernels in play.

## Generalization

One may try to generalize the microlocal formulation of index theory to [[global analytic geometry]], using [[derived microlocalization]], to get a [[global analytic index theory]] that generalizes Kashiwara and Schapira's approach to the analytic situation over an arbitrary Banach ring.

## Reference

* [[Masaki Kashiwara]] and [[Pierre Schapira]] _Microlocal Euler classes and Hochschild homology_ [arXiv](http://arxiv.org/abs/1203.4869).

[[!redirects microlocal formulation of index theory]]


microlocal sheaf theory

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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In _microlocal sheaf theory_ one generalizes the concepts of [[microlocal analysis]], such as the [[wave front set]] of a [[distributions]], to [[sheaf theory]], notably to a concept of _[[microsupport]]_ of a sheaf on a [[smooth manifold]], as a subset of the [[cotangent bundle]].

This microlocal point of  view was then extended to [[sheaf  theory]]  by
[Kashiwara-Schapira 82](#KashiwaraSchapira82), [Kashiwara-Schapira 82](#KashiwaraSchapira85)) who  introduced  the  notion  of _[[microsupport]]_  of  sheaves giving rise to _[[microlocal sheaf theory]]_ (see [Schapira 17](#Schapira17)). Beware that some authors still use the term "microlocal analysis" for this sheaf theoretic concept.

## References

The generalization of the concepts of [[microlocal analysis]]to microlocal sheaf theory via the concept of [[microsupport]] of sheaves was introduced in spring

* {#KashiwaraSchapira82} [[Masaki Kashiwara]], [[Pierre Schapira]], _Microlocal study of sheaves_, CAstérisque, vol. 128 (1985)

* {#KashiwaraSchapira85} [[Masaki Kashiwara]], [[Pierre Schapira]], _Microsupport des faisceaux; applications aux module différentielle_, C. R. Acad. Sci. Paris 295, 8 (1982)

Review of this incldues

* {#Schapira17} [[Pierre Schapira]], _Microlocal analysis and beyond_, in [[Gabriel Catren]], [[Mathieu Anel]] (eds.), _[New Spaces in Mathematics and Physics](https://ercpqg-espace.sciencesconf.org/resource/page/id/1)_ ([arXiv:1701.08955](https://arxiv.org/abs/1701.08955))

See also:

* [[Victor Ginzburg|V. Ginsburg]], _Characteristic varieties and vanishing cycles_, Invent. Math. __84__, 327--402 (1986), [MR87j:32030](http://www.ams.org/mathscinet-getitem?mr=833194), [doi](http://dx.doi.org/10.1007/BF01388811)

* Makoto Sakai, _Restriction, localization and microlocalization_, (expository paper) pp. 195--205 in collection "Quadrature domains and their application" (Operator theory: advances and applications __156__), [doi](http://dx.doi.org/10.1007/b137105)

* M. [[Kashiwara]], [[P. Schapira]], F. Ivorra, I. Waschkies, _Microlocalization of ind-sheaves_, in "Studies in Lie theory", Progress in Math. __243__, Birkhäuser 2006.

* [[Dmitry Tamarkin]], _Microlocal condition for non-displaceablility_, [arxiv/0809.1584](http://arxiv.org/abs/0809.1584) (application of microlocal analysis to symplectic/Lagrangean geometry).

* Goro Kato, Daniele Carlo Struppa, _Fundamentals of algebraic microlocal analysis_, M. Dekker 1999, [googB](http://books.google.hr/books?isbn=0824793277)

* [[Masaki Kashiwara]], _Systems of microdifferential equations_, Birkhäuser 1983, 87 pp.

* [[Masaki Kashiwara]], Takahiro Kawai, Tatsuo Kimura, _Foundations of algebraic analysis_, Transl. from Japanese by Goro Kato. Princeton Mathematical Series __37__, 1986. xii+255 pp.
[MR87m:58156](http://www.ams.org/mathscinet-getitem?mr=855641); [[Jean-Luc Brylinski]], Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) __18__ (1988), no. 1, 104–108, [doi](https://doi.org/10.1090/S0273-0979-1988-15622-4)

* [[Masaki Kashiwara]], Pierre Schapira, _Hochschild homology and microlocal Euler classes_, [arxiv/1203.4869](https://arxiv.org/abs/1203.4869)

* Vivek Shende, _Microsheaves from Hitchin fibers via Floer theory_, [arXiv:2108.13571](https://arxiv.org/abs/2108.13571)

See also [[microlocal category]].

category: geometry, analysis


microlocalization

[[!redirects Microlocalization]]

#Contents#
* table of contents
{:toc}

## Idea

Microlocalization is a tool invented by [[Mikio Sato]] to study linear [[partial differential equations]] (as a part of his [[algebraic analysis]] program) not only locally in space but also locally in momentum variable. It is a purely algebraic theory that was also continued in parallel by analysts, like Hormander, giving the domain of [[microlocal analysis]].

## Construction

The original construction is based on the use of the specialization functor and Fourier-Sato transformation. In this section, we will discuss the construction in a general setting, i.e., over an arbitrary field of characteristic $0$. In the real situation, one usually refines the construction by using conic sheaves (i.e., sheaves invariant with respect to the natural $\mathbb{R}_+$-action) to get information about the oriented direction of propagation of singularities of sheaves of solutions of analytic partial differential systems. The construction we describe does not treat this refined information.

Let $Z\hookrightarrow X$ be a closed subspace of a given analytic manifold, defined by a sheaf of ideals $\mathcal{I}$, with normal bundle denoted $T_Z X$ and conormal bundle denoted $T^*_Z X$. One defines the deformation to the normal bundle as the (analytic space associated to the) relative scheme over $X$ given by
$$\widetilde{T_Z X}:=Spec_X(\oplus_{i\in \mathbb{Z}} z^{-i} \mathcal{I}^i)^{an}$$
with $\mathcal{I}^i=\mathcal{O}_X$ for $i\leq 0$.
There is a projection $p:\widetilde{T_Z X}\to X$ and a projection $\tau:\widetilde{T_Z X}\to \mathbb{A}^1$. The fiber at $0$ of $\tau$ is denoted $s:T_Z X\to \widetilde{T_Z X}$, and its fiber at $t\neq 0$ is $X$. The fiber of $p$ on the open subset $(X\backslash Z)$ is $(X\backslash Z)\times \mathbb{A}^1-\{0\}$.

The specialization of a sheaf $F\in D^b(k_X)$ is the sheaf $\nu_Z(F)\in D^b(k_{T_Z X})$ defined as
$$\nu_Z(F):=s^*p^*F.$$

The Fourier-Sato transform is the functor
$$\Phi:D^b(k_{T_Z X})\to D^b(k_{T^*_Z X})$$
defined by
$$\Phi(G):=\mathbb{R}p_{2!}p_1^*G$$
where $p_1:T_Z X\times_Z T^*_Z X\to T_Z X$ and $p_2:T_Z X\times_Z T^*_Z X\to T^*_Z X$ are the two natural projections.

The $Z$-microlocalization functor is the functor
$$\mu_Z:=\Phi\circ \nu_Z:D^b(k_X)\to D^b(k_{T^*_Z X}).$$

The microlocalization functor on a variety $M$ is defined as the $Z$-microlocalization associated to the closed immersion $Z=M\subset M\times M=X$. Since $T^*_{\Delta_M} (M\times M)\cong T^*M$, this gives a functor
$$
\mu:D^b(k_M)\to D^b(k_{T^*M}).
$$

Denoting $q_1,q_2:M\times M\to M$ the natural projection, we defined the microlocal homomorphisms $\mu hom(F,G)$ between two complexes of sheaves $F$ on $G$ on $X$ by
$$\mu hom(F,G):=\mu_{\Delta_M}\mathbb{R} Hom(q_2^{-1} F,q_1^{!}G).$$

If $\pi:T^*_{\Delta_M}(M\times M)\to M$ is the natural projection, we have
$$\pi_*\mu hom(F,G)\cong \mathbb{R}Hom(F,G).$$

## Related subjects

[[index theory]]

[[microlocal formulation of index theory]]

[[global analytic index theory]]

[[derived microlocalization]]

[[algebraic microlocalization]]

## References

Sato's theory of microlocalization was first described in the setting of [[D-modules]]:

* [[M. Kashiwara]], Kawai, Kimura: foundations of algebraic analysis.

It was then extended to a purely sheaf theoretical theory in

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Sheaves on manifolds_, Grundlehren **292**, Springer (1990) [[doi:10.1007/978-3-662-02661-8](https://doi.org/10.1007/978-3-662-02661-8)]

This theory of microlocalization of (ind)-sheaves (and also sub-analytic sheaves) was developped in the following works:

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Ind-sheaves, distributions and microlocalization_, describes the program.
* [[Masaki Kashiwara]], [[Pierre Schapira]], [[Florian Ivorra]], [[Ingo Waschkies]] _Microlocalization of ind-sheaves_, gives the main results
and proofs.
* [[Masaki Kashiwara]], [[Pierre Schapira]] _Ind-sheaves_, SMF, gives a complete account of the theory.
* [[Luca Prelli]], _Microlocalization of sub-analytic sheaves_, gives the theory in the sub-analytic setting.

A good overview of the theory can by found at:

* [[Pierre Schapira]] Derived categories for the analyst (2010)


micrometer

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+-- {: .hide}
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=--
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=--

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* table of contents
{:toc}

## Idea

The _micrometer_ or _micron_ is a [[physical unit]] of [[length]].

## Definition

A **micrometer** (abbreviated $\mu m$) is one $10^{6}$th of a [[meter]]:

$$
  \mu m \coloneqq 10^{-6} m
  \,.
$$

The [[scale]] between about a [[nanometer]] and a micrometer is the scale of [[mesoscopic physics]].

## Related concepts

* [[meter]]

* [[nanometer]]

* [[femtometer]]

* [[Planck length]]

## References

See also

* Wikipedia, _[Micrometre](https://en.wikipedia.org/wiki/Micrometre)_

[[!redirects micrometers]]

[[!redirects micrometre]]
[[!redirects micrometres]]

[[!redirects micron]]
[[!redirects microns]]


midpoint algebra

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The idea of a midpoint algebra comes from [[Peter Freyd]].

## Definition

A __midpoint algebra__ is a [[magma]] $(M,\vert)$ that is commutative, idempotent, and medial:

* for all $a$ in $M$, $a \vert a = a$

* for all $a$ and $b$ in $M$, $a \vert b = b \vert a$

* for all $a$, $b$, $c$, and $d$ in $M$, $(a \vert b) \vert (c \vert d) = (a \vert c) \vert (b \vert d)$

## Properties

The [[currying]] of the midpoint operation $\vert$ results in the __contraction__ $(-)\vert : M \to (M \to M)$. Contractions are midpoint [[homomorphisms]]: for all $a$, $b$, and $c$ in $M$, $(a \vert) (b \vert c) = ((a \vert) b) \vert ((a \vert) c)$.

## Examples

The [[rational numbers]], [[real numbers]], and the [[complex numbers]] with $a \vert b \coloneqq \frac{a + b}{2}$ are examples of midpoint algebras.

The [[trivial group]] with $a \vert b = a \cdot b$ is a midpoint algebra.

## Related concepts

* [[commutative magma]]

* [[cancellative midpoint algebra]]

* [[symmetric midpoint algebra]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

* [[Martín Escardó]] and [[Alex Simpson]]. *A universal characterization of the closed Euclidean interval.* In 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachusetts, USA, June 16-19, 2001, Proceedings, pages 115–125. IEEE Computer Society, 2001. ([doi:10.1109/LICS.2001.932488](https://doi.org/10.1109/LICS.2001.932488), [pdf](http://www.cs.bham.ac.uk/~mhe/papers/interval.pdf))


Miguel A. Vázquez-Mozo

* [personal page](https://diarium.usal.es/vazquez/)

## Selected writings

On [[quantum anomalies]] and the [[Green-Schwarz mechanism]]:

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Anomalies and the Green-Schwarz Mechanism*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.06467](https://arxiv.org/abs/2211.06467)]

On ([[effective field theory]]) [[quantum field theory]] and the [[standard model of particle physics]]:

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Lectures on Field Theory and the Standard Model: A Symmetry-Oriented Approach* [[arXiv:2306.08097](https://arxiv.org/abs/2306.08097)]

category: people


Miguel Montero

* [Institute page](https://hetg.physics.harvard.edu/people/miguel-montero)

## Selected writings

On the [[swampland cobordism conjecture]]:

* [[Miguel Montero]], [[Cumrun Vafa]], _Cobordism Conjecture, Anomalies, and the String Lamppost Principle_, J. High Energ. Phys. 2021, 63 (2021) ([arXiv:2008.11729](https://arxiv.org/abs/2008.11729))

category: people


Miguel Sánchez

* [InSpire page](https://inspirehep.net/authors/990460)

* [arXiv entries](https://arxiv.org/a/sanchez_m_2.html)

## Selected writings

On [[spacetime]] [[causality]], [[globally hyperbolic spacetimes]] and [[Cauchy surfaces]]:

* {#MinguzziSánchez} E. Minguzzi, [[Miguel Sánchez]], *The causal hierarchy of spacetimes*, in: *Recent Developments in Pseudo-Riemannian Geometry*, EMS ESI Lectures in Mathematics and Physics **4** (2008) 299-358 [[arXiv:gr-qc/0609119](https://arxiv.org/abs/gr-qc/0609119), [ISBN:978-3-03719-051-7](https://bookstore.ams.org/view?ProductCode=EMSESILEC/4)]

* [[Antonio N. Bernal]], [[Miguel Sánchez]], _On smooth Cauchy hypersurfaces and Geroch's splitting theorem_, Commun. Math. Phys. **243** (2003) 461-470 [[arXiv:gr-qc/0306108v2](http://arxiv.org/abs/gr-qc/0306108), [doi:10.1007/s00220-003-0982-6](https://doi.org/10.1007/s00220-003-0982-6)]

* {#BernalSánchez05} [[Antonio N. Bernal]], [[Miguel Sánchez]], *Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes*, Commun. Math. Phys. **257** (2005) 43-50 [[arXiv:gr-qc/0401112](https://arxiv.org/abs/gr-qc/0401112), [doi:10.1007/s00220-005-1346-1](https://doi.org/10.1007/s00220-005-1346-1)]

* [[Antonio N. Bernal]], [[Miguel Sánchez]], *Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions*, Lett. Math. Phys. **77** (2006) 183-197 [[arXiv:gr-qc/0512095](https://arxiv.org/abs/gr-qc/0512095), [doi:10.1007/s11005-006-0091-5](https://doi.org/10.1007/s11005-006-0091-5)]

Review:

* [[Miguel Sánchez]], Globally hyperbolic spacetimes: slicings, boundaries and counterexamples*, Gen Relativ Gravit **54** 124 (2022) [[arXiv:2110.13672](https://arxiv.org/abs/2110.13672), [doi:10.1007/s10714-022-03002-6](https://doi.org/10.1007/s10714-022-03002-6)]

[[!redirects Miguel Sanchez]]

[[!redirects Miguel Sánchez-Caja]]
[[!redirects Miguel Sanchez-Caja]]


Mihhail Aizatulin

* [acm profile](https://dl.acm.org/profile/81464671169)

## Selected writings

On [[verified software]] for [[cryptography]]:

* [[Mihhail Aizatulin]], Franccois Dupressoir, Andrew D. Gordon, Jan Jürjens, *Verifying Cryptographic Code in C: Some Experience and the Csec Challenge* ([pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/MSR-TR-2011-118.pdf))

* [[Mihhail Aizatulin]], Andy Gordon, Jan Jürjens, *Extracting and Verifying Cryptographic Models from C Protocol Code by Symbolic Execution*, CCS '11 Proceedings of the 18th ACM Conference on Computer and Communications Security 2011 ([pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/2016/11/extracting.pdf))

* [[Mihhail Aizatulin]], *Verifying Cryptographic Security Implementations in C Using Automated Model Extraction* ([arXiv:2001.00806](https://arxiv.org/abs/2001.00806))

category: people


Mike Lieberman

* [personal page](http://www.math.muni.cz/~lieberman)

* [GoogleScholar page](https://scholar.google.com/citations?user=zl0rVzAAAAAJ&hl=th)

## Selected writings

On [[formal category theory]]:

* [[Ivan Di Liberti]], [[Simon Henry]], [[Mike Lieberman]], [[Fosco Loregian]], _Formal category theory_, course notes (2017) [[pdf](https://tetrapharmakon.github.io/stuff/course_muni-formal.pdf), [[DLHLL-FormalCategoryTheory.pdf:file]]]

category: people


Mike Pierce

Hi! You can find my website at [math.ucr.edu/~mpierce](http://math.ucr.edu/~mpierce/). Most of [my edits on the nLab](https://ncatlab.org/nlab/author/Mike%20Pierce) will probably be just adding links to external papers. :)


Mike Prest

__Mike Prest__ is a mathematician working at Manchester University. He especially works in representation theory and in [[model theory]] and particularly its applications in algebra.

He is one of the two (the other is Ted Voronov) London Mathematical Society regional organisers for the North of England / North Wales region.

* [homepage](http://www.maths.manchester.ac.uk/~mprest/)

Related entries include [[Ziegler spectrum]].

category: people


Mike Shulman

Mike Shulman is an Associate Professor at the [University of San Diego](http://www.sandiego.edu).  His personal web page is [here](http://home.sandiego.edu/~shulman/).

## Selected writings

On [[higher inductive types]]:

* {#LumsdaineShulman17} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], *Semantics of higher inductive types*, Math. Proc. Camb. Phil. Soc. **169** (2020) 159-208 [[arXiv:1705.07088](https://arxiv.org/abs/1705.07088), talk slides [pdf](http://home.sandiego.edu/~shulman/papers/cellcxs.pdf), [doi:10.1017/S030500411900015X](https://doi.org/10.1017/S030500411900015X)]

On [[modal type theory]]:

* [[Mike Shulman]], _Semantics of multimodal adjoint type theory_ ([arXiv:2303.02572](https://arxiv.org/abs/2303.02572))

On [[cohesive homotopy type theory]]:

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

* [[Mike Shulman]], *Homotopy type theory: the logic of space*, New Spaces in Mathematics: Formal and Conceptual Reflections, ed. Gabriel Catren and Mathieu Anel, Cambridge University Press, 2021 ([arXiv:1703.03007](https://arxiv.org/abs/1703.03007), [doi:10.1017/9781108854429](https://doi.org/10.1017/9781108854429))

On [[higher observational type theory]]:

* [[Thorsten Altenkirch]], [[Yorgo Chamoun]], [[Ambrus Kaposi]], [[Michael Shulman]], *Internal parametricity, without an interval* ([arXiv:2307.06448](https://arxiv.org/abs/2307.06448))

On [[displayed type theory]]:

* [[Astra Kolomatskaia]], [[Michael Shulman]], *Displayed Type Theory and Semi-Simplicial Types* ([arXiv:2311.18781](https://arxiv.org/abs/2311.18781))

On [[set theory]]:

* {#Shulman18} [[Michael Shulman]], _Comparing material and structural set theories_ ([arXiv:1808.05204](https://arxiv.org/abs/1808.05204))

## Talks

On [[higher observational type theory]]:

* [[Michael Shulman]], *Towards a Third-Generation HOTT:*, talk at *[Homotopy Type Theory at CMU](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (2022)  [part 1: [slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-04-28.pdf), [video](https://www.youtube.com/watch?v=FrxkVzItMzA); part 2: [slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-05-05.pdf), [video](https://www.youtube.com/watch?v=5ciDNfmvMdU); part 3: [slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-05-12.pdf), [video](https://www.youtube.com/watch?v=9pDddxB7LbE)]

category: people

[[!redirects Mike Shulman]]
[[!redirects MikeShulman]]
[[!redirects Michael Shulman]]
[[!redirects Michael A Shulman]]
[[!redirects Michael A. Shulman]]


Mike Stay

I was one of [John](John+Baez)'s students.  My [homepage](http://math.ucr.edu/~mike) and [blog](http://reperiendi.wordpress.com).

category: people

[[!redirects MikeStay]]


MikeShulman > history

< [[MikeShulman]]
[[!redirects MikeShulman -- history]]


Mikhail Bondarko


Mikhail Gromov

__Mikhail Gromov__ (Михаил Леонидович Громов) is a Russian-French mathematician, a Professor at IHES, with deep contributions to many areas of mathematics.

* [ihes: Gromov](http://www.ihes.fr/~gromov)

* wikipedia: [English](http://en.wikipedia.org/wiki/Mikhail_Gromov_%28mathematician%29); [Russian](http://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%BE%D0%BC%D0%BE%D0%B2,_%D0%9C%D0%B8%D1%85%D0%B0%D0%B8%D0%BB_%D0%9B%D0%B5%D0%BE%D0%BD%D0%B8%D0%B4%D0%BE%D0%B2%D0%B8%D1%87), [French](http://fr.wikipedia.org/wiki/Mikha%C3%AFl_Gromov)

* math [genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=14999)

## Selected writings

On the [[h-principle]] and [[microflexible sheaves]]:

* [[Mikhail Gromov]], _Partial differential relations_, Ergebn. Math. Grenzgeb. (3), 9, Springer (1986) ([doi:10.1007/978-3-662-02267-2](https://link.springer.com/book/10.1007/978-3-662-02267-2))

* [[Mikhail Gromov]], _A topological technique for the construction of solutions of differential equations and inequalities_, Proc. Int. Congress Math. Nice 1970, vol. 2, 221-225 ([djvu w OCR](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0221.0226.ocr.djvu), [pdf](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0221.0226.ocr.pdf))

* [[Mikhail Gromov]], Papers on h-principle: geometric methods for solving partial differential equations/inequalities and the homotopy structure in the spaces of their solutions, [list](http://www.ihes.fr/~gromov/topics/topic2.html) (with some links)

On the [[Oka principle]]:

* [[Mikhail Gromov]], _Oka's principle for holomorphic sections of elliptic bundles_, J. Amer. Math. Soc. __2__ (1989), 851–-897 ([doi:10.2307/1990897](https://doi.org/10.2307/1990897))

See also:

* _Pseudo holomorphic curves in symplectic manifolds_, Inventiones Mathematicae __82__, n. 2 (1985) 307–347.

Gromov also researches connections between [[mathematics]], [[biology]], and sees [[n-categories]] as the structures in mathematics closest to what is going on in the mind. see:

* [Ergologic and Interfaces Between Languages](https://www.youtube.com/watch?v=jRSqGYJQQM8)
* [Structures, Learning and Ergosystems: Chapters 1-4, 6](http://www.ihes.fr/~gromov/PDF/ergobrain.pdf)

## Related $n$Lab entries

* [[symplectic topology]]

* [[Gromov-Witten invariant]]

category: people

[[!redirects Mikhail Gromov]]
[[!redirects Mikhael Gromov]]
[[!redirects Mikhail Leonidovich Gromov]]
[[!redirects Михаил Громов]]
[[!redirects Михаил Леонидович Громов]]


Mikhail Kapranov

Mikhail Kapranov is a professor of mathematics at Kavli IPMU in Tokyo.

* [old Yale website](http://www.math.yale.edu/~mk486/); [old Toronto website](http://www.math.toronto.edu/kapranov/)

* A list of publications [html](http://www.math.yale.edu/~mk486/publ.html)

* lecture at msri 2000 _Noncommutative neighborhoods and noncommutative Fourier transform_: [link](http://msri.org/publications/ln/msri/2000/interact/kapranov/1/index.html)

## Selected writings

Introducing [[Serre functors]]:

* {#BondalKapranov90} [[Alexei I. Bondal]], [[Mikhail M. Kapranov]], _Representable functors, Serre functors, and mutations_, Mathematics of the USSR-Izvestiya **35** 3 (1990) 519-541 [[doi:10.1070/IM1990v035n03ABEH000716](https://doi.org/10.1070/IM1990v035n03ABEH000716)]

On [[Rozansky-Witten invariants]]:

* {#Kapranov99} [[Mikhail Kapranov]], _Rozansky–Witten invariants via Atiyah classes_,  Compositio Math.  __115__ (1999), no. 1, 71--113 ([MR2000h:57056](http://www.ams.org/mathscinet-getitem?mr=2000h:57056), [doi](http://dx.doi.org/10.1023/A:1000664527238), [alg-geom/9704009](http://arxiv.org/abs/alg-geom/9704009))

Discussion of [[superalgebra]] as induced from free groupal symmetric monoidal categories ([[abelian 2-groups]]) and hence ultimately from the [[sphere spectrum]] (cf. *[[super 2-algebra]]* and *[[spectral superscheme]]*):

* {#Kapranov13} [[Mikhail Kapranov]], _Categorification of supersymmetry and stable homotopy groups of spheres_, talk at _[Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)](https://web.archive.org/web/20130617191515/http://mathserver.neu.edu/~bwebster/ACRT/)_ (April 2013) [[[Webster-ACRTAbstracts2013.pdf:file]],  video:[YT](https://www.youtube.com/watch?v=WBsZrcYOrxI)]

  > **Abstract:**. The "minimal sign skeleton" necessary to formulate the [[signs in supergeometry|Koszul sign rule]] is a certain [[Picard category]], a [[symmetric monoidal category]] with all [[invertible object|objects]] and [[invertible morphism|morphisms invertible]]. It can be seen as the free Picard category generated by one object and corresponds, by [[Grothendieck]]'s dictionary, to the [[n-truncation|truncation]] of the [[sphere spectrum|spherical spectrum]] $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the [[first stable homotopy group of spheres]] $\pi_{n+1}(S^n)$. This suggest a "[[higher structures|higher]]" or [[categorification|categorified]] versions of [[super-algebra|super-mathematics]] which utilize deeper structure of [[sphere spectrum|$S$]]. The first concept on this path is that of a supersymmetric monoidal category which is [[categorification|categorified]] version of the concept of a [[supercommutative algebra]].

* {#Kapranov15} [[Mikhail Kapranov]], _Supergeometry in mathematics and physics_, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) _[[New Spaces for Mathematics and Physics]]_ ([arXiv:1512.07042](http://arxiv.org/abs/1512.07042))

* {#Kapranov15b} [[Mikhail Kapranov]], _Super-geometry_, talk at _[New Spaces for Mathematics & Physics](http://ercpqg-espace.sciencesconf.org/program)_, IHP Paris, Oct-Sept 2015 ([video recording](https://www.youtube.com/watch?v=bjsNwKYT8JE))

## Related entries

* [[Kapranov's noncommutative geometry]]

* [[Kapranov-Voevodsky 2-vector spaces]]

* [[superalgebra]]

* [[higher parallel transport]]

* [[Higher Kac-Moody algebras]]

category: people

[[!redirects Mikhail Kapranov]]
[[!redirects Kapranov]]
[[!redirects Михайл Капранов]]
[[!redirects Капранов]]
[[!redirects M. Kapranov]]

[[!redirects Mikhail M. Kapranov]]


Mikhail Katz

* [Webpage](http://u.cs.biu.ac.il/~katzmik/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Mikhail_Katz)

## Selected writings

On [[Cayley 4-forms]]:

* [[Mikhail Katz]], [[Steven Shnider]], _Cayley 4-form, comass, and triality isomorphisms_ ([arXiv:0801.0283](http://arxiv.org/abs/0801.0283))

On [[Gottfried Leibniz]]'s [[infinitesimals]]:

* {#KatzSherry13} [[Mikhail Katz]], David Sherry, _Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond_, D. Erkenn **78** (2013) 571 [[arXiv:1205.0174](https://arxiv.org/abs/1205.0174), [doi:10.1007/s10670-012-9370-y](https://link.springer.com/article/10.1007/s10670-012-9370-y)]

and on the history of [[nonstandard analysis]]:

* Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, [[Mikhail Katz]], [[Semen S. Kutateladze]], David Sherry, *Toward a history of mathematics focused on procedures*, Foundations of Science **22** (2017) 763–783 [[arxiv/1609.04531](http://arxiv.org/abs/1609.04531), [doi:10.1007/s10699-016-9498-3](https://doi.org/10.1007/s10699-016-9498-3)]

## Related entries

* [[Cayley 4-form]]

* [[nonstandard analysis]]

* [[infinitesimal number]]

category: people


Mikhail Khovanov

* [website](http://www.math.columbia.edu/~khovanov/)

## Selected writings

Introducing [[Khovanov homology]]:

* [[Mikhail Khovanov]], *A categorification of the Jones polynomial*, Duke Math. J. __101__ (2000) 359-426 [[arXiv:math/9908171](https://arxiv.org/abs/math/9908171), [doi:10.1215/S0012-7094-00-10131-7](https://projecteuclid.org/journals/duke-mathematical-journal/volume-101/issue-3/A-categorification-of-the-Jones-polynomial/10.1215/S0012-7094-00-10131-7.full), MR1740682 (2002j:57025)]

lecture notes (and on [[categorification]] generally):

* [[Mikhail Khovanov]] (notes by [[You Qi]]), *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]

Review:

* [[Mikhail Khovanov]], Robert Lipshitz, _Categorical lifting of the Jones polynomial: a survey_, Bulletin (New Series) of the American Mathematical Society (2022) [[doi:10.1090/bull/1772](https://doi.org/10.1090/bull/1772)]

On [[braid cobordism]] and [[triangulated categories]]:

* {#KhovanovThomas07} [[Mikhail Khovanov]], [[Richard Thomas]], *Braid cobordisms, triangulated categories, and flag varieties*, Homology, Homotopy and Applications **9**(2007) 19-94 [[arXiv:math/0609335](https://arxiv.org/abs/math/0609335), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-9/issue-2/Braid-cobordisms-triangulated-categories-and-flag-varieties/hha/1201127331.full)]

Introducing and discussing a [[cobordism category]] of "foams" for studying [[link homology]]:

* [[Mikhail Khovanov]], *$sl(3)$ link homology*, Algebr. Geom. Topol. **4** (2004) 1045-1081 [[arXiv:math/0304375](https://arxiv.org/abs/math/0304375), [doi:10.2140/agt.2004.4.1045](https://doi.org/10.2140/agt.2004.4.1045)]

* [[Mikhail Khovanov]], [[Louis-Hadrien Robert]], *Foam evaluation and Kronheimer--Mrowka theories*, Advances in Mathematics **376**  (2021) 107433 [[arXiv:1808.09662](https://arxiv.org/abs/1808.09662), [doi:10.1016/j.aim.2020.107433](https://doi.org/10.1016/j.aim.2020.107433)]

* [[Mikhail Khovanov]], [[Louis-Hadrien Robert]], *Conical $SL(3)$ foams*, Journal of Combinatorial Algebra **6** 1/2 (2022) 79-108 [[arXiv:2011.11077](https://arxiv.org/abs/2011.11077), [doi:10.4171/jca/61](https://doi.org/10.4171/jca/61)]

* [[Mikhail Khovanov]], *Universal construction, foams and link homology*, lecture series at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#KhovanovMar2023), video 1:[YT](https://www.youtube.com/watch?v=Urh1NNcbCcc)]

Discussion of [[non-deterministic automata]] as 1-dimensional [[defect QFT|defect]] [[TQFTs]]:

* Paul Gustafson, [[Mee Seong Im]], Remy Kaldawy, [[Mikhail Khovanov]], Zachary Lihn, *Automata and one-dimensional TQFTs with defects* [[arXiv:2301.00700](https://arxiv.org/abs/2301.00700)]

## Related entries

* [[Khovanov homology]]

category: people


Mikhail Movshev

* [webpage](http://www.math.sunysb.edu/~mmovshev/)

## Selected writings

On the [[Lie algebra cohomology]] of the [[super Poincaré Lie algebra]] ([[brane scan]] of [[Green-Schwarz sigma-models]]):

* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries_ ([arXiv:1011.4731](http://arxiv.org/abs/1011.4731))

* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra_, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 ([arXiv:1106.0335](http://arxiv.org/abs/1106.0335))

## Related $n$Lab entries

* [[10d super Yang-Mills theory]], [[IKKT matrix model]]

category: people

[[!redirects Michael Movshev]]


Mikhail O. Katanaev

* [MathNet page](https://m.mathnet.ru/php/person.phtml?personid=17481&option_lang=eng)

* [InSpire page](https://inspirehep.net/authors/1882389)

* [GoogleScholar page](https://scholar.google.ru/citations?user=uQ8Gfj0AAAAJ&hl=en)

## Selected writings

On [[differential geometry]] in [[mathematical physics]]:

* [[Mikhail O. Katanaev]], *Geometrical methods in mathematical physics* (in Russian) [[arXiv1311.0733](https://arxiv.org/abs/1311.0733)]

category: people

[[!redirects Mikhail Katanaev]]


Mikhail Shaposhnikov

* [webpage](https://lppc.epfl.ch/Shaposhnikov)

## Selected writings

On [[baryogenesis]]:

* V. A. Rubakov, [[Mikhail Shaposhnikov]], *Electroweak Baryon Number Non-Conservation in the Early Universe and in High Energy Collisions*, Usp. Fiz. Nauk **166** (1996) 493-537 Phys. Usp. 39  (1996) 461-502 7lbrack;[arXiv:hep-ph/9603208](https://arxiv.org/abs/hep-ph/9603208)]

Interpretation of the apparent [[Higgs field]]-instability in terms of [[asymptotic safety]]:

* {#ShaposhnikovWetterich09} [[Mikhail Shaposhnikov]], [[Christof Wetterich]], _Asymptotic safety of gravity and the Higgs boson mass_, Phys. Lett. B 683 (2010) 196 [[arXiv:0912.0208](https://arxiv.org/abs/0912.0208)]

On (the non-existence of) the [[hierarchy problem]] and the [[naturalness]]-issue in view of properly understood [[renormalization]]:

* {#Mooij-Shaposhnikov} [[Sander Mooij]], [[Mikhail Shaposhnikov]], _QFT without infinities and hierarchy problem_ [[arXiv:2110.05175](https://arxiv.org/abs/2110.05175)]

  > The aim of this work has been to stress that there is no physical sense in the [[naturalness]] criterion.

## Related entries

* [[baryogenesis]]

category: people


Mikhail Shifman

* [institute page](https://cse.umn.edu/physics/mikhail-shifman)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Mikhail_Shifman)

* Interview of Mikhail Shifman by David Zierler on July 7, 2021, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA [[aip:oral-histories/47523](https://www.aip.org/history-programs/niels-bohr-library/oral-histories/47523)]

## Selected writings

On the [Chern-Simons level renormalization](Chern-Simons+level#LevelRenormalization) (shift by the [[dual Coxeter number]]):

* [[Mikhail A. Shifman]], *Four-dimension aspect of the perturbative renormalization in three-dimensional Chern-Simons theory*, Nuclear Physics B **352** 1 (1991) 87-112 (doi:10.1016/0550-3213(91)90130-P)

On [[superalgebra]], [[supergeometry]], [[supersymmetry]], [[supergravity]] and  [[superstrings]]:

* [[Mikhail Shifman]] (ed.)

  _[[The Many Faces of the Superworld]]_

  [[Yuri Golfand]] memorial volume

  World Scientific, 2000

  [doi:10.1142/4332](https://doi.org/10.1142/4332)

On the history of the concept of [[supersymmetry]]:

* {#KaneShifman00} [[Gordon Kane]], [[Mikhail Shifman]], _The Supersymmetric World: The Beginnings of The Theory_, World Scientific 2000 ([doi:10.1142/4611](https://doi.org/10.1142/4611))

  * {#KaneShifman01} [[Gordon Kane]], [[Mikhail Shifman]], _Introduction to "The Supersymmetric world"_ ([arXiv:hep-ph/0102298](https://arxiv.org/abs/hep-ph/0102298)), chapter in: [Kane-Shifman 00](#KaneShifman00)

On [[quark-hadron duality]]:

* [[Mikhail Shifman]], _Quark-Hadron Duality_, in: _At The Frontier of Particle Physics_, pp. 1447-1494 (2001) ([doi:10.1142/9789812810458_0032](https://doi.org/10.1142/9789812810458_0032))

* [[Mikhail Shifman]], _Lectures on quark-hadron duality_, Czech J Phys 52, B102–B135 (2002) ([doi:10.1007/s10582-002-0080-6](https://doi.org/10.1007/s10582-002-0080-6))

On [[quantum field theory]] and [[string theory]]:

* [[Mikhail Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.),

  _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_

  Ian Kogan Memorial Collection (In 3 Volumes)

  World Scientific 2005

  [doi:10.1142/5621](https://doi.org/10.1142/5621)

On [[Evgeny Likhtman]]:

* encyclopedia entry: **[[Likhtman_Evgeny_Pinkhasovich.pdf:file]]** (by [[Mikhail Shifman]])

On [[vortex strings]] in $\mathcal{N}=2$ super QCD identified as [[superstrings]] in [[type II string theory]] [[KK-compactification|compactified]] on a [[conifold]]:

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories*, Rev. Mod. Phys. **79** 1139 (2007) [[doi:10.1103/RevModPhys.79.1139](https://doi.org/10.1103/RevModPhys.79.1139), [arXiv:hep-th/0703267](https://arxiv.org/abs/hep-th/0703267)

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons*, Cambridge University Press (2009) [[doi:10.1017/CBO9780511575693](https://doi.org/10.1017/CBO9780511575693)]

* [[Mikhail Shifman]], [[A. Yung]], *Critical String from Non-Abelian Vortex in Four Dimensions*,
Physics Letters B **750** (2015) 416-419
[[arXiv:1502.00683](https://arxiv.org/abs/1502.00683), [doi:10.1016/j.physletb.2015.09.045](https://doi.org/10.1016/j.physletb.2015.09.045)
]

* P. Koroteev, [[Mikhail Shifman]], [[A. Yung]], *Studying Critical String Emerging from Non-Abelian Vortex in Four Dimensions*, Phys.Lett. B **759** (2016) 154-158
[[arXiv:1605.01472](https://arxiv.org/abs/1605.01472), [doi:10.1016/j.physletb.2016.05.075](https://doi.org/10.1016/j.physletb.2016.05.075)
]

On [[non-perturbative effects]] in [[QCD]] ([[hadrodynamics]]) via [[operator product expansion]]:

* [[Mikhail Shifman]], *OPE-based Methods in Nonperturbative QCD*, Eur. Phys. J. C (2022) [[arXiv:2208.10600](https://arxiv.org/abs/2208.10600)]

category: people

[[!redirects Mikhail A. Shifman]]

[[!redirects Misha Shifman]]


Mikhail Vasiliev

Theoretical physicist at [Lebedev Physical Institute](http://www.lebedev.ru/en/structure.html)

## related $n$Lab entries

* [[higher spin gauge theory]]

category: people


Mikhail Yakovlevich Suslin

* [Wikipedia entry](https://en.wikipedia.org/wiki/Mikhail_Yakovlevich_Suslin)

## related $n$Lab entries

* [[descriptive set theory]]

* [[interactions of images and pre-images with unions and intersections]]

[[!redirects Mikhail Suslin]]

category: people


Mikica Kocic

* [webpage](http://www.su.se/profiles/miko1432-1.227355)

## writings

* _Overview of the fields in QFT_, 2016 ([[FieldSpecies.pdf:file]])

  (the basic [[field (physics)|field species]] in [[quantum field theory]])

* _Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions_, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])

  (the [[propagators]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]], see also _[[propagators - table]]_)

## related $n$Lab entries

* [[field (physics)]]

* [[quantum propagator]], [[Klein-Gordon equation]]

category: people


Mikio Nakahara

* [webpage](https://www.mikio-nakahara.com/)

## Selected writings

On [[geometry]], [[topology]] and [[physics]] (as in [[fiber bundles in physics]]):

* [[Mikio Nakahara]], _[[Geometry, Topology and Physics]]_, IOP (2003) [[doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf]

category: people


Mikio Sato

**Mikio Sato** is a Japanese algebraic geometer and number theorist. He founded a subject called algebraic analysis, which he introduced along with the sheaf theoretic technique of [[hyperfunction]]s, fundamental aspects of [[microlocalization]]. Sato studied [[D-module]]s and especially [[holonomic D-module|holonomic systems]].

Much of his research is related to the study of singularities and related research in Hodge theory where he substantially extended the foundations. The similarities of that research to [[B. A. Dubrovin]]'s noton of Frobenius manifolds  has been further developed by Klaus Hertling. He applied some of the methods to mathematical physics ([[integrable systems]], solitons and wave equations...).

* interview in [Notices AMS](http://www.ams.org/notices/200702/fea-sato-2.pdf)

* [wikipedia](http://en.wikipedia.org/wiki/Mikio_Sato)

## Selected writings

On [[holonomic quantum fields]]:

* [[Tetsuji Miwa]], [[Michio Jimbo]], *Introduction to holonomic quantum fields*, pp. 28--36 in: _The Riemann problem, complete integrability and arithmetic applications_, Lec. Notes in Math. **925**, Springer (1982) $[$[doi:10.1007/BFb0093497](http://dx.doi.org/10.1007/BFb0093497)$]$

* {#JMSM05} [[Michio Jimbo]], [[Tetsuji Miwa]], [[Mikio Sato]], Yasuko Môri, *Holonomic Quantum Fields — The unanticipated link between deformation theory of differential equations and quantum fields —*,  In: [[Konrad Osterwalder|K. Osterwalder]] (ed.), *Mathematical Problems in Theoretical Physics*, Lecture Notes in Physics, **116** Springer (2005) $[$[doi:10.1007/3-540-09964-6_310](https://doi.org/10.1007/3-540-09964-6_310)$]$

category:people


Mikkel Nygaard

* [dblp page](https://dblp.org/pid/30/3432.html)

## Selected writings

On [[category theory|category theoretic]] generalizations of [[domain theory]]:

* [[Mikkel Nygaard]], [[Glynn Winskel]], *Domain theory for concurrency*, Theoretical Computer Science
**316** 1–3 (2004) 153-190 [[doi:10.1016/j.tcs.2004.01.029](https://doi.org/10.1016/j.tcs.2004.01.029), [pdf](https://www.cl.cam.ac.uk/~gw104/DomThy.pdf)]

category: people


Miklos Rédei

* [personal page](https://www.lse.ac.uk/cpnss/people/miklos-redei)

## Selected writings

On the history of [[John von Neumann]]'s reasoning on the foundations of [[quantum mechanics]] via [[quantum logic]] which made him turn from [[Hilbert spaces]] to [[von Neumann algebra factors]]:

* [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

category: people

[[!redirects Miklos Redei]]


Miklós Rédei

* [webpage](http://personal.lse.ac.uk/redeim/)

## related $n$Lab entries

* [[Schrödinger representation]]

* [[quantum probability]]

* [[conditional expectation]]

* [[wavefunction collapse]]

category: people

[[!redirects Miklos Redei]]


Milad Niqui

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=99730)

## Selected writings

On [[constructive analysis]] with exact [[real numbers]] via [[type theory]]:

* [[Herman Geuvers]], [[Milad Niqui]], [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Mathematical Structures in Computer Science **17** 01 (2007) 3-36 [[doi:10.1017/S0960129506005834](https://doi.org/10.1017/S0960129506005834)]

category: people


mildly context-sensitive grammar

+-- {: .rightHandSide}
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### Context
#### Linguistics
+-- {: .hide}
[[!include linguistics - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

Mildly context-sensitive grammar is a kind of formal grammar with an expressivity between that of [[context-free grammar]] and [[context-sensitive grammar]].
They have enough expressive power to capture natural languages, but not too much so that they are still parsable efficiently.

## Definition

While there has been no consensus on the definition of “mild”, there is an
equivalence [VW94] between several independent definitions, see [Weir (1988)](#Weir88).
Borrowing the definition from [Genkin et al. (2010)](#GenkinEtAl10), a grammar formalism is mildly context-sensitive if the languages it
generates are [[semilinearity|semilinear]], it is parsable in polynomial time, it contains all context-free languages as well as the following:

* multiple agreement $L1 = \{ a^n b^n c^n | a, b, c \in V, n \in \mathbb{N} \},$
* crossed dependency $L2 = \{ a^n b^m c^n d^m | a, b, c, d \in V, m, n \in \mathbb{N} \}$,
* duplication $L3 = \{w w | w \in V^\star\}$.

## References

* {#Shieber85} Shieber, Stuart M. _Evidence against the context-freeness of natural language._ Philosophy, Language, and Artificial Intelligence. Springer, Dordrecht, 1985.
* {#Weir88} Weir, David Jeremy. _Characterizing mildly context-sensitive grammar formalisms_, 1988.
* {#GenkinEtAl10} Daniel Genkin, Nissim Francez, and Michael Kaminski. _Mildly Context-Sensitive Languages via Buffer Augmented Pregroup Grammars_, In Essays in Memory of Amir Pnueli, 2010.


Mile Gu

* [institute page](https://www.quantumcomplexity.org/milegu/)

* [GoogleScholar page](https://scholar.google.com/citations?user=ZBDAby4AAAAJ&hl=en)

## Selected writings

On [[quantum circuits]] for [[quantum computation]] as ([[geodesic]], if optimal) [computational paths](computation#ReferencesConceptualizationViaPathLifting):

* {#NielsenDowlingGuDoherty06} [[Michael A. Nielsen]], [[Mark R. Dowling]], [[Mile Gu]], [[Andrew C. Doherty]], *Quantum Computation as Geometry*, Science, **311** 5764 (2006) 1133-1135 [[doi:10.1126/science.1121541](https://doi.org/10.1126/science.1121541), [arXiv:quant-ph/0603161](https://arxiv.org/abs/quant-ph/0603161)]

* [[Mark R. Dowling]], [[Michael A. Nielsen]], *The geometry of quantum computation*, Quantum Information & Computation **8** 10 (2008) 861–899 [[doi:10.5555/2016985.2016986](https://dl.acm.org/doi/abs/10.5555/2016985.2016986), [arXiv:quant-ph/0701004](https://arxiv.org/abs/quant-ph/0701004)]

category: people


Miles Blencowe

* [website](https://home.dartmouth.edu/faculty-directory/miles-p-blencowe)

category: people


Miles P. Blencowe

* [Institute page](https://faculty-directory.dartmouth.edu/miles-p-blencowe)

* [Spire page](https://inspirehep.net/authors/1016160)

## Selected writings

On the emergence of [[conformal field theory]] in the perturbations of the [[super-membrane]] around a classical solution stretched along the [[asymptotic boundary]] of [[anti de Sitter spacetime]], predating the modern formulation of the [[AdS-CFT correspondence]], and the corresponding [[Green-Schwarz sigma-model|brane scan]] of [[super-conformal algebra|super-conformal]] branes:

* [[Miles P. Blencowe]], [[Mike Duff]], _Supersingletons_, Physics letters B, **203** 3(1988) 229-236 ([cds:184143](http://cds.cern.ch/record/184143), doi:10.1016/0370-2693(88)90544-8)

category: people

[[!redirects Miles Blencowe]]


Miles Reid

* [webpage](http://homepages.warwick.ac.uk/staff/Miles.Reid/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Miles_Reid)

## related $n$Lab entries

* [[ADE singularity]]

category: people


Milky Way

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#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Our [[galaxy]].

>  Democritus $[$holds$]$ that it is the splendor which ariseth from the coalition of many small stars, which, being firmly united amongst themselves, do mutually enlighten one another.

> Δημόκριτος πολλῶν καὶ μικρῶν καὶ συνεχῶν ἀστέρων συμφωτιζομένων ἀλλήλοις συναυγασμὸν διὰ τὴν πύκνωσιν.

> ([Pseudo Plutarch](#PseudoPlutarch) ~ 400 AD, attributing to [[Democritus]] $\sim$ 400 BC -- original Greek [here](http://articles.adsabs.harvard.edu/pdf/1901Obs....24..382L#page=2))

> $\,$

> The Milky Way is made up of a very large number of small, tightly-clustered stars, which, on account of their concentration and smallness, seem to be cloudy patches. Because of this, it was likened to milk in color.

> ([[Naṣīr al-Dīn al-Ṭūsī]] in *Tadhkira* 1261 AD, as translated in [Ragep 1993, p. 128](#Ragep93))

> $\,$

> the Galaxy [Milky Way] is nothing else than a congeries of innumerable stars distributed in clusters. To whatever region of it you direct your spyglass, an immense number of stars immediately offer themselves to view, of which very many appear rather large and very conspicuous but the multitude of small ones is truly unfathomable.

> ([[Galileo Galilei]] in _Sidereus Nuncius_ 1610, as translated in [Van Helder 1989](#vanHelder), [p. 34](http://people.reed.edu/~wieting/mathematics537/SideriusNuncius.pdf#page=35))

\linebreak

## Related concepts

* [[astronomy]], [[galaxy]]

* [[cosmology]], [[observable universe]]

## References

In antiquity and the middle ages:

* {#PseudoPlutarch} [Pseudo-Plutarch](https://en.wikipedia.org/wiki/Pseudo-Plutarch), *[Of the Galaxy, or the Milky Way](http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0404%3Abook%3D3%3Achapter%3D1)*, Book 3, Chapter I in:  *[Placita Philosophorum](http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2008.01.0404)*, ~300-400 AD

* {#Ragep93} F. J. Ragep, II.4 $[10]$ (p. 128) in: *[[Naṣīr al-Dīn al-Ṭūsī]]’s Memoir on Astronomy (al-Tadhkira fī cilm al-hay’a): Volume I: Introduction, Edition, and Translation*, Sources in the History of Mathematics and Physical Sciences 12, Springer 1993 ([doi:10.1007/978-1-4757-2241-3](https://link.springer.com/book/10.1007/978-1-4757-2241-3))

* {#vanHelder} Albert Van Helden, _Sidereal Messenger_, translation of: [[Galileo Galilei]], _Sidereus Nuncius_ 1610, University of Chicago Press 1989 ([pdf](http://people.reed.edu/~wieting/mathematics537/SideriusNuncius.pdf#page=35))

Modern survey

* Susan Gardner, Samuel D. McDermott, Brian Yanny, *The Milky Way, Coming into Focus: Precision Astrometry Probes its Evolution, and its Dark Matter* ([arXiv:2106.13284](https://arxiv.org/abs/2106.13284))

See also

* Wikipedia, _[Milky Way](https://en.wikipedia.org/wiki/Milky_Way)_


Milnor algebra

## Idea

Milnor algebra $M(f)$ of an analytic function $f$ has a role of describing a germ of singularity of a function modulo the ideal generated by all partial derivatives of the functions.

## Literature

Related notions: [[Tjurina algebra]], [[singularity]], [[hypersurface singularity]], [[Jacobian algebra]]

* Stephen ST Yau, _Milnor algebras and equivalence relations among holomorphic functions_ Bull. Amer. Math. Soc. 9 (1983) 235-239 [doi](https://doi.org/10.1090/S0273-0979-1983-15168-6)
* wikipedia: [Milnor number](https://en.wikipedia.org/wiki/Milnor_number)

There is also a related "Lie algebra of a singularity"(introduced by S S T Yau). See

* A. Elashvili and G. Khimshiashvi, _Algebras of simple hypersurface singularities_, Journal of Lie Theory __16__ (2006) 621--649 [pdf](https://www.heldermann-verlag.de/jlt/jlt16/elala2e.pdf)


Milnor construction

+-- {: .rightHandSide}
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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a ([[Hausdorff topological space|Hausdorff]]) [[compact topological group]] $G$, the _Milnor construction_ of the [[universal principal bundle]] for $G$ (also known as _Milnor's join construction_) constructs the [[join of topological spaces|join]] of infinitely many copies of $ G $, i.e., the colimit of [[join of topological spaces|joins]]

$$
  (E G)_{Milnor}
  \;\coloneqq\;
  \underset
    {\underset{n \in \mathbb{N}}{\longrightarrow}}
    {\lim}
  \;
  \big(
   \underset
     {n \; factors}
     {\underbrace{G \ast G \ast \ldots \ast G}}
  \big)
$$

and canonically equips it with a continuous and [[free action|free]] right [[group action]] of $ G $ that yields the structure of a [[G-CW-complex]]. Consequently, the natural [[quotient space]] projection $(E G)_{Milnor} \to (E G)_{Milnor}/G $ is a model for the [[universal principal bundle]] $ E G \to B G $ of locally trivial principal $ G $-bundles over [[paracompact Hausdorff spaces]], or equivalently, of [[numerable bundle]] $G$-[[principal bundles]] over all [[Hausdorff topological spaces]].

There is a generalisation of Milnor's construction that works for [[topological groupoids]], and reduces to the above case when the groupoid is $\mathbf{B}G$, the delooping of the group $G$.

## Related entries

* [[classifying space]]

* [[universal principal bundle]]

* [[bar construction]]

## References

* [[John Milnor]], *Construction of Universal Bundles I*, Ann. of Math. __63__ 2 (1956) 272-284 [[jstor:1969609](http://www.jstor.org/stable/1969609)]

* [[John Milnor]], _Construction of Universal Bundles II_, Ann. of Math. __63__ 3 (1956) 430-436 [[jstor:1970012](http://www.jstor.org/stable/1970012)]

  reprinted in: *Collected Works of John Milnor* [[gBooks](http://books.google.com/books?id=WyZeeEn8VFwC)]

* [[John Milnor]], [[James Stasheff]], _Characteristic Classes_, Princeton University Press.

* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]], _The Milnor Construction: Homotopy Classification of Principal Bundles_, [doi](http://dx.doi.org/10.1007/978-3-540-74956-1_8), in:
Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, Vol. 726 (2008) 75-81.

[[!redirects Milnor's construction]]

[[!redirects Milnor's classifying space]]

[[!redirects Milnor classifying space]]
[[!redirects Milnor classifying spaces]]


Milnor fiber

[[!redirects Milnor fibration]]
[[!redirects Milnor map]]
Milnor fiber, Milnor map and Milnor fibration

For a hypersurface in $\mathbf{C}^{n+1}$ given by an equation $f = 0$ for some $f\in\mathbf{C}[[z_0,z_1,\ldots,z_n]]$ with an isolated  singularity at zero...

* John Milnor, _Singular points of complex hypersurfaces_, Annals of Mathematics Studies __61__, Princeton Univ. Press 1968 iii+122 pp. [MR239612](https://www.ams.org/mathscinet-getitem?mr=239612)
* P. Seidel, _Fukaya categories and Picard-Lefschetz theory_, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp. [MR2009f:53143](https://www.ams.org/mathscinet-getitem?mr=2441780), [doi](https://doi.org/10.4171/063)
* Sec. I.1 in V. Kulikov, _Mixed Hodge structures and singularities_, Cambridge Univ. Press 1998, xxii+186 pp.
* Nero Budur, _Singularity invariants related to Milnor fibers: survey_, [arXiv:1012.3150](https://arxiv.org/abs/1012.3150)

Relation to [[local system]]s on the complement of a union of complex hyperplanes ([[arrangement of hyperplanes]])

* [[Daniel Cohen]], A. Suciu, _On Milnor ﬁbrations of arrangements_, J. London Math. Soc. 51 (1995) 105--119; MR96e:32034 [doi](https://doi.org/10.1112/jlms/51.1.105)

For meromorphic functions (instead of holomorphic)

* Arnaud Bodin, Anne Pichon, José Seade, _Milnor fibrations of meromorphic functions_, J. London Math. Soc. __51__:1 (1995) 105--119 [doi](https://doi.org/10.1112/jlms/jdp027)

There is also a [[rigid analytic geometry|rigid analytic]] version __analytic Milnor fiber__, which is a rigid variety over the [[nonarchimedean field]] of complex Laurent series associated to a complex hypersurface singularity. It has been used in the study of motivic zeta function via motivic integration on rigid varieties in

* J. Nicaise, J. Sebag, _Motivic Serre invariants, ramification, and the analytic Milnor fiber_, Invent. Math. 168:1, 133--173 (2007) [MR0239612](http://www.ams.org/mathscinet-getitem?mr=2285749)


Milnor K-theory

_The following text is copied from [MO/4246/4258](http://mathoverflow.net/questions/4246/why-is-milnor-k-theory-not-ad-hoc/4258#4258) by [[Denis-Charles Cisinski]]:_

**Milnor K-theory** gives a way to compute [[étale cohomology]] of [[fields]] (i.e. [[Galois cohomology]]): if E is a field of characteristic different from a prime l, there is a residue map from the nth Milnor K-group of E mod l to the nth étale cohomology group of E with coefficients in the sheaf of lth [[roots of unity]] to the n (i.e. tensored with itself n times). There is the [[Bloch-Kato conjecture]], which predicts that these residue maps are bijective. It happens that the case l=2 was conjectured by [[John Milnor|Milnor]] (up to a reformulation I guess). The [[Milnor conjecture]] has been proved by [[Vladimir Voevodsky|Voevodsky]] (and it was the first great achievements of [[homotopy theory of schemes]], which he initiated with [[Fabien Morel|Morel]] during the 90's), and he got his Fields medal in 2002 for this. Now [[Markus Rost|Rost]] and Voevodsky claimed they have a proof of the full Bloch-Kato conjecture for any prime l (which should appear some day, thanks to the work of quite a few people, among which [[Charles Weibel]] is not the least). Note also that the Bloch-Kato conjecture makes sense for l=p=char(E), but then, you have to replace étale cohomology by [[de Rham-Witt cohomology]] (and this has also been proved by Bloch and Kato). [[Andrei Suslin|Suslin]] and Voevodsky also proved that the Bloch-Kato conjecture implies the [[Beilinson-Lichtenbaum conjecture]], which predicts the precise relationship between torsion [[motivic cohomology]] of varieties with torsion [[étale cohomology]].

Milnor K-theory is related to [[motivic cohomology]] (i.e. [[higher Chow groups]]) in degree n and weight n H^n(X,Z(n)): for X=Spec(E), H^n(X,Z(n)) is the nth Milnor K-group. This is how [[homotopy theory of schemes]] enters in the picture (one of the main feature introduced by Voevodsky to study motivic cohomology with finite coefficients is the theory of [[motivic Steenrod operations]]). On the other hand, Rost studied Milnor K-theory for itself: among a lot of other things, he proved that, if you consider it as a functor from the category of fields, with all its extra structures (residue maps interacting well), you can reconstruct [[higher Chow groups]] of schemes (over a field), via some Gersten complex.

Milnor K-theory is also a crucial ingredient in Kato's [[higher class field theory]].

## See also

* [[Galois cohomology]]
* [[motivic cohomology]]
* [[Milnor conjecture]]
* [[Bloch-Kato conjecture]]
* [[motivic homotopy theory]]


Milnor mu-bar invariants

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* the following line creates the automatic table of contents
{: toc}

## Idea

In _[Link Groups](#jmLinkGroups)_ , [[John Milnor]] introduced the notion of the [[Link Group]] as a way to study [[links]].  The notion of equivalence of links that Milnor used is slightly different to that obtained by extending the usual notion of equivalence of [[knots]].  In Milnor's paper, the crucial aspect of links was the interactions between distinct components.  Thus for Milnor, a link in a manifold $M$ is a map $\coprod_n S^1 \to M$ such that the components have disjoint images.  Similarly, two links are **homotopic** if there is a [[homotopy]] between the maps which is a link at every time.  Thus links can be deformed in such a manner that individual components can pass through themselves, but not through other components.  Also link components can have self-intersections or the map on a component can be a constant map.  Milnor uses the term **proper link** to refer to a link in which the map is a homeomorphism onto its image.

The [[Whitehead link]] is a simple example of a link that is not trivial under ambient [[isotopy]] but is trivial under Milnor's notion of homotopy.

The $\mu$-invariants come from explicit descriptions of the link groups of particular links.  Specifically, Milnor calls a link _almost trivial_ if every proper sublink is trivial (see [[Brunnian link]]).  Such a link corresponds to an element in a particular link group which can be completely described by certain numbers.

## Link Group

Let us begin by describing the link group.  Milnor's alternative description is as follows.  Consider the complement of a link $L$ in an open $3$-manifold $M$.  We choose a [[basepoint]] in this complement and so have the [[fundamental group]].  We define a relation on this group as follows: two loops $\alpha$, $\beta$ are equivalent if the link $L \cup \alpha^{-1} \beta$ is homotopic in $M$ to one of the form $L' \cup 1$ (where $1$ is the constant loop at the basepoint).  The **link group** is the group of equivalence classes of such loops.

A more practical description is the following.

+-- {: .num_defn #linkgroup}
###### Definition
Let $L$ be a link in an open $3$-manifold $M$.
Let $G(L)$ be the fundamental group of the complement of $L$.  Let $L^i$ denote the sublink obtained by deleting the $i$th component of $L$.  Let $A_i(L)$ be the kernel of the natural inclusion $G(L) \to G(L^i)$ and $[A_i]$ its commutator subgroup.  Let $E(L) = [A_1] [A_2] \cdots [A_n]$.  This is a normal subgroup of $G(L)$.  The quotient, $\mathcal{G}(L) \coloneqq G(L)/E(L)$ is the **link group** of $L$.
=--

Milnor's first theorem on this group was to show that this group is an invariant of the homotopy class of the link, at least for proper links.

+-- {: .num_theorem #htyinv}
###### Theorem ([Milnor, Theorem 1](#jmLinkGroups)) ######
If two proper links are homotopic, then their link groups are isomorphic.
=--

To study this group for a particular link, we need to find some particular elements in it.  These are the **meridians** and the **parallels**.  Basically, a meridian goes around one component of the link once, in a specific direction, whilst a parallel goes along it.  Technically, the parallels of a link are not _elements_ of its link group, but cosets.

Choose a component $L_i$ of the link $L$.  Choose orientations of the ambient manifold, $M$, and of the circle.
To define the meridian and parallel of $L_i$ we need to choose a path from the basepoint, $x_0$, to a point on the image of $L_i$ which does not intersect the image of $L$ at any other time.  Let $p$ be such a path, so then $p(1)$ is a point on the image of $L_i$.

+-- {: .num_defn #meridian}
###### Definition
The **$i$th meridian** of $L$ is the element $\alpha_i \in \mathcal{G}(L)$ defined as follows.  Choose a small neighbourhood $N$ of $p(1)$.  Define a path by going along $p$ until we are inside $N$, then go around a closed loop in $N$ which has linking number $+1$ with the part of the image of $L_i$ inside $N$.  Then return to $x_0$ along $p$.
=--

+-- {: .num_defn #parallel}
The **$i$th parallel** of $L$ is the coset $\beta_i \mathcal{A}_i \in \mathcal{G}(L^i)$ defined as follows.  The subgroup $\mathcal{A}_i$ is the kernel of the homomorphism $\mathcal{G}(L) \to \mathcal{G}(L^i)$.  Go along $p$ to its end.  Then go around the image of $L_i$ according to the orientation of the circle.  Finally return to $x_0$ along $p$.  The preimage of this element defines a coset in $\mathcal{G}(L)$ which we write as $\beta_i \mathcal{A}_i$.
=--

The basic method of studying a link via link groups is to consider a link as an element of the link group of the link obtained by removing one of its components.  To show that this is a reasonable thing to do, Milnor proved the following theorem.

+-- {: .num_theorem #conjelts}
###### Theorem ([Milnor, Theorem 3](#jmLinkGroups)) ######
Let $L$ be a proper link with $n$ components.  Let $f$, $f'$ be closed loops in the complement of $L$.  If they represent conjugate elements of $\mathcal{G}(L)$ then the links $(L,f)$ and $(L,f')$ are homotopic.
=--

## $\mu$-Invariants

For [[Brunnian links]], which Milnor calls **almost trivial** links, the classification question reduces to looking at elements of the link group of trivial links.  It is important to note that the ambient space here is Euclidean space, $\mathbb{R}^3$.

Let $L$ be an $n$-component Brunnian link.  Then we consider the element $\beta'_n \in \mathcal{G}(L^n)$ corresponding to the $n$th parallel.  Upon removing a further component, say the $n-1$st, this element becomes trivial since we are then looking at $L^{n-1}$ which is trivial.  Thue $\beta'_n \in \mathcal{A}_{n-1}(L^n)$, the kernel of $\mathcal{G}(L^n) \to \mathcal{G}(L^{n-1,n})$ (here $L^{n-1,n}$ is $L$ with both the $n-1$st and $n$th components removed).  Now $\mathcal{A}_{n-1}(L^n)$ is the smallest normal subgroup containing the meridian $\alpha_{n-1}$ (since removing the $n-1$st component is the same thing as allowing the meridian $\alpha_{n-1}$ to collapse) and so every element of $\mathcal{A}_{n-1}$ can be written as a word in alphabet of powers and conjugates of $\alpha_{n-1}$.  Milnor uses the notation
$$
\alpha_{n-1}^\sigma, \quad \sigma \in \mathcal{R}(L^{n-1,n})
$$
to write this.  The exponent $\sigma$ itself decomposes as
\[
\label{muinvs}
\sigma = \sum \mu(i_1 \cdots i_{n-2}, n-1 n) k_{i_1} \cdots k_{i_{n-2}}
\]
where the summation is over all permutations $i_1 \cdots i_{n-2}$ of $1$, $2$, ..., $n-2$.

+-- {: .num_theorem #muinvs}
###### Theorem ([Milnor, Section 5](#jmLinkGroups))
The integers $\mu(i_1 \cdots i_{n-2}, n_1 n)$ are homotopy invariants of $L$.  The homotopy class of $L$ is completely specified by these integers.
=--

There was nothing special about the choice of components.  A similar procedure works for any pair of components.  The resulting integers obey the following rules:

\[
\begin{aligned}
\mu(i_1 i_2 \cdots i_{n-2}, i_{n-1} i_n) &= \mu(i_n i_1 i_2 \cdots i_{n-3}, i_{n-2} i_{n-2}) \\
\mu(i_1 \cdots i_{\nu} r j_1 \cdots j_{n-\nu-2} s) &= (-1)^{n-\nu} \sum \mu(h_1 \cdots h_{n-2} r s)
\end{aligned}
\]

In the second identity, the summation is over all shuffle products of $(i_1 \cdots i_{\nu})$ with $(j_1 \cdots j_{n - \nu - 2})$.

Let us expand on the definition of the $\mu$-invariants.  We start with the exponential notation.  The following holds for an arbitrary proper link, $L$, embedded in an open $3$-manifold $M$.

Let $J \mathcal{G}(L)$ be the integral group ring of $\mathcal{G}(L)$.  As mentioned above, any element of $\mathcal{A}_i(L)$ is a product of powers of conjugates of $\alpha_i$.  We can write such an element in the form $\alpha_i^s$ for $s \in J \mathcal{G}(L)$ by interpreting:
$$
\begin{aligned}
\alpha_i^{x + y} &= \alpha_i^x \alpha_i^y \\
\alpha_i^{k x} &= (\alpha_i^x)^k \\
\alpha_i^\beta &= \beta \alpha \beta^{-1}
\end{aligned}
$$
where $x, y \in J\mathcal{G}(L)$, $k \in \mathbb{Z}$, and $\beta \in \mathcal{G}(L)$.

We write the kernel of $J \mathcal{G}(L) \to J \mathcal{G}(L^i)$ as $\mathcal{K}_i(L)$.  Using these, we define:
$$
\mathcal{R}(L) \coloneqq J \mathcal{G}(L) / \mathcal{K}_1(L)^2 + \cdots + \mathcal{K}_n(L)^2
$$

Now the notation $\alpha_i^s$ for an element of $\mathcal{A}_i(L)$ does not provide an injective map from $J\mathcal{G}(L)$ to $\mathcal{A}_i(L)$.  The kernel is the ideal $\mathcal{K}_i(L) + (\mathcal{K}_1(L)^2 + \cdots + \mathcal{K}_n(L)^2)$ which is naturally isomorphic to $\mathcal{R}(L^i)$.

+-- {: .num_theorem #trivlink}
###### Theorem ([Milnor, Theorem 5](#jmLinkGroups))
Let $L$ be a link which is homotopic to one in with the $i$th component is constant.  Then every element of $\mathcal{A}_i(L)$ can be expressed uniquely in the form $\alpha_i^\sigma$ with $\sigma \in \mathcal{R}(L^i)$.
=--

Now let us suppose that $L$ is trivial.  Then $G(L)$ is the free product of the fundamental group of $M$ with the infinite cyclic groups generated by the (elements representing the) meridians of $L$.  Let these be $a_1$, ..., $a_n$ and let $k_i = a_i - 1$ in $J G(L)$.  Milnor defines a **canonical word** to be a product of the form $\phi_0 k_{j_1} \phi_1 k_{j_2} \phi_2 \cdots k_{j_p} \phi_p$ with $p \ge 0$, $\phi_i \in \pi_1(M)$, and $1 \le j_i \le n$.  A **canonical sentence** is a sum or difference of any number of canonical words.  It turns out ([Milnor, Theorem 7](#jmLinkGroups)) that each element of $\mathcal{R}(L)$ is represented by a unique canonical sentence.

Now let us return to the case of the almost trivial link in Euclidean space.  From above, we have the element $\beta'_n \in \mathcal{A}_i(L^n)$ corresponding to the $n$th parallel.  Removing any other component allows us to trivialise $\beta'_n$ since removing, say, the $i$th component leaves us with $L^i$ which is homotopic to the trivial link on $n-1$ components.  Removing the $i$ component corresponds to setting $a_i$ to $1$ in $J G(L^n)$, equivalently to setting $k_i = 0$.  So upon setting $k_i = 0$ we must have that $\beta'_n \mapsto 1$ and thus (by uniqueness) $\sigma \mapsto 0$.  Hence $k_i$ divides $\sigma$, and so every canonical word in $\sigma$ is of the form $k_{i_1} \cdots k_{i_{n-2}}$ for some permutation of $1$, $2$, ..., $n-2$.  Sorting them out by permutation, we get the expression in \eqref{muinvs}.

Now, how do we interpret or calculate these invariants?  We need to work out what an expression of the form in \eqref{muinvs} is saying.  Consider a canonical word, $k_{i_1} \cdots k_{i_{n-2}}$.  The corresponding element is:
$$
\alpha_{n-1}^{k_{i_1} \cdots k_{i_{n-2}}}
$$
Let us write $\alpha = \alpha_{n-1}$.  Now $\alpha^{k_1}$ is $\alpha^{a_1 - 1} = a_1 \alpha a_1^{-1} \alpha^{-1}$.  Thus this tells us to go around $L_1$, then $L_{n-1}$, back around $L_1$, and finally back around $L_{n-1}$.  Each time we introduce a new power, we do the same except that we replace the loop around $L_{n-1}$ with the loop so far constructed.

So the general method is as follows: choose two components of the link.  Write one of them as a word in the meridians of the others.  Then simplify this word using the other chosen link as the "base": namely, write everything in terms of conjugates of that base.  This will then separate out into the desired form and, hopefully, the link invariants can be read off.

## References

* [[John Milnor]] (1954). Link groups. _Ann. of Math. (2)_, _59_, 177–195. [MR](http://www.ams.org/mathscinet-getitem?mr=71020){: #jmLinkGroups}


Milnor slide trick

> This page meant to recall the proof of the local recognition of [[Hurewicz fibrations]] (see [there](Hurewicz+fibration#LocalRecognition)). But it didn't and doesn't.

Let $\pi: E \to B$ be a [[principal bundle|principal]] $G$-[[fiber bundle]] which has a [[numerable cover]] (this condition obtains if for example $B$ is [[paracompact space|paracompact]]). Suppose given a commutative diagram in [[Top]]:

$$\array{
X & \overset{f}{\to} & E \\
i_0 \downarrow & & \downarrow \pi \\
X \times I & \underset{\phi}{\to} & B
}$$

where $i_0$ is the composite inclusion $X \cong X \times \{0\} \hookrightarrow X \times I$. We are trying to show that $\phi$ lifts through $\pi$.

As I recall, the trick proceeds by considering the bundle

$$\array{
(\phi_0)^*E \times (-\infty, 0] \cup \phi^* E \cup (\phi_1)^* E \times [1, \infty) \\
\downarrow \\
(X \times (-\infty, 0]) \cup (X \times I) \cup (X \times [1, \infty)
}$$

where the base is $X \times \mathbb{R}$ and $\phi_t$ is the restriction of $\phi$ along $X \times \{t\} \hookrightarrow X \times I$, and then one constructs a bundle [[lifting]] of the homeomorphism

$$X \times \mathbb{R} \to X \times \mathbb{R}: (x, t) \mapsto (x, t + 1)$$

This bundle lifting is "the slide". Now the bundle is [[trivial bundle|trivial]] over $X \times [-1, 0]$ (see below), so it has a [[section]], and one transports this section along the slide to give a section $\sigma$ over the part

$$\array{
\phi^* E\\
\downarrow \\
X \times [0, 1]
}$$

and then the composition

$$X \times I \overset{\sigma}{\to} \phi^* E \to E$$

gives the desired homotopy [[lifting]].

To see that the bundle restricted over $X \times (-\infty, 0]$ is trivial, we just need to check that $(\phi_0)^* E$ is trivial over $X$. However, by the original commutative square, $\phi_0$ equals the composite

$$X \overset{f}{\to} E \overset{\pi}{\to} B$$

and already $\pi^* E$ is trivial (over $E$) essentially because $\pi$ is a $G$-[[torsor]]: there is a bundle isomorphism

$$\pi^* E \cong E \times_B E \to E \times G$$

over $E$.

The construction of the slide is where [[transfinite composition]] comes in. The details are at this moment a little hazy, but the rough idea is to construct a [[partition of unity]] $\rho_\alpha$ subordinate to the pulled back (locally finite) numerable cover $U_\alpha$ of $X \times I$. One is supposed to [[well-ordering theorem|well-order]] the $\alpha$, and then transfinitely compose a bunch of mini-slides over $(x, t) \mapsto (x, t + \rho_{\alpha}(x))$. The transfinite composition is well-defined on the fiber over any $x$ because the arrows in the composite are non-identity only for those $U_\alpha$ which contain $x$, and there are only finitely many of these by local finiteness.

To be continued.


Milnor's conjecture

See the 1996 [preprint](http://www.math.uiuc.edu/K-theory/0170) by [[Vladimir Voevodsky]] and the [wikipedia](http://en.wikipedia.org/wiki/Milnor's_conjecture) article.

* [[Fabien Morel]], _Voevodsky's proof of Milnor's conjecture_, Bull. Amer. Math. Soc. 35 (1998), 123-143 ([web](http://www.ams.org/journals/bull/1998-35-02/S0273-0979-98-00745-9/home.html))

A self-contained proof of Milnor's conjecture, and more generally the [[norm residue isomorphism theorem]] is given by

* [[Christian Haesemeyer]] and [[Charles Weibel]], _The Norm Residue Theorem in Motivic Cohomology_ ([pdf](http://sites.math.rutgers.edu/~weibel/BK.pdf))

[[!redirects Milnor conjecture]]
[[!redirects Milnor's conjecture]]


Milnor's theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Milnor's theorem_ describes the dual mod-$p$ [[Steenrod algebra]] in terms of [[generators and relations]].

## Statement

In the following, we use for $p = 2$ the notation

$$
  P^n \coloneqq Sq^{2n}
$$

$$
  \beta \coloneqq Sq^1
  \,.
$$

This serves to unify the expressions  for $p = 2$ and for $p \gt 2$ in the following. Notice that for all $p$

* $P^n$ has even degree $deg(P^n) = 2n(p-1)$;

* $\beta$ has odd degree $deg(\beta) = 1$.

+-- {: .num_theorem #MilnorTheoremOnDualSteenrodAlgebra}
###### Theorem
**(Milnor's theorem)**

The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ (def. \ref{DualSteenrodAlgebraForHPf}) is, as an [[associative algebra]], the free [[graded commutative algebra]]

$$
  \mathcal{A}^\ast_{\mathbb{F}_p}
    \simeq
  Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)
$$

on generators:

* $\xi_n$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$;

* $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$.

Moreover, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_p}$ is given by

$$
  \Psi(\xi_n)
  =
  \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k
$$

and

$$
  \Psi(\tau_n)
  =
  \tau_n \otimes 1
  +
  \underoverset{k=0}{n}{\sum}
  \xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k
  \,,
$$

where we set $\xi_0 \coloneqq 1$.

=--

This is due to ([Milnor 58](#Milnor58)). See for instance ([Kochman 96, theorem 2.5.1](#Kochman96))

## References

* {#Milnor58} [[John Milnor]], _The Steenrod algebra and its dual_, Ann. of Math. 67 (1958), 150–171.

* {#Kochman96} [[Stanley Kochman]], section 2.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

[[!redirects Milnor theorem]]


Milnor-Moore theorem

#Contents#
* table of contents
{:toc}

## Idea

The [[universal enveloping algebra]] of a [[Lie algebra]] naturally becomes a [[Hopf algebra]] in which all the elements of the original Lie algebra are [[primitive elements]]. The _Milnor-Moore theorem_ states conditions under which, conversely, any Hopf algebra generated by primitive elements is the universal enveloping algebra of the Lie algebra structure on these elements.

## References

The result is due to

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], _On the structure of Hopf algebras_, Annals of Math. __81__ (1965), 211-264. ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf))

Discussion for the special case of abelian [[restricted Lie algebras]] (with an eye towards its use in the [[May spectral sequence]]) is in

* {#May66} [[Peter May]],  _The cohomology of restricted Lie algebras and of Hopf algebras_, Journal of Algebra 3, 123-146 (1966) ([pdf](http://math.uchicago.edu/~may/PAPERS/3.pdf))

* [[Peter May]], _Some remarks on the structure of Hopf algebras_, Proceedings of the AMS, vol 23, No. 3 (1969) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/9.pdf))