Radon–Nikodym derivative

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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# Radon--Nikodym derivatives
* table of contents
{: toc}

## Idea

Given two [[measures]] $\mu, \nu$ on the same [[measurable space]], their Radon--Nikodym derivative is essentially their ratio $\mu/\nu$, although this is traditionally written $\mathrm{d}\mu/\mathrm{d}\nu$ because of analogies with [[differentiation]].  This ratio or derivative is a [[measurable function]] which is defined up to [[almost equality|equality almost everywhere]] with respect to the divisor $\nu$.  It only exists iff $\mu$ is [[absolutely continuous measure|absolutely continuous]] with respect to $\nu$.

Integration on a general [[measure space]] can be seen as the process of multiplying a measure by a function to get a measure.  Then the Radon--Nikodym derivative is the reverse of this: dividing two measures to get a function.

## The Radon–Nikodym theorem

\begin{definition}
Suppose $X$ is a [[set]], $\Sigma$ is a [[σ-algebra]] of subsets of $X$,
$\mu\colon\Sigma\to[0,\infty]$ is a countably additive measure,
and $\nu\colon\Sigma\to\mathbf{R}$ is a finitely additive map.
We say that $\nu$ is _absolutely continuous_ with respect to $\mu$
if for any $\epsilon\gt0$ there is $\delta\gt0$ such that $|\nu E|\lt\epsilon$
for any $E\in\Sigma$ such that $\mu(E)\lt\delta$.
We say that $\nu$ is _truly continuous_ with respect to $\mu$
if for any $\epsilon\gt0$ we can find $E\in\Sigma$ and $\delta\gt0$ such that $\mu(E)$ is finite and $|\nu(F)|\lt\epsilon$ whenever $F\in\Sigma$
and $\mu(E\cap F)\lt\delta$.
\end{definition}

\begin{remark}
If we assume $\nu$ to be countably additive, then the definition
of absolutely continuity simplifies as follows:
$\nu$ is absolutely continuous with respect to $\mu$ if and only if
for any $E\in\Sigma$ we have $\nu(E)=0$ whenever $\mu(E)=0$.
\end{remark}

\begin{remark}
$\nu$ is truly continuous with respect to $\mu$ if and only if
$\nu$ it is countable additive,
$\nu$ absolutely continuous with respect to $\mu$,
and for any $E\in\Sigma$ such that $\nu(E)\gt0$
we can find $F\in\Sigma$ such that $F\subset E$, $\mu(F)$ is finite,
and $\nu(F)\gt0$.
\end{remark}

\begin{theorem}
Suppose $X$ is a [[set]], $\Sigma$ is a [[σ-algebra]] of subsets of $X$,
$\mu\colon\Sigma\to[0,\infty]$ is a countably additive measure,
and $\nu\colon\Sigma\to\mathbf{R}$ is a function.
Then there is a $\mu$-integrable function $f$ such that $\nu(E)=\int_E f$ for all $E\in\Sigma$ if and only if $\nu$ is finitely additive and truly continuous with respect to $\mu$.
\end{theorem}

\begin{corollary}
In the context of the above theorem,
if $(X,\Sigma,\mu)$ is [[σ-finite]], then there is a $\mu$-integrable function $f$ such that $\nu(E)=\int_E f$ for all $E\in\Sigma$ if and only if
$\nu$ is countably additive and absolutely continuous with respect to $\mu$.
If $\mu(X)$ is finite, it suffices to require that $\nu$ is finitely additive.
\end{corollary}

\begin{example}
The following example shows that the condition of being truly continuous
is necessary in the non-σ-finite case.
Consider an uncountable set $X$,
a [[σ-algebra]] $\Sigma$ of subsets of $X$ that are countable or have countable complement,
the counting measure $\mu\colon\Sigma\to[0,\infty]$,
and the measure $\nu\colon\Sigma\to[0,1]$ that vanishes on countable sets
and takes value 1 on all uncountable subsets.
Then $\nu$ is absolutely continuous with respect to $\mu$ because $\mu$ only vanishes on the empty set.
However, $\nu$ is not truly continuous with respect to $\mu$
because $\mu$ takes finite values only on finite sets,
but for any such a set $E$ we have $\nu(X\setminus E)=1$,
which yields a contradiction if $\epsilon\lt1$.
\end{example}

\begin{example}
The [[measure space]] in the last example is quite pathological: it is not [[localizable]].
A [[localizable]] example can be constructed if and only if [[real-valued-measurable cardinals]] exist.
In this case, take $X$ to be a [[real-valued-measurable cardinal]],
$\mu\colon 2^X\to[0,\infty]$ to be the counting measure,
and $\nu\colon 2^X\to[0,1]$ a probability measure that vanishes on all countable subsets of $X$.
Then $\nu$ is absolutely continuous with respect to $\mu$ because only the empty subset of $X$ has $\mu$-measure 0.
However, $\nu$ is not truly continuous with respect to $\mu$
because $\mu$ takes finite values only on finite sets,
but for any such a set $E$ we have $\nu(X\setminus E)=1$,
which yields a contradiction if $\epsilon\lt1$.
\end{example}

\begin{remark}
Truly continuous countably (or finitely) additive measures
on a measure space $(X,\Sigma,\mu)$
form a module over the complex [[*-algebra]]
of measurable maps $X\to\mathbf{R}$ modulo equality almost everywhere.
The Radon–Nikodym theorem then says that this module
is a free module of rank 1.
Furthermore, generators of this module can be identified
with truly continuous measures $\nu$ such that $\mu$ is absolutely continuous with respect to $\nu$.
\end{remark}

## The Lebesgue decomposition

\begin{theorem}
Suppose $X$ is a [[set]], $\Sigma$ is a [[σ-algebra]] of subsets of $X$,
$\mu\colon\Sigma\to[0,\infty]$ is a countably additive measure,
and $\nu\colon\Sigma\to\mathbf{R}$ is a countably additive function.
Then there is a unique decomposition
$$\nu=\nu_s+\nu_t+\nu_e,$$
where $\nu_t$ is truly continuous with respect to $\mu$,
$\nu_e$ is absolutely continuous with respect to $\mu$ and vanishes
on every measurable set of finite $\mu$-measure,
and $\nu_s$ is singular with respect to $\mu$,
meaning there is a set $F\in\Sigma$ such that $\mu(F)=0$
and $\nu(E)=0$ for all $E\in\Sigma$ such that $E\subset X\setminus F$.
In particular, setting $\nu_a=\nu_t+\nu_e$ yields a unique decomposition $\nu=\nu_s+\nu_a$ of $\nu$ as a sum of a singular and absolutely continuous measure.
\end{theorem}

## Definitions

Let $X$ be a [[measurable space]] (so $X$ consists of a [[set]] ${|X|}$ and a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{M}_X$), and let $\mu$ and $\nu$ be [[measures]] on $X$, valued in the [[real numbers]] (and possibly taking infinite values) or in the [[complex numbers]] (and taking only finite values).  Let $f$ be a [[measurable function]] $f$ (with real or complex values) on $X$.

+-- {: .un_defn}
###### Definition
The function $f$ is a __Radon--Nikodym derivative__ of $\mu$ with respect to $\nu$ if, given any [[measurable subset]] $A$ of $X$, the $\mu$-measure of $A$ equals the integral of $f$ on $A$ with respect to $\nu$:
$$ \mu(A) = \int_A f \nu = \int_{x \in A} f(x) \mathrm{d}\nu(x) .$$
(The latter two expressions in this equation are different notations for the same thing.)
=--

## Properties

These properties are basic to the concept; the notation is as in the definition above.

+-- {: .num_theorem}
###### Theorem

Let $f$ be a Radon--Nikodym derivative of $\mu$ with respect to $\nu$, and let $g$ be a measurable function on $X$.  Then $g$ is a Radon--Nikodym derivative of $\mu$ with respect to $\nu$ if and only if $f$ and $g$ are equal almost everywhere with respect to $\nu$.
=--

+-- {: .num_theorem}
###### Theorem
If a Radon--Nikodym derivative of $\mu$ with respect to $\nu$ exists, then $\mu$ is [[absolutely continuous measure|absolutely continuous]] with respect to $\nu$.
=--

+-- {: .num_theorem}
###### Theorem
If $\mu$ is [[absolutely continuous measure|absolutely continuous]] with respect to $\nu$ and both $\mu$ and $\nu$ are $\sigma$-[[sigma-finite measure|finite]], then a Radon--Nikodym derivative of $\mu$ with respect to $\nu$ exists.
=--

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

For fairly elementary proofs, see [Bartels (2003)](#Bartels2003).
=--

(This last theorem is not as general as it could be.)

Note the repetition of 'with respect to $\nu$' in various guises; let us fix $\nu$ (assumed to be $\sigma$-finite) and take everything with respect to it.  Then it is convenient to treat all measurable functions up to [[almost equality|equality almost everywhere]]; and given any absolutely continuous $\mu$ (also assumed to be $\sigma$-finite), we speak of [[the]] Radon--Nikodym derivative of $\mu$.

## Notation

See also the discussion of notation at [[measure space]].

Using the simplest notation for integrals, the definition of Radon--Nikodym derivative reads
$$ \mu(A) = \int_A f \nu ,$$
or equivalently
$$ \int_A \mu = \int_A f \nu .$$
In other words, the measure $\mu$ is the product of the function $f$ and the measure $\nu$:
$$ \mu = f \nu ;$$
and so $f$ is the ratio of $\mu$ to $\nu$:
$$ f = \mu/\nu .$$
So this is the simplest notation for the Radon--Nikodym derivative.

However, this notation for integrals is uncommon; one is more likely to see
$$ \int_A \mathrm{d}\mu = \int_A f \,\mathrm{d}\nu ,$$
which leads to
$$ f = \mathrm{d}\mu/\mathrm{d}\nu $$
for the Radon--Nikodym derivative.  But none of these '$\mathrm{d}$'s are really necessary.

We can also use a fuller notation with a dummy variable as the object of the symbol '$\mathrm{d}$':
$$ \int_{x \in A} \mu(\mathrm{d}x) = \int_{x \in A} f(x) \,\nu(\mathrm{d}x) ;$$
this leads to
$$ f(x) = \mu(\mathrm{d}x)/\nu(\mathrm{d}x) ,$$
which does not give a symbol for $f$ directly.  If instead of $\mu(\mathrm{d}x)$ one unwisely writes $\mathrm{d}\mu(x)$, then this gives the previous notation for the Radon--Nikodym derivative.

Now let $\nu$ be [[Lebesgue measure]] on the [[real line]] and let $F$ be an upper [[semicontinuous function]] on the real line, so that $F$ defines a [[Borel measure]] $\mu$ generated by
$$ \mu({]-\infty,a]}) \coloneqq F(a) .$$
Then $F$ is [[absolutely continuous function|absolutely continuous]] if and only if $\mu$ is absolutely continuous, in which case the [[derivative]] $F'$ exists almost everywhere and is a Radon--Nikodym derivative of $\mu$.  That is,
$$ \mu/\nu = F' = \mathrm{d}F/\mathrm{d}t .$$
The presence of '$\mathrm{d}$' on the right-hand side inspires people to put it on the left-hand side as well; but this is spurious, since we really want to write
$$ \mu = \mathrm{d}F $$
and
$$ \nu = \mathrm{d}t ,$$
where $t$ is the [[identity function]] on the real line.

## References

A comprehensive treatment can be found in Chapter 23 of

* [[David H. Fremlin]], _Measure Theory_.

Some fairly elementary proofs prepared for a substitute lecture in [[John Baez]]\'s introductory measure theory course are here:

*  [[Toby Bartels]] (2003): _The Radon Nikodym Theorem_; [web](http://tobybartels.name/notes/#Radon).
   {#Bartels2003}

The strategy there is based on:

*  Richard Bradley (1989): An Elementary Treatment of the Radon-Nikodym Derivative, American Mathematical Monthly 96(5), 437--440.

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Radovan Dermisek

* [webpage](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-4-4p.pdf)

## related $n$Lab entries

* [[Schwinger-Dyson equation]]

* [[Dirac field]]

* [[QED]]

category: people


Radu Diaconescu

## Selected writings

The [[Diaconescu-Goodman-Myhill theorem]] in [[logic]]:

* {#Diaconescu75} [[Radu Diaconescu]], _Axiom of choice and complementation_, Proceedings of the American Mathematical Society 51:176-178 (1975) ([doi:10.1090/S0002-9939-1975-0373893-X](https://doi.org/10.1090/S0002-9939-1975-0373893-X))

Introducing [[Diaconescu's theorem]] in [[topos theory]]:

* [[Radu Diaconescu]], _Change of base for toposes with generators_ , J. Pure Appl. Algebra __6__ (1975), no. 3, 191-218 (doi:10.1016/0022-4049(75)90015-8)

On [[atomic toposes]]

* {#Barr-Diaconescu80}[[Michael Barr]], [[Radu Diaconescu]], *Atomic Toposes*, J. Pure Appl. Algebra **17** (1980) 1-24 [doi:10.1016/0022-4049(80)90020-1, [pdf](http://www.math.mcgill.ca/barr/papers/atom.top.pdf), [[BarrDiaconescu-Atomictoposes.pdf:file]]]

category: people


Radu Roiban

* [webpage](http://www.phys.psu.edu/people/rsr10)

## Selected writings

Discussion of [[D4-branes]] [[polarized brane|polarizing]] into [[NS5-branes]]:

* [[Iosif Bena]], [[Radu Roiban]], _$\mathcal{N}=1^\ast$ in 5 dimensions: Dijkgraaf-Vafa meets Polchinski-Strassler_, 	JHEP 0311 (2003) 001 ([arXiv:hep-th/0308013](https://arxiv.org/abs/hep-th/0308013))

On [[twistor string theory]]:

* {#RoibanVolovich04} [[Radu Roiban]], [[Anastasia Volovich]], *All Googly Amplitudes from the B-model in Twistor Space*, Phys. Rev. Lett. 93 (2004) 131602 ([arXiv:hep-th/0402121](https://arxiv.org/abs/hep-th/0402121))

* {#RoibanSpradlinVolovich04} [[Radu Roiban]], [[Marcus Spradlin]], [[Anastasia Volovich]], *On the Tree-Level S-Matrix of Yang-Mills Theory*, Phys. Rev. D70 : 026009, 2004 ([arXiv:hep-th/0403190](https://arxiv.org/abs/hep-th/0403190))

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation operator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:

* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))

On [[scattering amplitudes]] in [[perturbative quantum gravity|perturbative]] [[D=4 N=8 supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Lance Dixon]], [[Henrik Johansson]], [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_, Fortschr. Phys. **59** (2011) 7-8 [[arXiv:1103.1848](http://arxiv.org/abs/1103.1848), [doi:10.1002/prop.201100037](https://doi.org/10.1002/prop.201100037)]

On the "[[double copy]]" [[KLT relations]]-approach to [[scattering amplitudes]] in [[perturbative quantum gravity]] and [[supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Marco Chiodaroli]], [[Henrik Johansson]], [[Radu Roiban]], *Supergravity amplitudes, the double copy and ultraviolet behavior*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2304.07392](https://arxiv.org/abs/2304.07392)]

## Related entries

* [[D=4 supergravity]]

category: people


Raf Cluckers

* [web](http://wis.kuleuven.be/algebra/Raf)

Related pages: [[motivic integration]], [[rigid analytic space]]

* Avraham Aiyenbud, Raf Cluckers, _Wave front holonomicity of $C^{exp}$-class distribution and non-archimedean local fields_, Forum of Mathematics, Sigma __8__ (2020) e35 [doi](https://doi.org/10.1017/fms.2020.27)
* R. Cluckers, L. Lipshitz, _Fields with analytic structure_, J. Eur. Math. Soc. __13__, 1147–1223, [pdf](http://wis.kuleuven.be/algebra/Raf/prints/JEMS.pdf)
and several articles (in various formalisms) in collection
* R. Cluckers, J. Nicaise, J. Sebag (Editors), _Motivic integration and its interactions with [[model theory]] and non-archimedean Geometry_, 2 vols.
London Mathematical Society Lecture Note Series __383__, __384__

[[!redirects R. Cluckers]]


Rafael Borowiecki

My main interest from very early on in life is science. Within science i choose to specialize in all kinds of mathematics (except numerical methods and statistics) and mathematical physics (especially theories of quantum gravity). This is because it is pure, true, universal, useful, fun, beautiful etc. I have been wandering all across this landscape since 1991 and found now that category theory ties together all the things that i love the most. Hence now i am most interested in a categorical foundation of mathematics, or any foundation that can structure the branches of mathematics in detail. This is because i like order so much, and to simplify things. I have put a lot of hard work to find out what i want to know, bit by bit. It is like laying the biggest puzzle there can be. I hope to be able to apply all this to understand systems such as higher life, the universe and reality.

I don't have a homepage but i will put up a domain from my home soon, dedicated to my project and to publish the work that has been done.

To make it short i am a platonist, perspectivist, realist, minimalist, idealist and optimist. It might also be fun to know that i am a cosmopolit and i am learning to meditate.

category: people

[[!redirects Ra]]


Rafael Sivera

* website?

## related $n$Lab entries

* [[Nonabelian Algebraic Topology]]

  * [[strict omega-groupoid]]

  * [[crossed complex]]

category: people


Rafael Sorkin

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

## Selected writings

Introducing [[Kaluza-Klein monopoles]]:

* {#Sorkin83} [[Rafael Sorkin]], _Kaluza-Klein monopole_ Phys. Rev. Lett. 51 (1983) 87 ([doi:10.1103/PhysRevLett.51.87](https://doi.org/10.1103/PhysRevLett.51.87))

* {#GrossPerry83} [[David Gross]], [[Malcolm Perry]], _Magnetic Monopoles in Kaluza-Klein Theories_  Nucl. Phys. B226 (1983) 29 (doi:10.1016/0550-3213(83)90462-5)

## Related $n$Lab entries

* [[causal set]]

category: people


Rafaël Bocquet

* [website](https://rafaelbocquet.gitlab.io/)

## Publications

* [[Rafaël Bocquet]], [[Ambrus Kaposi]], [[Christian Sattler]], *Relative induction principles for type theories*, ([arXiv:2102.11649](https://arxiv.org/abs/2102.11649))

## Talks

* [[Rafaël Bocquet]], *Coherence of definitional equality in type theory*, [[Homotopy Type Theory Electronic Seminar Talks]], 23 September 2021 ([video](https://www.youtube.com/watch?v=WKCxBygnYqE), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Bocquet-2021-09-23-HoTTEST.pdf))

* [[Rafaël Bocquet]], *Towards coherence theorems for equational extensions of type theories*, Strength of Weak Type Theory, [[DutchCATS]], 12 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_bocquet.pdf))

[[!redirects Rafael Bocquet]]

category: people


Raffael Stenzel

* [website](https://sites.google.com/mail.muni.cz/rstenzel/home)

## Selected writings

On [[comprehension schemes]] and [[elementary (infinity,1)-toposes|elementary $(\infty,1)$-toposes]]:

* [[Raffael Stenzel]], _(∞,1)-Categorical Comprehension Schemes_ [[arXiv:2010.09663](https://arxiv.org/abs/2010.09663), talk: [YT](https://www.youtube.com/watch?v=qVhRruh_1jI)]

On [[(infinity,1)-sites|$(\infty,1)$-sites]]:

* [[Raffael Stenzel]], *Higher sites and their higher categorical logic*, talk at [[Homotopy Type Theory Electronic Seminar Talks|HoTT Electronic Seminar]], (18 November 2021) [video:[YT](https://www.youtube.com/watch?v=hUJlK8XgZxQ), slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Stenzel-2021-11-18-HoTTEST.pdf)]

* [[Raffael Stenzel]], *Notions of $(\infty,1)$-sites and related formal structures* [[arXiv:2306.06619](https://arxiv.org/abs/2306.06619)]

category: people


Raffaele Savelli

* [inspire page](https://inspirehep.net/authors/1056297)

* [institute page](https://www.theorie.physik.uni-muenchen.de/MATH/members/former_mem/former_mpi/savelli_raffaele/index.html)

## Selected writings

On [[bundle gerbes]] [[connections on bundle gerbes|with connection]] and [[Čech cohomology|Čech]]-[[Deligne cohomology]] for the description of [[B-fields]] in [[string theory]]:

* [[Loriano Bonora]], [[Fabio Ferrari Ruffino]], [[Raffaele Savelli]], *Classifying A-field and B-field configurations in the presence of D-branes*, JHEP 0812:078 (2008) [[arXiv:0810.4291](https://arxiv.org/abs/0810.4291), [doi:10.1088/1126-6708/2008/12/078](https://doi.org/10.1088/1126-6708/2008/12/078)]

On [[shifted C-field flux quantization]] in [[F-theory]]:

* [[Andres Collinucci]], [[Raffaele Savelli]], *On Flux Quantization in F-Theory*,  J. High Energ. Phys. **2012** 15 (2012)   [[arXiv:1011.6388](https://arxiv.org/abs/1011.6388), doi:10.1007/JHEP02(2012)015]

* [[Andres Collinucci]], [[Raffaele Savelli]], *On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups*,  J. High Energ. Phys. **2012** 94 (2012) [[arXiv:1203.4542](https://arxiv.org/abs/1203.4542), doi:10.1007/JHEP08(2012)094]

* [[Raffaele Savelli]], *Flux Quantization in F-theory and Freed-Witten anomaly*, talk at StringMath (2012) [[pdf](https://www.hcm.uni-bonn.de/fileadmin/stringmath2012/parallel_talks/savelli.pdf), [[Savelli-FTheoryFluxQuantization.pdf:file]]]

category: people


Rahmi Gueven

* [webpage](http://bilimakademisi.org/bilim-akademisi/kurucu-kisa-ozgecmisleri/prof-dr-rahmi-guven/)

## Selected writings

Introducing [[black brane|black]] [[M-brane]]-solutions to [[D=11 supergravity]] ([[M2-brane]], [[M5-brane]]):

* {#Gueven92} [[Rahmi Gueven]], _Black $p$-brane solutions of $D = 11$ supergravity theory_, Phys. Lett. B276 (1992) 49 (doi:10.1016/0370-2693(92)90540-K, [spire:338203](http://inspirehep.net/record/338203))

## Related $n$Lab entries

* [[M5-brane]], [[black brane]]

category: people


Rahul Pandharipande

* [webpage](https://people.math.ethz.ch/~rahul)

## Selected writings

On the [[Gromov-Witten/Donaldson-Thomas correspondence]]:

* [[Davesh Maulik]], [[Nikita Nekrasov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov–Witten theory and Donaldson–Thomas theory, I_. Compositio Mathematica. 142 (5): 1263–1285. (2003) ([arXiv:math/0312059](https://arxiv.org/abs/math/0312059). [doi:10.1112/S0010437X06002302](http://www.journals.cambridge.org/abstract_S0010437X06002302))

* [[Davesh Maulik]], [[Alexei Oblomkov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds_ ([arXiv:0809.3976](https://arxiv.org/abs/0809.3976v1))

category: people


Raimund Vogl

* [institute page](https://www.uni-muenster.de/IT/en/cit/leitung.shtml)

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

## Selected writings

On unconventional paradigms for [[quantum computation]]:

* [[Steven Duplij]], [[Raimund Vogl]], *Innovative Quantum Computing*, IOP (2023) [[ISBN:978-0-7503-5281-9](https://iopscience.iop.org/book/mono/978-0-7503-5281-9)]

category: people


Rainald Flume

* [institute page](http://www.th.physik.uni-bonn.de/th/People/flume/)

* [research group page](http://www.th.physik.uni-bonn.de/Groups/Flume/)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=139663)

## Selected writings

A precursor result to the [[hypergeometric construction of KZ solutions]], specifically of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW-model]]-[[2d CFTs]]:

* [[Philippe Christe]], [[Rainald Flume]], *The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term*, Nuclear Physics B **282** (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$

category: people


Rainer Verch

* [website](http://personal-homepages.mis.mpg.de/verch/)

## Selected writings

On [[quantum measurement]] in the refined context of ([[local field theory|local]] [[algebraic quantum field theory|algebraic]]) [[quantum field theory]] [[AQFT on curved spacetimes|on Lorentzian manifolds]]:

* [[Christopher J. Fewster]], [[Rainer Verch]], *Measurement in Quantum Field Theory* [[arXiv:2304.13356](https://arxiv.org/abs/2304.13356)]

## Related pages

* [[AQFT]]

category: people


Rainer Vogt

**Rainer Vogt** (1942--2015) was  a mathematician who specialised in [[homotopy theory]].

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=86037)

* obituary at his former research group in Osnabrück.

* {#VogtMemory} Rainer Vogt, _My time as Mike Boardman's student and our work on infinite loop spaces_ ([[VogtMemory.pdf:file]])

He received his Ph.D. from the University of Warwick and his thesis advisor was [[Michael Boardman]]. He was a a full professor at Osnabrück University in Germany.

He coauthored with Boardman the famous book

* [[Michael Boardman]], [[Rainer Vogt]], *Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics **347** Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]

This book introduced the notion of *[[weak Kan complexes]]* that was later popularized by [[André Joyal]] under the name *[[quasi-categories]]* as a natural basis for the  [[higher category theory]] of [[(∞,1)-category|(∞,1)-categories]].

An early application of the ideas from that book was in the proof of what is sometimes refereed to as Vogt's theorem on categories of homotopy coherent maps, between homotopy coherent diagrams. This is briefly discussed at [[homotopy coherent diagram]]. It occurs in his paper

* {#Vogt73} [[Rainer Vogt]], _Homotopy limits and colimits_, Math. Z., **134** (1973) 11-52 [[doi:10.1007/BF01219090](https://doi.org/10.1007/BF01219090), [eudml:171965](https://eudml.org/doc/171965)]

which gave a neat explicit construction/description of [[homotopy limit]]s.

## Selected writings

On [[convenient categories of topological spaces]] in the generality subsuming [[compactly generated topological spaces]] and [[Delta-generated topological spaces]]:

* {#Vogt71} [[Rainer M. Vogt]], *Convenient categories of topological spaces for homotopy theory*,  Arch. Math 22, 545–555 (1971) ([doi:10.1007/BF01222616](https://doi.org/10.1007/BF01222616))

On a [[model structure on simplicial algebras]] for [[simplicial Lawvere theories]]:

* {#SchwänzlVogt91} [[Roland Schwänzl]], [[Rainer Vogt]], _The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories_, Archives of Mathematics 56 (1991) 405-411 ([doi:10.1007/BF01198229](https://doi.org/10.1007/BF01198229))

On the [[iterated algebraic K-theory]] of ([[connective spectrum|connective]]) [[ring spectra]]:

* {#SchwaenzlVogt94} [[Roland Schwänzl]], [[Rainer Vogt]], *Basic Constructions in the K-Theory of Homotopy Ring Spaces*, Transactions of the American Mathematical Society, Vol. 341, No. 2 (Feb., 1994), pp. 549-584 ([jstor:2154572](https://www.jstor.org/stable/2154572), [doi:10.2307/2154572](https://doi.org/10.2307/2154572))

## Related entries

* [[stable homotopy category]], [[smash product of spectra]]

* [[quasi-category]]

* [[Boardman-Vogt resolution]]

* [[Boardman-Vogt tensor product]]

## References

Some of the above material is taken from [[joyalscatlab:Vogt, Rainer|Joyal's CatLab -- Rainer Vogt]].

category:people

[[!redirects Rainer M. Vogt]]

[[!redirects Vogt, Rainer ]]
[[!redirects Vogt]]
[[!redirects R. Vogt]]
[[!redirects R. M. Vogt]]


Rajan Mehta

* [webpage](http://www.math.smith.edu/~rmehta/)

## related $n$Lab entries

* [[curved L-infinity algebra]]

category: people


Rajesh Gopakumar

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

## Selected writings

On [[open/closed string duality]] in interpreting [['t Hooft double line notation]] in the [[large N limit]] and in [[AdS/CFT duality]]:

* {#GopakumarVafa98} [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

* {#Gopakumar04} [[Rajesh Gopakumar]], _Free Field Theory as a String Theory?_, Comptes Rendus Physique 5 (2004) 1111-1119 ([hep-th/0409233](https://arxiv.org/abs/hep-th/0409233))

Introducing [[Gopakumar-Vafa duality]]:

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _Topological Gravity as Large N Topological Gauge Theory_, Adv. Theor. Math. Phys. 2:413-442, 1998 ([arXiv:hep-th/9802016](https://arxiv.org/abs/hep-th/9802016))

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

On [[AdS3/CFT2]]:

* {#EberhardtGaberdielGopakumar19} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _Deriving the $AdS_3/CFT_2$ Correspondence_ ([arXiv:1911.00378](https://arxiv.org/abs/1911.00378))

## Related $n$Lab entries

* [[topological string]]

* [[Gopakumar-Vafa invariant]]

* [[higher spin gauge theory]]

category: people


Ralf Kemper

* [Institute page](https://ekvv.uni-bielefeld.de/pers_publ/publ/PersonDetail.jsp?personId=34159269&lang=EN)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=125545)

## Selected writings

On [[superconvex spaces]]:

* {#BörgerKemper94a} [[Reinhard Börger]], [[Ralf Kemper]], *There is no cogenerator for total convex spaces*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome **35** 4 (1994) 335-338 [[numdam:CTGDC_1994__35_4_335_0](http://www.numdam.org/item/CTGDC_1994__35_4_335_0)]

* {#BörgerKemper94b} [[Reinhard Börger]], [[Ralf Kemper]], *Cogenerators for convex spaces*, Applied Categorical Structures **2** (1994) 1-11 [[doi:10.1007/BF00878499](https://doi.org/10.1007/BF00878499)]

category: people


Ralf Krömer

*  [homepage at Wuppertal](http://www2.math.uni-wuppertal.de/~kroemer/)

## Selected writings

On the history of developments in [[category theory]]:

* [[Ralf Krömer]], *Tool and object: A history and philosophy of category theory*, Science Networks. Historical Studies **32**, Springer (2007) [[doi:10.1007/978-3-7643-7524-9](https://doi.org/10.1007/978-3-7643-7524-9)]

On [[paths]]

* Ralf Krömer, _The set of paths in a space and its algebraic structure. A historical account_, Annales mathématiques de la Faculté des Sciences de Toulouse, 22, (2013), 915 – 968.

On [[duality]]:

* Ralf Krömer and [[David Corfield]], _The Form and Function of Duality in Modern Mathematics_, [article](http://philosophiascientiae.revues.org/976?lang=en)

category:people

[[!redirects Ralf Kromer]]


Ralf Meyer

* [webpage](http://xwww.uni-math.gwdg.de/template/?userid=3864&username=rameyer)

## Related entries

* [[KK-theory]]

category: people

[[!redirects R. Meyer]]


Ralf Schiffler

* [institute page](https://schiffler.math.uconn.edu/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=115170)

## Selected writings

On [[quiver representations]]:

* [[Ralf Schiffler]], *Quiver Representations*, CMS Books in Mathematics, Springer  (2014) $[$[doi:10.1007/978-3-319-09204-1](https://doi.org/10.1007/978-3-319-09204-1)$]$

category: people


Ralph Abraham

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

category: people


Ralph Blumenhagen

* [webpage](http://wwwth.mpp.mpg.de/members/blumenha/)

## Selected writings

On [[intersecting D-brane models]]:

* [[Ralph Blumenhagen]], [[Mirjam Cvetic]], [[Paul Langacker]], [[Gary Shiu]], *Towards Realistic Intersecting D-Brane Models*, Ann. Rev. Nucl. Part. Sci. **55** (2005) 71-139 [[arXiv:hep-th/0502005](http://arxiv.org/abs/hep-th/0502005), [doi:10.1146/annurev.nucl.55.090704.151541](https://doi.org/10.1146/annurev.nucl.55.090704.151541)]

On [[conformal field theory]]:

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

On [[string theory]]:

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], *Basic Concepts of String Theory*, Theoretical and Mathematical Physics, Springer (2013) [[doi:10.1007/978-3-642-29497-6](https://doi.org/10.1007/978-3-642-29497-6)]

## Related entries

* [[string phenomenology]]

category: people


Ralph Cohen

> not to be confused with *[[Fred Cohen]].*

***

* [website](http://math.stanford.edu/~ralph/)

## Selected writings

On [[cyclic loop spaces]]:

* {#Cohen1985} [[Ralph Cohen]], p. 194 of: *A Model for the Free Loop Space of a Suspension*, in: [[Haynes Miller]], [[Douglas Ravenel]] (eds.), *Algebraic Topology*, Lecture Notes in Mathematics **1286**, Springer 1985 ([doi:10.1007/BFb0078743](https://link.springer.com/chapter/10.1007/BFb0078743))

* [[Gunnar Carlsson]], [[Ralph Cohen]], *The cyclic groups and the free loop space*, Commentarii Mathematici Helvetici **62** (1987) 423–449  ([doi:10.1007/BF02564455](https://doi.org/10.1007/BF02564455), [dml:140092](https://eudml.org/doc/140092))

On [[Yang-Mills monopoles]] and [[braid groups]]:

* [[Ralph Cohen]], [[John D. S. Jones]], _Monopoles, braid groups, and the Dirac operator_, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 ([euclid:cmp/1104254240](https://projecteuclid.org/euclid.cmp/1104254240))

On [[string topology]]:

*  [[Ralph Cohen]], John R. Klein, [[Dennis Sullivan]], _The homotopy invariance of the string topology loop product and string bracket_, J. of Topology 2008 __1__(2):391-408; [doi](http://dx.doi.org/10.1112/jtopol/jtn001)

* [[Ralph Cohen]], _Homotopy and geometric perspectives on string topology_ ([pdf](http://math.stanford.edu/~ralph/skyesummary.pdf))

* {#CohenJones} [[Ralph Cohen]], [[John David Stuart Jones]], _A homotopy theoretic realization of string topology_ , Math. Ann. 324
(2002), no. 4, ([arXiv:0107187](http://arxiv.org/abs/math/0107187))

* {#CohenGodin03} [[Ralph Cohen]], [[Veronique Godin]], _[[A Polarized View of String Topology]]_ ([arXiv:math/0303003](http://arxiv.org/abs/math/0303003))

* [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_, in: [[Ralph Cohen]], [[Kathryn Hess]], [[Alexander Voronov]], _String topology and cyclic homology_, Advanced courses in mathematics CRM Barcelona, Birkhäuser 2006 ([math.GT/05036259](http://arxiv.org/abs/math/0503625), [doi:10.1007/3-7643-7388-1](https://doi.org/10.1007/3-7643-7388-1), [pdf](http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7))

* [[Ralph Cohen]], [[John Jones]], _Gauge theory and string topology_ ([arXiv:1304.0613](http://arxiv.org/abs/1304.0613))

On the [[homotopy type of spaces of rational maps]] and [[moduli spaces of monopoles]] related to [[braid groups]]:

* [[Fred Cohen]], [[Ralph Cohen]], [[Benjamin M. Mann]], [[R. James Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))

* {#CohenShimamoto91} [[Ralph L. Cohen]], [[Don H. Shimamoto]], *Rational Functions, Labelled Configurations, and Hilbert Schemes*, Journal of the London Mathematical Socienty **43** 2 (1991) 509-528 ([doi:10.1112/jlms/s2-43.3.509](https://doi.org/10.1112/jlms/s2-43.3.509))

* [[Ralph Cohen]], [[John D. S. Jones]], _Monopoles, braid groups, and the Dirac operator_, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 ([euclid:cmp/1104254240](https://projecteuclid.org/euclid.cmp/1104254240))

* [[Ralph L. Cohen]], [[John D. S. Jones]], [[Graeme B. Segal]], *Stability for holomorphic spheres and Morse theory*, in: K. Grove, I. H. Madsen, E. K. Pedersen (eds.) *Geometry and Topology: Aarhus*, Contemporary Mathematics
Volume: 258 (2000)  ([arXiv:math/9904185](https://arxiv.org/abs/math/9904185), [ISBN:978-0-8218-2158-9](https://bookstore.ams.org/conm-258))

and more generally on [[moduli spaces]]:

* [[Ralph Cohen]], _Stability phenomena in the topology of moduli spaces_ ([arXiv:0908.1938](https://arxiv.org/abs/0908.1938))

On [[string topology]] (and [[Pontrjagin rings]] of [[based loop spaces]]) of [[complex projective spaces]] and [[n-spheres]]:

* [[Ralph L. Cohen]], [[John D. S. Jones]], Jun Yan, *The loop homology algebra of spheres and projective spaces*, in *Categorical Decomposition Techniques in Algebraic Topology*, Progress in Mathematics **215**, Birkhäuser (2003) [[arXiv:math/0210353](https://arxiv.org/abs/math/0210353), [doi:10.1007/978-3-0348-7863-0_5](https://doi.org/10.1007/978-3-0348-7863-0_5)]

category: people

[[!redirects Ralph L. Cohen]]


Ralph Fox

Ralph Fox was an American topologist, well known for his contribution to [[knot theory]], both at a research level and for popularising it in talks and articles.

He was the doctoral supervisor of [[John Milnor]], [[John Stallings]] and [[Barry Mazur]], amongst others.

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

* [St. Andrews history entry](http://www-history.mcs.st-andrews.ac.uk/Biographies/Fox_Ralph.html)

## Selected writings

On [[braid groups]] and [[configuration spaces of points]]:

* {#FoxNeuwirth62} [[Ralph H. Fox]], [[Lee Neuwirth]], *The braid groups*, Math. Scand. **10** (1962) 119-126
$[$[doi:10.7146/math.scand.a-10518](https://doi.org/10.7146/math.scand.a-10518), [pdf](https://www.mscand.dk/article/view/10518/8539), [MR150755](http://www.ams.org/mathscinet-getitem?mr=150755)$]$

[[!redirects R. H. Fox]]
[[!redirects Ralph H. Fox]]

category: people

category: knot theory


Ralph Kaufmann

[[!redirects Ralph  Kaufmann]]
[[!redirects R. Kaufmann]]
##Ralph M. Kaufmann

[Homepage](https://www.math.purdue.edu/~rkaufman/)

###Publications

* [Publication list](https://www.math.purdue.edu/~rkaufman/pubs.html)

category: people

[[!redirects Ralph M. Kaufmann]]


Ralph Matthes

* [personal page](https://www.irit.fr/~Ralph.Matthes/)

## Selected writings

On [[monoidal category|monoidal]] [[univalent categories]]:

* [[Kobe Wullaert]], [[Ralph Matthes]], [[Benedikt Ahrens]], *Univalent Monoidal Categories* [[arXiv:2212.03146](https://arxiv.org/abs/2212.03146)]

## Related entries

* [[UniMath project]]

category: people


Ram Kanwal

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=2292)

## related $n$Lab entries

* [[generalized solution of a PDE]]

* [[Cauchy principal value]]

category: people


Ramamurti Shankar

* [webpage](http://pantheon.yale.edu/~rshankar/)

## Selected writings

On [[anyon]]-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]]:

* {#MurthyShankar09} [[M.V.N. Murthy]], [[Ramamurti Shankar]], *Exclusion Statistics: From Pauli to Haldane* (1999, 2009) $[$[dspace:123456789/334](https://dspace.imsc.res.in/xmlui/handle/123456789/334) [pdf](https://www.imsc.res.in/xmlui/bitstream/handle/123456789/334/MR120.pdf?sequence=1)$]$

* {#DMV03} G. Date, [[M.V.N. Murthy]], [[Radhika Vathsan]], *Classical and Quantum Mechanics of Anyons* $[$[arXiv:cond-mat/0302019](https://arxiv.org/abs/cond-mat/0302019)$]$

## Related entries

* [[S-matrix]]

* [[effective field theory]]

* [[condensed matter physics]]

category: people

[[!redirects R. Shankar]]


Raman Sundrum

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

## Selected writings

Introducing the [[Randall-Sundrum model]]:

* {#RandallSundrum99a} [[Lisa Randall]], [[Raman Sundrum]], _A Large Mass Hierarchy from a Small Extra Dimension_, Phys.Rev.Lett.83:3370-3373, 1999 ([arXiv:hep-ph/9905221](https://arxiv.org/abs/hep-ph/9905221))

* {#RandallSundrum99b} [[Lisa Randall]], [[Raman Sundrum]], _An Alternative to Compactification_, Phys.Rev.Lett.83:4690-4693,1999 ([arXiv:hep-th/9906064](https://arxiv.org/abs/hep-th/9906064))

category: people


Rami Grossberg

__Rami Grossberg__ is a mathematician at Carnegie Mellon University. He specialized in [[model theory]], especially the classification theory of [[elementary class|nonelementary classes]], infinitary logics, combinatorial set theory and applications to algebra. He wrote his thesis at Hebrew University, Jerusalem in 1986 supervised by [[Saharon Shelah]].

* [webpage](http://www.math.cmu.edu/~rami) with a list of publications
* Rami Grossberg, _Classification theory for abstract elementary classes_, Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics __302__, AMS, (2002) 165--204, [pdf](http://www.math.cmu.edu/~rami/Rami-NBilgi.pdf)
* Rami Grossberg, Monica VanDieren, _Galois-stability for tame abstract elementary classes_, Journal of Mathematical Logic __6__, No. 1 (2006) 25--49, [pdf](http://www.math.cmu.edu/~rami/tame.pdf)


ramification of ideals

#Contents#
* table of contents
{:toc}

## Idea

Ramification of [[ideals]] through [[ring]]/[[algebra]] [[homomorphisms]] is the [[Isbell duality|dual]] incarnation of [[branch points]] of [[branched covering spaces]].

## Definition

Given a [[ring]] [[injection]] $i \colon B \to A$ (for instance the inclusion of $\mathcal{O}_K$ in $\mathcal{O}_L$, where $K \to L$ is a [[field extension]] and $\mathcal{O}_K$ (resp. $\mathcal{O}_L$) is the [[ring of integers]] of $K$ (resp. $L$)), then a [[prime ideal]] $J \subset B$ is said to be _ramified in $A$_ if $\iota(J) A\subset A$ is not a prime ideal anymore.

Here $\iota(J)A$ will be a product of powers of [[prime ideal]]s of $A$, and the _ramification index_ of $J$ at a prime ideal of $A$ is the power with which this appears.

## Examples

* Under the [[function field analogy]] ramifications of [[algebraic number fields]] as [[field extensions]] of the [[rational numbers]] are [[analogy|analogous]] to [[branched covers of the Riemann sphere]] in [[complex geometry]].

## Related concepts

* [[Kummer's theorem]]

* [[unramified morphism]]

* [[formally unramified morphism]]

## References

* {#Snyder02} [[Noah Snyder]], section 1.5.2 of _Artin L-Functions: A Historical Approach_, 2002 ([pdf](http://www.math.columbia.edu/~nsnyder/thesismain.pdf))

[[!redirects ramified ideal]]
[[!redirects ramified ideals]]

[[!redirects ramified extension]]
[[!redirects ramified extensions]]


Ramkumar Ramachandra

* Website: [artagnon.com](https://artagnon.com)

## Related entries

* [[derived algebraic geometry]]
* [[Coq]]

category:people


Ramzi Khuri

* [Institute page Baruch college](https://www.baruch.cuny.edu/wsas/academics/natural_science/rkhuri.htm)

* [InSpire page](https://inspirehep.net/authors/1003052)

## Selected publications

On [[black branes]] and [[solitonic branes]]:

* [[Michael Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String and Fivebrane Solitons: Singular or Non-singular?*, Nucl.Phys. B **377** (1992) 281-294 [[arXiv:hep-th/9112023](https://arxiv.org/abs/hep-th/9112023), doi:10.1016/0550-3213(92)90025-7]

* {#DuffKhuriLu94} [[Mike Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String Solitons*, Phys. Rept. **259** (1995) 213-326 [[arXiv:hep-th/9412184](https://arxiv.org/abs/hep-th/9412184), doi:10.1016/0370-1573(95)00002-X]

On [[string-string duality]]:

* {#DuffKhuri93} [[Michael Duff]], [[Ramzi Khuri]], _Four-Dimensional String/String Duality_, Nucl.Phys. B411 (1994) 473-486 ([arXiv:hep-th/9305142](https://arxiv.org/abs/hep-th/9305142))

On [[M-theory on Calabi-Yau manifolds]]:

* [[Sergio Ferrara]], [[Ramzi Khuri]],  [[Ruben Minasian]], _M-theory on a Calabi-Yau manifold_, Phys. Lett.B 375:81-88, 1996 ([arXiv:hep-th/9602102](https://arxiv.org/abs/hep-th/9602102))

## Related $n$Lab entries

* [[string-string duality]]

* [[M-theory on Calabi-Yau manifolds]]

category: people

[[!redirects Ramzi R. Khuri]]


Ran space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Ran space_ $Ran X$ of a [[space]], $X$, is the space of all nonempty [[finite set|finite]] [[subsets]] of $X$, equipped with geometric structure that knows about several coincident points merging to a single point.

The Ran space is defined, in [[topology|topological]] and [[algebraic geometry|algebro-geometric]] contexts, for instance in ([Lurie 14, lectures 7-9](#LurieTamagawaNumbers)). It is naturally a [[stratified space]] ([Ayala-Francis-Tanaka 14, section 3.7](#AyalaFrancisTanaka14))

For cardinality $i$

$$
Ran_{\leq i} X = \text{colim} ( Fin_{surj,\leq i}^{op} \to Strat \to StTop )
$$

where $Fin_{surj,\leq i}$ has objects that are finite sets of cardinality at most $i$. When $i$ is infinite the subscript is left off as above.

It is named after [[Ziv Ran]].

## Properties

The Ran space of a [[connected space|connected]] [[manifold]] is [[weakly contractible space|weakly contractible]] ([BeilDrin04, p. 173](#BeilDrin04)).

## Constructions

### Algebraic case: Ran Prestack

Let $X: CAlg_k \to Spaces$ be a derived prestack. Consider the diagram in prestacks
$$ X^{(-)}: N(fSet) \to PrSt_k, I \mapsto X^I $$
where $fSet$ is the category of non-empty finite sets, with morphisms being surjective maps. The functor sends morphisms to the corresponding diagonal maps. The Ran prestack by definition is the (homotopy) colimit of this diagram:

$$Ran_X := colim_I X^I$$

## Examples

$$
  Ran_{\leq 2k+1}(S^1) \;\simeq\; S^{2k+1}
$$

([Bott 52](#Bott52), [Tuffley 02](#Tuffley02))

## Related concepts

* [[nonabelian Poincaré duality]]

* [[factorization algebra]]

## References

### General

Apparently the concept of the Ran space was first introduced in

* {#BeilDrin04} [[Alexander Beilinson]], [[Vladimir Drinfeld]] , _Chiral algebras_, American Mathematical Society Colloquium Publications *51*, (2004).

Review includes

* {#AyalaFrancisTanaka14} [[David Ayala]], [[John Francis]], [[Hiro Lee Tanaka]], section 3.7 of _Local structures on stratified spaces_, Adv. Math. 307, 903-1028 (2017) ([arXiv:1409.0501](https://arxiv.org/abs/1409.0501))

Textbook accounts include

* [[Jacob Lurie]], section 5.5.1 of _[[Higher Algebra]]_

See also

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

Ran spaces of the [[circle]] are discussed in

* {#Bott52} [[Raoul Bott]], _On the third symmetric potency of $S^1$, Fund. Math. 39 (1952), 264–268 (1953)

* {#Tuffley02} Christopher Tuffley, _Finite subsets of $S^1$_, Algebraic & Geometric Topology, Volume 2 (2002) 1119-1145 ([arXiv:math/0209077](https://arxiv.org/abs/math/0209077))

Rational functions as functions on the Ran space in the sense of functions on the complement of finitely many points as these range over all points, is discussed in

* {#Gaitsgory11} [[Dennis Gaitsgory]], _Contractibility of the space of rational maps_ ([arXiv:1108.1741](http://arxiv.org/abs/1108.1741))

Acyclicity of (some version of) the Ran space of a connected, [[quasi-projective scheme|quasi-projective]] [[scheme]] over an [[algebraically closed field]] is shown in lecture 10 of

* {#LurieTamagawaNumbers} [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincare Duality (282y)_, lecture notes, 2014, ([website](http://www.math.harvard.edu/~lurie/282y.html))

Any flat [[quasicoherent sheaf]] on the Ran space of a smooth [[algebraic variety]] over $k$ canonically acquires a [[D-module]] structure as shown in

* James Tao, _$n$-Excisive Functors, Canonical Connections, and Line Bundles on the Ran Space_ ([arXiv:1906.07976] (https://arxiv.org/abs/1906.07976))

Discussion in the context of [[conformal field theory]] includes

* [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], [[David Morrison]] and [[Edward Witten]] (eds.) volume II, part III of _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

### Relation to configuration spaces of points

Expressing the [[rational cohomology]] of [[ordered configuration spaces of points]] via [[factorization homology]] and [[Ran spaces]]:

* Quoc P. Ho, _Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras_ ([arXiv:2004.00252](https://arxiv.org/abs/2004.00252))

[[!redirects Ran spaces]]

[[!redirects ran space]]
[[!redirects ran spaces]]


Randall-Sundrum model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Randall-Sumdrum model_ ([Randall-Sundrum 99a](#RandallSundrum99a), [99b](#RandallSundrum99b)) is a class of [[string theory inspired model|string theory inspired]] [[model (physics)|models]] in combined [[cosmology]] and [[particle physics]], which assume that the [[observable universe]] constitutes the [[asymptotic boundary]] of an ambient [[anti de Sitter spacetime]]: the [[force]] of [[gravity]] would pertain to the full [[anti de Sitter spacetime|anti de sitter]] "[[bulk]]" [[spacetime]], but the [[gauge fields]] and [[fermion field|fermion]] [[matter fields]] would be constrained to reside on that boundary, as would hence be all observations made via [[electromagnetic radiation]] by [[observers]] inside this cosmology.

Hence the extra [[bulk]] [[dimensions]] in these models need not be small (technically: the [[fiber]] spaces need not be [[compact topological spaces]] with tiny [[Riemannian geometry|Riemannian]] [[volume]]) in order to be unobservable for [[observers]]. This is in contrast to the (historically much older) [[Kaluza-Klein compactification]] models for [[physics]] with extra [[dimensions]]. Therefore Randall-Sumdrum-like models are also referred to as _large extra dimension models_.

One original motivation for the RS-models was the idea that this setup could address the [[hierarchy problem]].

Something at least close to Randall-Sundrum models naturally appears in [[string phenomenology]]: In [[intersecting D-brane models]] the [[standard model of particle physics]] is part of the [[worldvolume]] [[quantum field theory]] of [[D6-branes]] [[intersection|intersected]] to a 3+1-dimensional subspace. By the general lore of [[black holes in string theory]], [[non-perturbative effect|non-perturbatively]] these [[D6-branes]] are [[black branes]] with [[anti de Sitter spacetime|anti de Sitter]] [[near horizon geometry]]. Hence the [[cosmology]] of [[intersecting D-brane models]] is qualitatively of Randall-Sundrum form, even if details may differ. For the quick idea see [Teraguchi 07, around slide 21](#Teraguchi07), for review and further pointers see [Uranga 02, section 18](#Uranga02).

For more on this see at _[[intersecting D-brane models]]_ the section _[Cosmology and Holography](intersecting+D-brane+model#Cosmology)_.

## Related concepts

* [[D=5 Yang-Mills theory]], [[D=5 Einstein-Maxwell theory]]

* [[hadron Kaluza-Klein theory]], [[hadron supersymmetry]]

  * [[flavor brane]]

* [[anti de Sitter gravity]]

* [[AdS-CFT duality]]

* [[intersecting D-brane model]]

* [[ekpyrotic cosmology]]

* [[standard model of cosmology]]

## References

### General

The model was introduced in

* {#RandallSundrum99a} [[Lisa Randall]], [[Raman Sundrum]], _A Large Mass Hierarchy from a Small Extra Dimension_, Phys.Rev.Lett.83:3370-3373, 1999 ([arXiv:hep-ph/9905221](https://arxiv.org/abs/hep-ph/9905221))

* {#RandallSundrum99b} [[Lisa Randall]], [[Raman Sundrum]], _An Alternative to Compactification_, Phys.Rev.Lett.83:4690-4693,1999 ([arXiv:hep-th/9906064](https://arxiv.org/abs/hep-th/9906064))

Different but similar approaches are

* [[Nima Arkani-Hamed]], [[Savas Dimopoulos]], [[Gia Dvali]], _The Hierarchy Problem and New Dimensions at a Millimeter_, Phys.Lett.B429:263-272 (1998) ([arXiv:hep-ph/9803315](https://arxiv.org/abs/hep-ph/9803315))

and the GRS model, due to

* {#GRS00} Ruth Gregory, Valery A. Rubakov, Sergei M. Sibiryakov, _Opening up extra dimensions at ultra-large scales_, Phys.Rev.Lett. 84 (2000) 5928-5931 ([arXiv:hep-th/0002072](https://arxiv.org/abs/hep-th/0002072))

* Csaba Csaki, Joshua Erlich, Timothy J. Hollowood, _Quasi-Localization of Gravity by Resonant Modes_, Phys. Rev. Lett.84:5932-5935, 2000 ([arXiv:hep-th/0002161](https://arxiv.org/abs/hep-th/0002161))

a well as [[intersecting D-brane models]].

More on the localized [[gravity]] in RS-models:

* {#KarchRandall00} [[Andreas Karch]], [[Lisa Randall]], *Locally Localized Gravity*, JHEP 0105:008 (2001) [[arXiv:hep-th/0011156](https://arxiv.org/abs/hep-th/0011156), [doi:10.1088/1126-6708/2001/05/008](https://doi.org/10.1088/1126-6708/2001/05/008)]

Review:

* Yoonbai Kim, Chong Oh Lee, Ilbong Lee, JungJai Lee, _Brane World of Warp Geometry: An Introductory Review_, J. Kor. Astron. Soc. 37 (2004) 1-14 ([arXiv:hep-th/0307023](https://arxiv.org/abs/hep-th/0307023))

* [[André Juan Ferreira-Martins]], *Gravity and its wonders: braneworlds and holography* ([arXiv:2105.10062](https://arxiv.org/abs/2105.10062))

Possible relation to [[flavour anomalies]] via potential [[leptoquarks]]:

* {#Crivellin18} [[Andreas Crivellin]], _Explaining the Flavour Anomalies with the Pati-Salam Vector Leptoquark_, PoS LHCP2018 (2018) 269 ([spire:1713260](http://inspirehep.net/record/1713260), [doi:10.22323/1.321.0269](https://doi.org/10.22323/1.321.0269))

* {#BlankeCrivellin18} Monika Blanke, [[Andreas Crivellin]], _$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background_, Phys. Rev. Lett. 121, 011801 (2018) ([arXiv:1801.07256](https://arxiv.org/abs/1801.07256))

Discussion of [[standard model of particle physics|standard model]]-building in [[heterotic M-theory|Horava-Witten]]-type compactifications:

* Y. H. Ahn, Sin Kyu Kang, Hyun Min Lee, *Towards a Model of Quarks and Leptons* ([arXiv:2112.13392](https://arxiv.org/abs/2112.13392))

### As intersecting brane models in string theory

Discussion of relation to [[string theory]] originates around

* {#Witten00} [[Edward Witten]], from p. 8 on in _The Cosmological Constant From The Viewpoint Of String Theory_, in _Sources and detection of dark matter and dark energy in the universe_, Springer 2001. 27-36. ([arXiv:hep-ph/0002297](https://arxiv.org/abs/hep-ph/0002297))

*  [[Steven Giddings]], [[Shamit Kachru]], [[Joseph Polchinski]], _Hierarchies from Fluxes in String Compactifications_, Phys. Rev. D66:106006, 2002 ([arXiv:hep-th/0105097](https://arxiv.org/abs/hep-th/0105097))

Review of the relation to [[string theory]]/[[intersecting D-brane models]] includes

* {#Uranga02} [[Angel Uranga]], section 18 of _TASI lectures on String Compactification, Model Building, and Fluxes_ ([arXiv:hep-th/0201209](https://arxiv.org/abs/hep-th/0201209))

* {#Teraguchi07} Shunsuke Teraguchi, around slide 21 of:  _String theory and its relation to particle physics_, 2007 ([pdf](http://phys.cts.ntu.edu.tw/ppp7/talks/PPP7_Shunsuke_Teraguchi.pdf),  [[PPP7teraguchi.pdf:file]])

based on

* {#KlebanovStrassler00} [[Igor Klebanov]], [[Matthew Strassler]], _Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi^{SB}$-Resolution of Naked Singularities_, JHEP 0008:052, 2000 ([arXiv:hep-th/0007191](https://arxiv.org/abs/hep-th/0007191))

* {#KlebanovWitten98} [[Igor Klebanov]], [[Edward Witten]], _Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity_, Nucl.Phys.B536:199-218, 1998 ([arXiv:hep-th/9807080](https://arxiv.org/abs/hep-th/9807080))

with further developments including

* {#Kaloper04} Nemanja Kaloper, _Origami World_, JHEP 0405 (2004) 061 ([arXiv:hep-th/0403208](https://arxiv.org/abs/hep-th/0403208))

* {#FlachiMinamitsuji09} Antonino Flachi, Masato Minamitsuji, _Field localization on a brane intersection in anti-de Sitter spacetime_, Phys.Rev.D79:104021, 2009 ([arXiv:0903.0133](https://arxiv.org/abs/0903.0133))

Relation to [[AdS-CFT duality]]:

* {#Soda10} Jiro Soda, _AdS/CFT on the brane_, Lect.Notes Phys.828:235-270, 2011 ([arXiv:1001.1011](https://arxiv.org/abs/1001.1011))

Review and further relation to [[black hole entropy]]:

* {#Karch22} [[Andreas Karch]], *Randall-Sundrum Branes and Holography*, talk at *[[Strings 2022]]* [[indico:4940841](https://indico.cern.ch/event/1085701/contributions/4940841)]

### Cosmology

Discussion of [[cosmology]] on branes in $\sim$ [[anti de Sitter spacetime|anti de Sitter]]-[[near horizon geometry|throat]] geometries:

* Tetsuya Shiromizu, Daisuke Ida, _Anti-de Sitter no-hair, AdS/CFT and the brane-world_, Phys. Rev. D64 (2001) 044015 ([arXiv:hep-th/0102035](https://arxiv.org/abs/hep-th/0102035))

* {#Padilla02} Antonio Padilla, _Braneworld Cosmology and Holography_ ([arxiv:hep-th/0210217](https://arxiv.org/abs/hep-th/0210217))

On  [[de Sitter spacetime]] [[cosmology]] realized in [[brane world models]] in ambient $\sim$[[anti de Sitter spacetime|AdS]]-[[bulk]] [[spacetime]]:

* {#BDDGS18} Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _Emergent de Sitter cosmology from decaying AdS_ ([arXiv:1807.01570](https://arxiv.org/abs/1807.01570))

* Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _de Sitter Cosmology on an expanding bubble_ ([arXiv:1907.04268](https://arxiv.org/abs/1907.04268))

More generally on gravity on the brane:

* A. A. Zheltukhin, _Brane mechanism of spontaneously generated gravity_ ([arXiv:1912.11520](https://arxiv.org/abs/1912.11520))

### In gauge-Higgs grand unification

RS-like models for [[gauge-Higgs unification|gauge-Higgs]] [[grand unified theories]]:

with [[Spin(11)]] ("[[SO(11)]]") [[gauge group]]:

* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([doi:10.1093/ptep/ptv153](https://doi.org/10.1093/ptep/ptv153), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))

* {#HosotaniYamatsu17} [[Yutaka Hosotani]], Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))

with [[Spin(12)]] ("[[SO(12)]]") [[gauge group]]:

* S. Rajpoot and P. Sithikong, _Implications of the $SO(12)$ gauge symmetry for grand unification_, Phys. Rev. D 23, 1649 (1981) ([doi:10.1103/PhysRevD.23.1649](https://doi.org/10.1103/PhysRevD.23.1649))

* Takaaki Nomura and Joe Sato, _Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space_, Nucl.Phys.B811:109-122, 2009 ([arXiv:0810.0898](https://arxiv.org/abs/0810.0898))

* Takaaki Nomura, _Physics beyond the standard model with $S^2$ extra-space_, 2009 ([pdf](http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf), [[NomuraSO12GUT.pdf:file]])

* Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, _Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry_ ([arXiv:1109.5835](https://arxiv.org/abs/1109.5835))

### Observation

Observational hints for brane world models:

From observation of the [[event horizon]] of the [[black hole]] in the center of the [[galaxy]] [[Messier 87]]:

* Indrani Banerjee, Sumanta Chakraborty, Soumitra SenGupta, _Silhouette of M87${}^*$: A new window to peek into the world of hidden dimensions_ ([arXiv:1909.09385](https://arxiv.org/abs/1909.09385))

Form observation of [[gravitational waves]] from [[relativistic binary]] mergers:

* Zi-Chao Lin, Hao Yu, Yu-Xiao Liu, _Constraint on the radius of five-dimensional dS spacetime with GW170817 and GRB 170817A_ ([arXiv:2001.06581](https://arxiv.org/abs/2001.06581))

[[!redirects Randall-Sundrum models]]

[[!redirects RS model]]
[[!redirects RS models]]
[[!redirects RS-model]]
[[!redirects RS-models]]

[[!redirects GRS model]]
[[!redirects GRS models]]
[[!redirects GRS-model]]
[[!redirects GRS-models]]


random matrix

#Contents#
* table of contents
{:toc}

## Idea

A __random matrix__ is a matrix-valued [[random variable]]. __Random matrix theory__ studies mainly the behaviour of eigenvalues and various functions of random matrices; as such it has large importance in physics.

## Related entries

* [[Calogero-Moser system]],

* [[large N limit]]

## References

### General

Review:

* {#Petrov19} [[Leonid Petrov]], *Random Matrices*, lecture notes 2019 ([pdf slides](https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf), [[PetrovRandomMatrices.pdf:file]], [webpage](https://lpetrov.cc/2019/07/rmt-announce))

* Madan Lal Mehta, _Random matrices_, 3rd ed. Pure and Applied Math. (Amsterdam) __142__, Elsevier/Academic Press 2004. xviii+688 pp. [MR2129906](http://www.ams.org/mathscinet-getitem?mr=2129906), [gBooks](http://books.google.com/books?id=Kp3Nx03_gMwC&lpg=PP1&pg=PP1#v=onepage&q&f=false)

* Wikipedia, _[Random matrix](http://en.wikipedia.org/wiki/Random_matrix)_

* [[Terrence Tao]], _Topics in random matrix theory_ [pdf](https://www.math.hkbu.edu.hk/~ttang/UsefulCollections/matrix-book-2011-08.pdf)

See also:

* V. L. Girko, _Theory of random determinants_,
Transl. from Russian (Višča Škola, Kiev 1980, [MR82h:60002](http://www.ams.org/mathscinet-getitem?mr=575525)) Mathematics and its Applications (Soviet Series) __45__, Kluwer 1990, [MR1080966](http://www.ams.org/mathscinet-getitem?mr=1080966)

* [[Freeman Dyson]], _Statistical theory of the energy levels of complex systems_, I, J. Math. Phys. __3__ 1962 140–156, [MR143556](http://www.ams.org/mathscinet-getitem?mr=143556), [doi](http://dx.doi.org/10.1063/1.1703773); II, JMP __3__ 1962 157–165, [MR143557](http://www.ams.org/mathscinet-getitem?mr=143557), [doi](http://dx.doi.org/10.1063/1.1703774); III,  JMP __3__ 1962 166–175, [MR143558](http://www.ams.org/mathscinet-getitem?mr=143558), [doi](http://dx.doi.org/10.1063/1.1703775); _A Brownian-motion model for the eigenvalues of a random matrix_, JMP __3__ 1962 1191–1198, [MR148397](http://www.ams.org/mathscinet-getitem?mr=148397), [doi](http://dx.doi.org/10.1063/1.1703862); _[[Fredholm determinant]]s and inverse scattering problems_, Comm. Math. Phys. 47, 171–183 (1976) [MR406201](http://www.ams.org/mathscinet-getitem?mr=406201) [euclid](http://projecteuclid.org/euclid.cmp/1103899727)

* J J M Verbaarschot, M R Zirnbauer, _Critique of the replica trick_, J. Phys. A: Math. Gen. 17 (1985) 1093-1109, [pdf](http://www.thp.uni-koeln.de/zirn/jav18i7p1093.pdf)

* Patrik L. Ferrari, _Why random matrices share universal processes with interacting particle systems?_, [arxiv/1312.1126](http://arxiv.org/abs/1312.1126)

* [[Bertrand Eynard]], [[Taro Kimura]], Sylvain Ribault, _Random matrices_, lecture notes ([arxiv/1510.04430](http://arxiv.org/abs/1510.04430))

*  Greg W. Anderson, Alice Guionnet, Ofer Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics

### In string/M-theory
  {#ReferencesStringTheory}

Random matrix theory applies to [[black holes in string theory]]:

via the [[SYK model]]:

* Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, [[Joseph Polchinski]], [[Phil Saad]], [[Stephen Shenker]], [[Douglas Stanford]], Alexandre Streicher, Masaki Tezuka, _Black Holes and Random Matrices_,  JHEP 1705:118, 2017 ([arXiv:1611.04650](https://arxiv.org/abs/1611.04650))

* Yiyang Jia, Jacobus J. M. Verbaarschot, _Spectral Fluctuations in the Sachdev-Ye-Kitaev Model_ ([arXiv:1912.11923](https://arxiv.org/abs/1912.11923))

via the [[BFSS matrix model]]:

* {#DuSahakian18} Haoxing Du, Vatche Sahakian, _Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory_ ([arXiv:1812.05020](https://arxiv.org/abs/1812.05020))

via [[AdS/CFT]] for [[Jackiw-Teitelboim gravity]]:

* {#SaadShenkerStanford19} [[Phil Saad]], [[Stephen Shenker]], [[Douglas Stanford]], _JT gravity as a matrix integral_ ([arXiv:1903.11115](https://arxiv.org/abs/1903.11115))

* {#StanfordWitten19} [[Douglas Stanford]], [[Edward Witten]], _JT Gravity and the Ensembles of Random Matrix Theory_ ([arXiv:1907.03363](https://arxiv.org/abs/1907.03363))

category: physics

[[!redirects random matrix theory]]
[[!redirects random matrices]]


random variable

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **random variable**, or _stochastic variable_, is  a quantity that is subject to 'random' variation.

Some authors in [[probability theory]] distinguish between random variables and **random elements**: a *random element* is a generic entity (such as a [[point]] of a [[set]] or [[space]]) subject to random variation, while a *random variable* is specifically a [[number]] subject to random variation.

## Definition

The formalization of this idea in modern [[probability theory]] ([Kolmogorov 33, III](#Kolmogorov33)) is to take a random variable to be a [[measurable function]] $f$ on a [[probability space]] $(X,\mathcal{F},\mu)$ (e.g. [Grigoryan 08, 3.2](#Grigoryan08), [Dembo 12, 1.2.1](#Dembo12)).

One thinks of $X$ as the space of all possible configurations (all the "[[possible worlds]]" with respect to the idealized situation under consideration), thinks of the measure $\mu(U)$ of any [[subset]] of it as the [[probability]] that one of the configurations $x \in U \subset X$ is _randomly_ realized, and thinks of $f(x)$ as the value of the given random variable in the situation of that configuration.

Often, a *random element* is a [[measurable function]] $X\to Y$, where $Y$ is a generic [[measurable space]], while the term *random variable* is reserved for the case where $Y$ is $\mathbb{R}$, $\mathbb{R}^n$, or a more general [[vector space]].
One speaks for example of *random elements of a set*, and of *real-valued* or *vector-valued random variables*.

In particular, the *random variable* tends to be used for those situation where there is a notion of [[expectation value]].
For example the [[expectation value]] of a (real-valued) random variable $f$ is the [[integral]]

$$
  \langle f \rangle \coloneqq \int_X f \cdot \mu
$$

of $f$ against the [[probability measure]] $\mu$, i.e. the average value of the random variable over all possible configuration, weighted by their [[probability]].

## Properties

### Relation to type theoretic constructions

There is at least some similarity of the concept of random variables to usage of the [[function monad]] ("[[reader monad]]") in the context of [[monad (in computer science)|monads in computer science]].

In [Verdier 14](#Verdier14) it says:

> The intuition behind the Reader monad, for a mathematician, is perhaps stochastic variables. A stochastic variable is a function from a probability space to some other space. So we see a stochastic variable as a monadic value.

and in [Toronto-McCarthy 10b, slide 35](#TorontoMcCarthy10b):

> you could interpret this by regarding random variables as reader monad computations.

See also ([Toronto-McCarthy 10b, slide 24](#TorontoMcCarthy10b)). [Toronto-McCarthy 10a, 2.2](#TorontoMcCarthy10a), [Toronto 14](#Toronto14) call the [[function monad]] the _random variable idiom_.

### Random variables and Dedekind reals

Given a measure space $(X,\Sigma,\mu)$, a _random variable_ is also often  defined as an _equivalence class_ of measurable _real_-valued functions on $X$ where two such functions are identified when they differ only on a subset of measure zero.

In this context, it has been observed by [[Pierre Deligne|P. Deligne]][^Del] that the po-set of measurable subsets $\Sigma$ can be equipped with a suitable [[Grothendieck topology]].

[^Del]:  [[SGA4]].I, p.412.

In the resulting [[Grothendieck topos]] $Meas(X,\Sigma,\mu)$ the object of [[Dedekind real numbers]] $R_D$ corresponds to the _sheaf of random variables_ on $X$ in this sense.

A Dedekind real in a topos $Sh(X)$ of sheaves on a _topological space_ is just a continuous real-valued function on $X$. This suggests the view that the sheaf-theoretic perspective on $(X,\Sigma,\mu)$ sweeps the measure-theoretic details under the rug and brings out the conceptual essence of a random variable as simply a real-valued 'function' or 'variable real number' on $X$ and goes in the same direction as the connection to the function monad mentioned in the previous section.

The details of this example, due to [[Dana Scott|D. Scott]], are described in ([Johnstone 1977](#Johnstone77), p.213).

## Related concepts

* [[probability theory]]

* [[probability monad]]

## References

* Wikipedia, _[Random variable](http://en.wikipedia.org/wiki/Random_variable)_

The modern formal concept originates around

* {#Kolmogorov33} [[Andrey Kolmogorov]], _Grundbegriffe der Wahrscheinlichkeitsrechnung_, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Berlin Heidelberg 1933.

Surveys and lecture notes include

* {#Grigoryan08} Alexander Grigoryan, _Measure theory and probability_, 2008. ([pdf](https://www.math.uni-bielefeld.de/~grigor/mwlect.pdf))

* {#Dembo12} [[Amir Dembo]], _Probability theory_, 2012. ([pdf](http://statweb.stanford.edu/~adembo/stat-310a/lnotes.pdf))

For more information on the above topos-theoretic example consult

* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014)

Discussion from a point of view of [[type theory]]/[[computer science]] includes

* {#TorontoMcCarthy10a} [[Neil Toronto]], [[Jay McCarthy]], _From Bayesian notation to pure Racket, via measure-theoretic probability_, in _Implementation and Application of Functional Languages_, 2010. ([article](https://link.springer.com/chapter/10.1007/978-3-642-24276-2_6))

* {#TorontoMcCarthy10b} [[Neil Toronto]], [[Jay McCarthy]], _From Bayesian Notation to Pure Racket_, talk notes 2010. ([pdf](http://jeapostrophe.github.io/home/static/toronto-2010ifl-slides.pdf))

* {#Toronto14} [[Neil Toronto]], _Useful Languages for Probabilistic Modeling and Inference_, PhD Thesis, 2014. ([pdf](http://cs.umd.edu/~ntoronto/papers/toronto-2014diss.pdf), [slides](http://cs.umd.edu/~ntoronto/papers/toronto-2014diss-slides.pdf))

* {#Verdier14} [[Olivier Verdier]], _[The Reader and Writer Monads and Comonads](http://www.olivierverdier.com/posts/2014/12/31/reader-writer-monad-comonad/)_, 2014.

category: probability

[[!redirects random variables]]

[[!redirects random element]]
[[!redirects random elements]]

[[!redirects stochastic variable]]
[[!redirects stochastic variables]]


Randy McCarthy

* [webpage](http://www.math.illinois.edu/~randy/)

## Selected writings

On [[algebraic K-theory]]:

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], *The local structure of algebraic K-theory*, Springer (2013) [[doi:10.1007/978-1-4471-4393-2](https://doi.org/10.1007/978-1-4471-4393-2)]

category: people


Randy Pollack

## Selected writings

On [[mathematical structures]] formulated in [[dependent type theory]], specifically in [[Coq]]:

* [[Herman Geuvers]], [[Randy Pollack]], [[Freek Wiedijk]], [[Jan Zwanenburg]], *A Constructive Algebraic Hierarchy in Coq*, Journal of Symbolic Computation **34** 4 (2002) 271-286 [[doi:10.1006/jsco.2002.0552](https://doi.org/10.1006/jsco.2002.0552), [pdf](http://www.cs.ru.nl/~herman/PUBS/JSC2002-GeuversPollackWiedijkZwanenburg-alghier1.pdf)]

## Related

* [LEGO](https://www.dcs.ed.ac.uk/home/lego/) [[proof assistant]]

category: people


Ranee K. Brylinski

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

## Selected writings

On [[quantum computation]]:

* [[Ranee K. Brylinski]], [[Goong Chen]] (eds.), *Mathematics of Quantum Computation*, Chapman and Hall/CRC 2002 ([doi:10.1201/9781420035377](https://doi.org/10.1201/9781420035377))

category: people

[[!redirects Ranee Brylinski]]


range

Classically, the __range__ of a [[function]] $f$ with [[domain]] $A$ is the [[set]] $\{f(x) \;|\; x \in A\}$ (whose existence, in [[material set theory]], is given by the [[axiom of replacement]]).  As we came to realise that a function should be given with a [[codomain]] (which is automatic in [[structural set theory]]), the term 'range' generalised in two ways:

*  as the [[codomain]] itself, so that the earlier terminology is then preserved only for [[surjections]];
*  as the [[image]]
   $$ \{y \colon B \;|\; \exists x\colon A,\; y = f(x)\} $$
   (whose existence, in axiomatic set theory, is given by the much weaker axiom of [[bounded separation]]) of $f\colon A \to B$.

The former generalisation was historically common (and is sometimes still used) in [[groupoid]] theory; the latter is what we usually mean today.

Note that the axiom of replacement is still needed for a function (such as a [[family of sets]]) whose codomain is a [[proper class]], to prove that its image is small when its domain is small.

[[!redirects range]]
[[!redirects ranges]]

[[!redirects range of a function]]
[[!redirects ranges of functions]]


rank

# Rank
* table of contents
{: toc}

## Idea

The term 'rank' is used in many contexts to number levels within a hierarchy.

## Rank of a module

Let $A$ be a [[ring]] and $N$ a [[module]] over $A$. If $A$ is a [[field]], then $N$ is a [[vector space]] and we speak of the _[[dimension]]_ of $N$; in the general case, we may speak of the _rank_:

A collection of elements $(w_i)_{i \in I}$ of $N$ is called a [[basis]] of $N$ (over $A$) if for every $x \in N$ there is a unique collection $(a_i)_{i \in I}$ of elements of $A$ such that $a_i = 0$ for all but finitely many $i \in I$ and $x = \sum_{i \in I} a_i w_i$.

If $N$ has a basis it is called a _[[nLab:free module]]_ (over $A$). For many examples of $A$ (the __invariant basis number rings__), the [[cardinality]] $# I$ only depends on $N$ and not on the choice of basis. It is called the **rank** of $N$ over $A$, notation: $rank_A(M)$. In any case, $N$ is called the __free module of rank $# I$__. If $N$ is a [[finitely generated]] free module then the rank is a [[finite number]].

All of the following are invariant basis rings (source: [Wikipedia](http://en.wikipedia.org/wiki/Invariant_basis_number)):

*  any nontrivial [[commutative ring]] $K$,

*  the [[group ring]] $K(G)$ for $K$ any [[field]] (or nontrival commutative ring?) and $G$ any [[group]],

*  any [[Noetherian ring]].

Besides the [[trivial ring]] (over which any module is free with any set as basis), an example of a ring without invariant basis number is the ring of $\aleph_0$-dimensional square [[matrix|matrices]] (over any ring) in which each column has only finitely many nonzero entries (which allows multiplication to be defined).  As a module over itself, this ring is free on any inhabited finite set, as may be shown by using the equation $\aleph_0 = n \aleph_0$ (applied to the columns).

## Rank of a linear map
 {#RankOfALinearMap}

Given a [[linear map]], hence a [[homomorphism]] of [[modules]], its rank is the rank of its [[image]]-module.

Often this is considered for the case that the linear map is represented by a [[matrix]] and one speaks of the *rank of a matrix*.

## Rank of a sheaf of modules

Let $(X,\mathcal{O})$ be a [[locally ringed space]] and $\mathcal{E}$ a $\mathcal{O}$-module. Then its _rank_ at a point $x \in X$ is the vector space dimension of the [[fiber]] $\mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x)$ over the [[residue field]] $k(x)$.

If $\mathcal{E}$ is of [[coherent sheaf|finite type]], then the rank at $x$ can equivalently be defined as the minimal number of elements needed to generate the stalk $\mathcal{E}_x$ as a $\mathcal{O}_x$-module (by [[Nakayama's lemma]]). In this case, the rank is a [[semicontinuous map|upper semicontinuous]] function $X \to \mathbb{N}$.

In the [[internal logic|internal language]] of the sheaf topos $\mathrm{Sh}(X)$, the rank of $\mathcal{E}$ can internally quite simply be defined as the minimal number of elements needed to generate $\mathcal{E}$ (taken as an element of the suitably completed natural numbers, i.e. the poset of inhabited upper sets). Under the correspondence of internal inhabited upper sets in $\mathrm{Sh}(X)$ and upper semicontinuous functions $X \to \mathbb{N}$ (details at _[[one-sided real number]]_), this definition coincides with the usual one if $\mathcal{E}$ is of finite type; see [this MathOverflow question](http://mathoverflow.net/questions/34412/the-upper-semi-continuous-rank-of-a-module-sheaf).

See also _[[rank of a coherent sheaf]]_.

## Rank of a vector bundle

As a simple special case of the above, a [[vector bundle]] is said to have _rank $n$_ if each [[fiber]] is a [[vector space]] of [[dimension]] $n$.

## Hereditary rank of a pure set

Every [[pure set]] within the [[von Neumann hierarchy]] appears first at some level given by an [[ordinal number]]; this number is its __hereditary rank__.

We may define this rank explicitly (and [[recursion|recursively]]) as follows:

$$ rank S = \bigcup_{x \in S} (rank x)^+ ,$$

where $\bigcup$ is the [[supremum]] operation on ordinals (literally the [[union]] for [[von Neumann ordinals]]) and $(-)^+$ is the [[successor]] operation (which is $a \mapsto a \cup \{a\}$ for von Neumann ordinals).

## Rank of a functor

Recall that a [[cardinal number]] $\alpha$ is said to be _regular_ if $|\bigcup_{i\in I} X_i |$<$\alpha$ whenever $|I|$<$\alpha$ and $|X_i|$<$\alpha$ for all $i\in I$.

A functor $F:\mathcal{A}\to \mathcal{B}$  _has rank_ $\alpha$ for some regular cardinal $\alpha$ if $F$ preserves $\alpha$-[[filtered colimit|filtered colimits]]. $F$ _has rank_ when it has rank $\alpha$ for some regular cardinal $\alpha$. A monad has rank ($\alpha$) when its underlying endofunctor does.

The properties of functors with rank are discussed in section 5.5 of Borceux ([1994](#Borceux2)).

## Rank of a Lie group

* [[rank-nullity theorem]]

* [[rank of a Lie group]]

* [[rank of a coherent sheaf]],

  [[degree of a coherent sheaf]]

  [[slope of a coherent sheaf]]

* [[stable coherent sheaf]], [[Bridgeland stability condition]]

## Related concepts

* [[rank-nullity theorem]]

## References

* {#Borceux2}[[Francis Borceux]], _[[Handbook of Categorical Algebra]] vol. 2_ , Cambridge UP 1994.

[[!redirects ranks]]

[[!redirects rank of a vector bundle]]
[[!redirects rank of vector bundles]]

[[!redirects monad rank]]
[[!redirects monad with rank]]

[[!redirects rank of a linear map]]
[[!redirects ranks of linear maps]]

[[!redirects rank of a matrix]]
[[!redirects ranks of matrices]]


rank of a coherent sheaf

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## Idea

As the concept of _[[coherent sheaf]]_ is a generalization of that of ([[complex vector bundle|complex]], [[algebraic vector bundle]]) [[vector bundle]], the concept of _rank_ of a coherent sheaf is the corresponding generalization of that of _[[rank of a vector bundle]]_.

## Definition

e.g. ([Huybrechts-Lehn 96, def. 1.2.2](#HuybrechtsLehn96))

## Related concepts

* [[degree of a coherent sheaf]]

* [[slope of a coherent sheaf]]

* [[stable coherent sheaf]], [[Bridgeland stability condition]]

## References

This appeared in the Grothendieck's FGA in order to treat the moduli problems leading to [[Quot scheme]]s. See Nitsure's contribution at [[FGA explained]] for a modern remake.

* {#HuybrechtsLehn96} [[Daniel Huybrechts]], [[Manfred Lehn]], _The Geometry of the Moduli Spaces of Sheaves_, 1996 ([[HuybrechtsLehn.pdf:file]])


rank of a Lie group

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## Definition

The _rank_ of a [[Lie group]] is the [[dimension]] of any one of its [[Cartan subgroups]], hence equivalently the dimension of any one of the [[Cartan subalgebras]] of its [[Lie algebra]].

For [[connected topological space|connected]] [[compact Lie groups]], a Cartan subgroup is a [[maximal torus]], and hence in this case the rank of the Lie group is the dimension of any one of its maximal tori.

## Related concepts

* [[Young tableau]]

## References

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

[[!redirects ranks of Lie groups]]


rank-nullity theorem

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## Idea

In [[linear algebra]], what is known as the *rank-nullity theorem* (e.g. 3.22 in [Axler (2015)](#Axler15), who calls it the *fundamental theorem of linear maps*) is the statement that for any [[linear map]] $f \colon V \to W$ out of a [[finite-dimensional vector space]], the [[sum]] of

* the [[rank of a linear map|rank]] $rk(f) \coloneqq dim\big(im(f)\big)$, i.e. the [[dimension of a vector space|dimension]] of the [[image]];

with

* the *nullity* $nl(f) \coloneqq dim\big( ker(f) \big)$, i.e. the [[dimension of a vector space|dimension]] of the [[kernel]]

equals

* the [[dimension]] $d_V \coloneqq dim(V)$ of the [[domain]] space $V$:

$$
  d_V \;=\; rk(f) + nl(f)
  \,.
$$

This rank-nullity theorem is the [[decategorification]] (under the [[dimension of a vector space|dimension]] [[functor]] $dim \colon FinDimVect \to \mathbb{Z}$) of the stronger statement that $V$ itself is the [[direct sum]] of its [[kernel]] and [[image]] [[vector spaces]]:

$$
  V \;\simeq\; im(f) \oplus ker(f)
  \,.
$$

This may be understood as an instance of the [[splitting lemma]] for [[vector spaces]], or more precisely of the statement ([here](split+exact+sequence#OfVectorSpaces)) that every [[short exact sequence]] of [[vector spaces]], such as

$$
  0 \to ker(f) \longrightarrow V \longrightarrow im(f) \to 0
$$

is a [[split exact sequence]], hence of the form

$$
  0 \to ker(f) \longrightarrow ker(f) \oplus im(f)  \longrightarrow im(f) \to 0
  \,.
$$

## References

Textbook accounts:

* {#Axler15} Sheldon Axler, 3.22 in: *Linear Algebra Done Right*, Springer (2015) [[doi:10.1007/978-3-319-11080-6](https://doi.org/10.1007/978-3-319-11080-6)]

A [[formal proof]] of the rank-nullity theorem in the [[Isabelle]] [[proof assistant]]:

* [Archive of Formal Proof](https://www.isa-afp.org/), *[Rank-Nullity Theorem in Linear Algebra](https://www.isa-afp.org/entries/Rank_Nullity_Theorem.html)*

See also:

* Wikipedia, *[Rank-nullity theorem](https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem)*

[[!redirects fundamental theorem of linear maps]]


Rankin-Cohen bracket

## Overview and definition

Let $\Gamma$ be a congruence subgroup of $SL(2,\mathbf{Z})$ and $\mathcal{M}(\Gamma)$ the graded (by the weight) algebra of modular forms with respect to $\Gamma$. All bidifferential operators which leave that space invariant are linear combinations of Rankin-Cohen brackets $[-,-]_n\colon(f, g)\mapsto [f,g]_n$. By definition, $n$-th bracket between elements $f\in\mathcal{M}(\Gamma)_{2k}$ and $g\in\mathcal{M}(\Gamma)_{2l}$ is given by the formula
$$
[f,g]_n := \sum_{r=0}^n (-1)^r\binom{n+2k-1}{n-r}\binom{n+2l-1}{r}f^{(r)}g^{(n-r)}\,\in \,\mathcal{M}(\Gamma)_{2k+2l+2n}
$$
where $f^{(r)} := \left(\frac{1}{2\pi i}\frac{\partial}{\partial z}\right)^r f$.
They are directly related to invariant differential operators used to produce new $sl(2)$-invariant bilinear forms from old ones, so
called [[transvectant]]s found by Gordan,

* P. Gordan, _Das Zerfallen der Curven in gerade Linien_, Math. Ann., 45 (1894), pp. 411-427 [doi](https://doi.org/10.1007/BF01446687)

Moscovici and Connes have constructed a sequence of Hopf algebras $\mathcal{H}_q$ related to geometry of [[foliation]]s. Hopf algebra $\mathcal{H}_q$ has deformations which may be given by universal deformation formulas, or in other words, by [[Drinfeld twist]]s which are power series in formal variable with unit free term. These 2-cocycles have the structure appearing in Rankin-Cohen brackets and are called Rankin-Cohen deformations and are akin in structure to what is in quantum group context known as [[Jordanian twist]]s, coming from the work of Gurevich on (generalized) Jordanian R-matrices, and of Ogievetsky, Coll-Gerstenhaber-Giaquinto and in a closer symmetrized form by Giaquinto and Zhang. An isomorphism between (reduced) Rankin-Cohen deformation and Jordanian deformation has been exhibited by Samsonov.

## Literature

* R. A. Rankin, _The construction of automorphic forms from the derivatives of a given form_, J. Indian Math.Soc., (N.S.) 20 (1956) 103–116
* H. Cohen, _Sums involving the values at negative integers of L-functions of quadratic characters_, Math.Ann. 217 (1975) 271-285
* P. Cohen, Yu. Manin, D. Zagier, _Automorphic pseudodifferential operators_, Algebraic aspects of integrable systems, 17–47, Progr. Nonlinear Differential Equations Appl.26, Birkauser 1997
* [[Alain Connes]], [[Henri Moscovici]], _Rankin-Cohen brackets and the Hopf algebra of transverse geometry_, Mosc. Math. J. __4__:1 (2004) 111–130
* R. Rochberg, X. Tang, Y. Yao, _A survey of Rankin-Cohen deformations_, Perspectives on Noncommutative Geometry, Fields Inst. Commun. 61:7 (2011) [arXiv:0909.4364](https://arxiv.org/abs/0909.4364)
* P. Bieliavsky, X. Tang, Y. Yao, _Rankin–Cohen brackets and formal quantization_, Adv. Math. __212__:1 (2007) 293-314 [doi](https://doi.org/10.1016/j.aim.2006.10.007)
* M. Samsonov, _Quantization of semi-classical twists and noncommutative geometry_, Lett. Math. Phys. __75__ (2006) 63–77 [doi](https://doi.org/10.1007/s11005-005-0038-2)
* Y.–J. Choie, W. Eholzer, _Rankin–Cohen operators for Jacobi and Siegel forms_, J. Number Theory 68 (1998) 160–177


Rankin-Selberg method

[[!redirects Rankin-Selberg-Zagier method]]

## Idea

A method for constructing [[automorphic L-functions]].

## References

* Wikipedia, _[Rankin-Selberg method](https://en.wikipedia.org/wiki/Rankin%E2%80%93Selberg_method)_

* {#Selberg40} [[Atle Selberg]], _Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist_, Arch. Math. Naturvid. 43: 47–50, (1940)  MR 0002626

* Rankin, Robert A. (1939), "Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions. I. The zeros of the function Σn=1∞τ(n)/ns on the line R s=13/2. II. The order of the Fourier coefficients of integral modular forms", Proc. Cambridge Philos. Soc. 35: 351–372, doi:10.1017/S0305004100021095, MR 0000411


Raoul Bott

**Raoul Bott** (1923--2005) was one of the great 20th century topologists and geometers. Among his famous works, one should mention the [[Bott periodicity theorem]] (of importance in [[K-theory]]), studies in [[Morse theory]] (including the study of Bott--Morse functions), the Borel--Weil--Bott theorem in geometric representation theory, the study of fixed point (localization) formulas (the Atiyah--Bott fixed point theorem) and the Atiyah-Bott-Patodi slick proof of the index theorem via the heat kernel expansion.

* [Wikipedia entry](http://en.wikipedia.org/wiki/Raoul_Bott)

* [[Loring Tu]]: *[The life and works of Raoul Bott](https://legacy-www.math.harvard.edu/history/bott/bottbio/index.html)* (2002)

* [[Michael Atiyah]]: *[Raoul Harry Bott: 24 September 1923 – 20 December 2005](https://celebratio.org/Bott_R/article/296/)*, Celebratio Mathematica (2007)

## Selected writings

Introducing the concept and terminology of the [[Pontrjagin product]] on the [[ordinary homology]] of [[based loop spaces]]:

* [[Raoul Bott]], [[Hans Samelson]], *On the Pontryagin product in spaces of paths*, Commentarii Mathematici Helvetici **27** (1953) 320–337 [[doi:10.1007/BF02564566](https://doi.org/10.1007/BF02564566)]

and application to the case of [[Lie groups]]:

* [[Raoul Bott]], *The space of loops on a Lie group*, Michigan Math. J. **5** 1 (1958) 35-61 [[doi:10.1307/mmj/1028998010](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-5/issue-1/The-space-of-loops-on-a-Lie-group/10.1307/mmj/1028998010.full)]

Introducing [[Bott periodicity]]:

* [[Raoul Bott]], *The Stable Homotopy of the Classical Groups*, Proceedings of the National Academy of Sciences of the United States of America **43** 10 (1957) 933-935 [[jstor:89403](https://www.jstor.org/stable/89403)]

Introducing the [[Atiyah-Bott-Shapiro orientation]] [[MSpin]]$\to$[[KO]] and [[MSpin^c|MSpinc]]$\to$[[KU]]:

* {#AtiyahBottShapiro64} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (doi:10.1016/0040-9383(64)90003-5, [pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))

On [[topological K-theory]]:

* [[Raoul Bott]], *Lectures on $K(X)$*, Benjamin (1969) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/bottk.pdf), [[Bott-KTheory.pdf:file]]]

  Russian transl. by B. Yu. Sternin, Matematika **11** 2 (1967) 32–56 [[mathnet:mat424](https://www.mathnet.ru/eng/mat424)]

On the [[Chern-Weil homomorphism]]:

* [[Raoul Bott]], _On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups_, Advances in Mathematics Volume 11, Issue 3, December 1973, Pages 289-303 (doi:10.1016/0001-8708(73)90012-1)

On [[differential forms]] in [[algebraic topology]]:

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], _Differential forms in algebraic topology_, Graduate Texts in Mathematics __82__, Springer 1982. xiv+331 pp. ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf))

On the [[simplicial de Rham complex]] and [[equivariant de Rham cohomology]]:

* {#BottShulmanStasheff76} [[Raoul Bott]], [[Herbert Shulman]], [[Jim Stasheff]], _On the de Rham theory of certain classifying spaces_, Advances in Mathematics, Volume 20, Issue 1, April 1976, Pages 43-56 (doi:10.1016/0001-8708(76)90169-9, [pdf](https://core.ac.uk/download/pdf/82496263.pdf))

On the [[rigidity theorem for elliptic genera]]:

* [[Raoul Bott]], [[Clifford Taubes]], _On the Rigidity Theorems of Witten_, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 ([doi:10.2307/1990915](https://doi.org/10.2307/1990915))

* [[Raoul Bott]], _On the Fixed Point Formula and the Rigidity Theorems of Witten_, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) _Nonperturbative Quantum Field Theory_. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) ([doi:10.1007/978-1-4613-0729-7_2](https://doi.org/10.1007/978-1-4613-0729-7_2))

category: people

[[!redirects R. Bott]]
[[!redirects Bott]]


Raphael Bousso

* [webpage](http://www.physics.berkeley.edu/research/faculty/bousso.html)

category: people


Raphael Flauger

* [webpage](http://www.ias.edu/people/cos/users/10685)

category: people


Raphael Zahler

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6624)

## Selected writings

On the [[Adams-Novikov spectral sequence]]:

* {#Zahler72} [[Raphael Zahler]], _The Adams-Novikov Spectral Sequence for the Spheres_, Annals of Mathematics Second Series, Vol. 96, No. 3 (Nov., 1972), pp. 480-504 ([doi:10.2307/1970821](https://doi.org/10.2307/1970821))

category: people


Raphaël Rouquier

__Raphaël Rouquier__ is a professor of mathematics at UCLA with principal interests in representation and Lie theory, [[categorification in Lie theory]] and homological algebra, especially including triangulated (especially of the type of the derived categories of coherent sheaves).

* [homepage](http://people.maths.ox.ac.uk/rouquier),

*  [wikipedia](http://en.wikipedia.org/wiki/Rapha%C3%ABl_Rouquier)

category: people
[[!redirects Raphaël Rouquier]]
[[!redirects R. Rouquier]]


rapidity

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## Idea

For [[vectors]] in [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ (regarded as the [[inner product space]] $(\mathbb{R}^{p+1}, \eta)$ for $\eta$ the [[Minkowski metric]]) _rapidity_ is the analog in [[Lorentzian geometry]] of what in [[Euclidean geometry]] would be the _[[angle]]_ to the [[time]]-axis. If vectors in $\mathbb{R}^{p,1}$ are thought of as [[tangent vectors]] to [[trajectories]] in [[Minkowski spacetime]], their rapidity is a measure for the _[[velocity]]_ of the trajectory, which is well-adapted to [[Lorentz transformations]].

## Definition

Let $x = (x^0, \vert x) \in \mathbb{R}^{p,1}$ be a [[vector]] with [[Minkowski metric|Minkowski norm]]-square

$$
  -{\vert x\vert}_\eta^2 \coloneqq (x^0)^2 - {\vert \vec x\vert}^2 \;\geq 0\;
  \,.
$$

and

$$
  x^0 \geq 0
  \,.
$$

Them its _rapidity_ is the unique $z \in \mathbb{R}$ such that

$$
  x^0 = \sqrt{-{\vert x\vert}^2_\eta} \, \cosh(z)
$$

and

$$
  {\vert \vec x\vert} = \sqrt{-{\vert x\vert}^2} \, \sinh(z)
$$

where $\cosh, \sinh$ denote the [[hyperbolic cosine]] and [[hyperbolic sine]], respectively.

In contrast, the [[velocity]] of $x$ is the product of the [[speed of light]] $c$ with the [[hyperbolic tangent]] of the rapidity:

$$
  \begin{aligned}
    v
    & \coloneqq
    \frac{ { \vert \vec x\vert} }{t}
    \\
    & =
    c \frac{ {\vert \vec x\vert} }{x^0}
    \\
    & =
    c \frac{ \sinh(z) }{ \cosh(z)}
    \\
    & =
    c \tanh(z)
  \end{aligned}
$$

## Related concepts

* [[velocity]], [[speed]]

* [[special relativity]]

* [singular support of causal propagator](causal+propagator#SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone)

## References

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

[[!redirects rapidities]]


rapidly decreasing function

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## Idea

A [[function]] on a [[Cartesian space]] is called _rapidly decreasing_ if every the product with any power of the canonical [[coordinate functions]] is a [[bounded function]] (def. \ref{RapidlyDecreasingFunction} below).

Of particular interest are the [[smooth functions]] with are rapidly decreasing and all whose [[partial derivatives]] are rapidly decreasing, too (def. \ref{FunctionsWithRapidlyDecreasingPartialDerivatives} below). These _[[Schwartz functions]]_ enjoy the special property that the operation of [[Fourier transform]] is an [[endomorphism]] on the _[[Schwartz space]]_ of all these functions. This gives them a central place in [[harmonic analysis]]. A _[[tempered distribution]]_ is a [[continuous linear functional]] on this [[Schwartz space]].

## Definition

### Rapidly decreasing functions

+-- {: .num_defn #RapidlyDecreasingFunction}
###### Definition
**(rapidly decreasing function)**

For $n \in \mathbb{N}$, an _[[integrable function]]_

$$
  f \colon \mathbb{R}^n \to \mathbb{R}
$$

on a [[Cartesian space]] $\mathbb{R}^n$ is called _rapidly decreasing_ if for all $\alpha \in \mathbb{N}^n$ the product function

$$
  x \mapsto x^\alpha f(x)
$$

is a [[bounded function]]. Here

$$
  x^\alpha \;\coloneqq\; (x^1)^{\alpha_1} \cdots (x^n)^{\alpha_n}
$$

is any given product of powers of the canonical [[coordinate functions]].

=--

### Functions with rapidly decreasing partial derivatives

+-- {: .num_defn #FunctionsWithRapidlyDecreasingPartialDerivatives}
###### Definition
**(function with rapidly decreasing [[partial derivatives]])**

A [[smooth function]]

$$
  f \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}
$$

is a _function with rapidly decreasing partial derivatives_ or _Schwartz function_ for short, if all its [[partial derivatives]] are rapidly decreasing functions in the sense of def. \ref{RapidlyDecreasingFunction}, hence if for all $\alphe, \beta \in \mathbb{N}^n$ we have that

$$
  x \mapsto x^\alpha \partial_{\beta} f
$$

is a [[bounded function]], where

$$
  \partial_\beta
  \;\coloneqq\;
  \frac{\partial^{\beta_1}}{\partial (x^1)^{\beta_1}}
  \cdots
  \frac{\partial^{\beta_n}}{\partial (x^n)^{\beta_n}}
$$

are the given [[partial derivatives]].

=--

+-- {: .num_prop #Smooth0TypeIsSheavesOnSmoothMfd}
###### Remark

These Schwartz functions (def. \ref{FunctionsWithRapidlyDecreasingPartialDerivatives}) form the _[[Schwartz space]]_ $\mathcal{S}(\mathbb{R}^n)$ used in the definition of [[tempered distributions]] in [[functional analysis]].

=--

## Properties

+-- {: .num_prop #RapidlyDecreasingFunctionsAreIntegrable}
###### Proposition
**([[rapidly decreasing functions]] are [[integrable functions]])**

If $f \colon \mathbb{R}^n \longrightarrow \mathbb{R}$ is a [[rapidly decreasing function]], then its [[integral]] exists

$$
  \int_{x \in \mathbb{R}^n} f(x) \, dvol(x)
   \;\lt \;
   \infty
  \,.
$$

In fact for all $\alpha \in \mathbb{N}^n$ the integral of $x \mapsto x^\alpha f(x)$ exists:

$$
  \int_{x \in \mathbb{R}^n} x^\alpha f(x) \, dvol(x)
   \;\lt \;
   \infty
  \,.
$$

=--

## Examples

+-- {: .num_example #CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}
###### Example
**([[compactly supported function|compactly supported]] [[smooth funtions]] are [[functions with rapidly decreasing partial derivatives]])**

Every [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R}^n)$ is rapidly decreasing (def. \ref{RapidlyDecreasingFunction}) has rapidly decreasing partial derivatives (def. \ref{FunctionsWithRapidlyDecreasingPartialDerivatives}):

$$
  C^\infty(\mathbb{R}^n)
   \hookrightarrow
  \mathcal{S}(\mathbb{R}^n)
  \,.
$$

=--

+-- {: .num_example}
###### (Non-)Example

Not every rapidly decreasing function (def. \ref{RapidlyDecreasingFunction}) has rapidly decreasing partial derivatives (def. \ref{FunctionsWithRapidlyDecreasingPartialDerivatives}).

For example the function $f \colon \mathbb{R} \to \mathbb{R}$

$$
  f(x) \coloneqq e^{-x^2}\sin(e^{x^2})
$$

is rapidly decreasing, but its first [[derivative]]

$$
  f'(x) = -2xe^{-x^2}\sin(e^{x^2})+2x\cos(e^{x^2})
$$

is asymptotically linearly increasing, due to the second term.

=--

## Related concepts

* [[continuous function]], [[differentiable function]], [[smooth function]]

* [[integrable function]], [[square-integrable function]]

* [[bounded function]]

* [[compactly supported function]]

* [[measurable function]]

[[!redirects rapidly decreasing functions]]

[[!redirects rapidly decaying function]]
[[!redirects rapidly decaying functions]]

[[!redirects function with rapidly decreasing partial derivatives]]
[[!redirects functions with rapidly decreasing partial derivatives]]

[[!redirects function with rapidly decaying partial derivatives]]
[[!redirects functions with rapidly decaying partial derivatives]]

[[!redirects Schwartz function]]
[[!redirects Schwartz functions]]