matrix group

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#### Group Theory
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#### Linear algebra
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[[!include higher linear algebra - contents]]
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#Contents#
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## Idea

A [[group]] of [[invertible matrix|invertible]] [[matrices]] under [[matrix multiplication]], hence a [[subgroup]] of a [[general linear group]].

Often: _[[matrix Lie group]]_.

For instance: [[orthogonal group]], [[unitary group]].

## Related concepts

* [[matrix Lie algebra]]

[[!redirects matrix groups]]


matrix Hopf algebra

Let $B$ be a [[bialgebra]], possibly noncommutative,
over a field $k$ and
$G = (g^i_j)^{i = 1,\ldots, n}_{j = 1,\ldots, n}$
an $n\times n$-matrix over $B$.
$B$ is a **matrix bialgebra** with basis $G$ if

* the set of entries of $G$ generates $B$ and

* the comultiplication $\Delta$ and counit $\epsilon$ satisfy
the matrix equations $\Delta G = G \otimes G$ (i.e. in components
$\Delta g^i_j = \sum_{k = 1}^n g^i_k \otimes g^k_j$)
and $\epsilon G = 1$ (reading in components $\epsilon (g^i_j) = \delta^i_j$).

According to a result of Redford every finite-dimensional Hopf algebra over a [[field]] is a matrix Hopf algebra with respect to some basis.

The free (noncommutative) associative algebra $F$
on $n^2$ generators $f^i_j$
has a unique coalgebra structure making it a
matrix bialgebra with basis $(f^i_j)^{i = 1,\ldots, n}_{j = 1,\ldots, n}$.
We call it the **free matrix bialgebra of rank $n^2$**.
Every bialgebra quotient of that bialgebra is a matrix bialgebra.

A **matrix Hopf algebra** $\mathcal{H}$ with basis $T = (t^i_j)$ is a Hopf algebra which possess a matrix subbialgebra $B$ with basis $T$ such that the map $H(id_B):H(B)\rightarrow\mathcal{H}$
is onto (where $H(B)$ denotes the [[Hopf envelope]] of $B$ and $H$ is understood as a [[functor]]).

A matrix Hopf algebra $\mathcal{H}$ with basis $T$ is often not a matrix bialgebra with basis $T$: e.g. the commutative
coordinate ring of $GL(n,k)$ is not a matrix bialgebra
with respect to the obvious basis $T$; in this example this can be repaired by enlarging the basis by one group-like element: the inverse of the determinant. On the other hand, the coordinate algebra of the special linear group $O(SL(n,k))$ is a matrix bialgebra and a matrix Hopf algebra with the same standard basis $T$.

[[!redirects matrix bialgebra]]
[[!redirects matrix Hopf algebras]]


matrix Lie algebra

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# Contents
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## Definition

A _matrix Lie algebra_ is the [[Lie algebra]] that is canonically a sub-Lie algebra of the [[general linear Lie algebra]] $Mat(n) = \mathfrak{gl}(n)$ of $n \times n$ [[matrices]].

See also [[matrix Lie group]].

## Properties

By **Ado's theorem**, every finite-[[dimension|dimensional]] Lie algebra over a [[field]] of [[characteristic zero]] is a matrix Lie algebra.

A similar statement fails for [[Lie groups]]. Ado's theorem has been used as a major step in the traditional proofs of the [[Lie's three theorems|Cartan–Lie theorem]] on the existence of [[Lie integration|integration of Lie algebras]] to Lie groups.

[[!redirects matrix Lie algebra]]
[[!redirects matrix Lie algebras]]

[[!redirects Ado theorem]]
[[!redirects Ado's theorem]]
[[!redirects Ado's theorem]]


matrix Lie group

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A _matrix Lie group_ is a [[Lie group]] that is canonically a [[matrix group]], hence a [[subgroup]] of the [[general linear group]] $GL_n$ on $\mathbb{R}^n$ or $\mathbb{C}^n$.

See also [[matrix Lie algebra]].

[[!redirects matrix Lie group]]
[[!redirects matrix Lie groups]]


matrix mechanics

[[classical mechanics|Classical]], [[quantum mechanics|quantum]] and [[statistical mechanics|statistical]] [[mechanics]] may all be seen as varieties of [[matrix]] mechanics for different [[rig|rigs]].  In quantum mechanics we use linear algebra over the ring $\mathbb{C}$ of [[complex numbers]]; in classical mechanics everything is formally the same, but we instead use the rig $\mathbb{R}_{min} = \mathbb{R} \cup \{+\infty\}$, where the addition is $min$ and the multiplication is $+$.

Statistical mechanics (or better, 'thermal statics') is matrix mechanics over a rig $\mathbb{R}^T$ that depends on the [[temperature]] $T$. As $T \to 0$, the rig $\mathbb{R}^T$ reduces to $\mathbb{R}_{min}$ and thermal statics reduces to classical statics, just as quantum dynamics reduces to classical dynamics as Planck's constant approaches zero.

The [[dynamics]] of particles becomes the [[statics]] of [[string theory|strings]] after [[Wick rotation]].

+-- {: .query}
I suppose matrix mechanics over a distributive lattice could be thought as a kind of minimax (or maximin) calculation - David.
=--

* [Quantization and Cohomology: Week 12](http://golem.ph.utexas.edu/category/2007/01/quantization_and_cohomology_we_11.html)
* [Quantization and Cohomology: Week 13](http://golem.ph.utexas.edu/category/2007/02/quantization_and_cohomology_we_12.html#c007500)
* [superposition in classical mechanics](http://golem.ph.utexas.edu/category/2006/12/talks_at_higher_categories_and.html#c006955)
* [Baez-Corfield discussion](http://math.ucr.edu/home/baez/qg-spring2004/discussion.html#idempotent)
* [discussion continued - homotopy theory as groupoid valued mechanics](http://golem.ph.utexas.edu/category/2007/06/cohomology_and_quantization_we.html#c022847)
* [Geometric Representation Theory (Lecture 12)](http://golem.ph.utexas.edu/category/2007/11/geometric_representation_theor_11.html)


matrix model

#Contents#
* table of contents
{:toc}

## Idea

Matrix models are [[model (physics)|physical models]] in which the dynamical quantities are [[square matrices]] (in certain class of matrices, e.g. [[hermitian matrix|hermitian]]), in other words, a [[Lagrangian]]/[[Hamiltonian]] depends on matrix quantities and is usually taken at the limit when the size of matrix tends to infinity.

Matrix models are studied mainly in the context of [[statistical mechanics]] (see [[random matrix theory]]) or in [[quantum field theory]].

## Related entries

* [[Fredholm determinant]], [[random matrix theory]], [[determinental process]], [[Kontsevich matrix model]], [[large N limit]]

* [[integrable system]], [[topological recursion]]

Matrix models that have been argued to capture [[D-brane]] dynamics and [[nonperturbative effects]] in [[string theory]] include

* [[BFSS matrix model]]

* [[IKKT matrix model]]

* [[BMN matrix model]]

* [[membrane matrix model]]

* [[K-matrix model]]

* [matrix model for the 3-brane in 6d](3-brane+in+6d#KachruOzYin98)

## References

First inkling of matrix models from the [[large N limit]] of [[QCD]]:

* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))

* A. Gonzalez-Arroyo, M. Okawa, _A twisted model for large $N$ lattice gauge theory_, Physics Letters B Volume 120, Issues 1–3, 6 January 1983, Pages 174-178 (doi:10.1016/0370-2693(83)90647-0)

* A. Gonzalez-Arroyo, M. Okawa, _Twisted-Eguchi-Kawai model: A reduced model for large-
$N$ lattice gauge theory_, Phys. Rev. D 27, 2397 (1983) ([doi:10.1103/PhysRevD.27.2397](https://doi.org/10.1103/PhysRevD.27.2397))

Discussion of [[topological recursion]] for matrix models originates with

* [[Leonid Chekhov]], [[Bertrand Eynard]], [[Nicolas Orantin]], _Free energy topological expansion for the 2-matrix model_, JHEP 0612:053,2006 ([arXiv:math-ph/0603003](https://arxiv.org/abs/math-ph/0603003))

See also

* Raghav G. Jha, *Introduction to Monte Carlo for Matrix Models* ([arXiv:2111.02410](https://arxiv.org/abs/2111.02410))

category: physics

[[!redirects matrix models]]


matrix product state

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#### Quantum Field Theory
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#Contents#
* table of contents
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## Idea

What are called _matrix product states_ (MPS) in [[quantum physics]] (specifically in [[solid state physics]] and in [[AdS/CFT]]) are those [[tensor network states]] of the form of a ring of [[tensors]] all of rank 3.

For example, given a [[metric Lie algebra]] $\mathfrak{g}$ (with [[string diagram]]-notation as discussed [there](metric+Lie+algebra#Definition)) its [[Lie bracket]]-[[tensor]] $f \coloneqq [-,-] \in \mathfrak{g}^\ast \otimes \mathfrak{g}^\ast \otimes \mathfrak{g}$ gives rise to matrix product states of the following form:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Related concepts

[[!include states and observables -- content]]

## References

For more see the references at _[[tensor network state]]_.

### General

Review and exposition:

* [[Roman Orus]], Section 5.1 of: _A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States_, 	Annals of Physics 349 (2014) 117-158  ([arXiv:1306.2164](https://arxiv.org/abs/1306.2164))

* [[Jens Eisert]], Section 2 of: _Entanglement and tensor network states_, Modeling and Simulation 3, 520 (2013) ([arXiv:1308.3318](https://arxiv.org/abs/1308.3318))

* [[Jacob Biamonte]], Ville Bergholm, Section 5 of: _Tensor Networks in a Nutshell_, Contemporary Physics ([arxiv:1708.00006](https://arxiv.org/abs/1708.00006))

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 8 of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

* {#Biamonte19} [[Jacob Biamonte]], Chapter II of: _Lectures on Quantum Tensor Networks_ ([arXiv:1912.10049](https://arxiv.org/abs/1912.10049))

* Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein, Section 2.3.2 of: _Tensor Network Contractions_, Lecture Notes in Physics, Springer (2020) ([arXiv:1708.09213](https://arxiv.org/abs/1708.09213), [doi:10.1007/978-3-030-34489-4](https://link.springer.com/book/10.1007/978-3-030-34489-4))

* [[Roman Orus]], Chapter II of: _Tensor networks for complex quantum systems_, Nature Reviews Physics 1, 538-550 (2019) ([arXiv:1812.04011](https://arxiv.org/abs/1812.04011), [doi:10.1038/s42254-019-0086-7](https://doi.org/10.1038/s42254-019-0086-7))

See also:

* Wikipedia, _[Matrix product state](https://en.wikipedia.org/wiki/Matrix_product_state)_

### In quantum computation

In the context of [[quantum computation]]:

* Yiqing Zhou, E. Miles Stoudenmire, Xavier Waintal, _What limits the simulation of quantum computers?_ ([arXiv:2002.07730](https://arxiv.org/abs/2002.07730))

[[!redirects matrix product states]]


matrix theory

> for matrix theory in the sense of [[string theory]] see at _[[BFSS matrix model]]_ and _[[IKKT matrix model]]_

***

Let $A$ be a [[Lawvere theory]] with generic object $T$. For $n \in \mathbb{N}$, the [[full subcategory]] of $A$ generated by the cartesian powers of $T^n$ is also a Lawvere theory, which we denote by $M_n(A)$. In the case of an annular theory (the theory of [[module]]s over a ring that we also call $A$), this is the construction of $n\times n$ matrices over $A$. If we denote by $M_n$ the application of this construction to the initial theory (the theory of [[set]]s), then we may identify $M_n(A)$ with the [[tensor product theory]] $M_n\otimes A$.

It is an amusing exercise to present $M_n$ in terms of generating operations and relations between them.


matroid

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#### Modalities, Closure and Reflection
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# Contents #
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{: toc}

## Idea

The concept of *matroid* ([Whitney 1935](#Whitney35)) is fundamental to [[combinatorics]], giving several different ways of encoding/defining and presenting a general notion of "independence", e.g., [[linearly independent subset|linear independence]] in a [[vector space]], algebraic independence in a [[field extension]], etc.

There is also a similar concept of an *[[oriented matroid]]*; every oriented matroid has an underlying matroid.

## Definitions

### Plain definition

\begin{definition}\label{Matroid}
A **matroid** on a [[set]] $X$ is a [[Moore closure|closure operator]] $cl \colon P(X) \to P(X)$ satisfying the _exchange axiom_: if $a \in cl(S \cup\{b\}) \setminus cl(S)$, then $b \in cl(S \cup\{a\}) \setminus cl(S)$.
\end{definition}

\begin{remark}
When [[combinatorics|combinatorialists]] speak of matroids as such, $X$ is usually taken to be a [[finite set]]. A typical example is $X$ some finite [[subset]] of a [[vector space]] $V$, taking $cl(S) \,\coloneqq\, X \cap Span(S)$ for any $S \subseteq X$.
\end{remark}

Further terminology:

\begin{definition}\label{Bases}
Under Def. \ref{Matroid}:

1. A [[subset]] $S \subseteq X$ is called *independent* if there is a strict inclusion $cl(T) \subset cl(S)$ for every strict inclusion $T \subset S$ (this is the same as requiring $x \notin cl(S\setminus \{x\})$ for every $x \in S$).

1. A [[subset]] $S \subset X$ is called a _basis_ if $cl(S) = X$ and $S$ is independent.

1. A _hyperplane_ is a closed subset $S$ (meaning $cl(S) = S$) that is maximal among proper closed subsets of $X$.

\end{definition}

It is possible to axiomatize the notion of matroid by taking bases (Def. \ref{Bases}) as the primitive notion, or independent sets as the primitive notion, or hyperplanes as the primitive notion, etc. -- Rota (after Birkhoff) speaks of _cryptomorphism_ between these differing definitions. Much of the power and utility of matroid theory comes from this multiplicity of definitions and the possibility of moving seamlessly between them; for example, a matroid structure might be easy to detect from the viewpoint of one definition, but not from another.

+-- {: .num_prop #dim1}
###### Proposition

Any two bases (Def. \ref{Bases}) of a matroid $X$ have the same [[cardinality]], provided that one of them is finite.

=--

The cardinality of such a basis is called of course the _[[dimension]]_ of the matroid. Clearly then a finite matroid has a well-defined dimension.

+-- {: .proof}
###### Proof
First, suppose $A$ is an independent set and $B$ is a finite basis, and suppose there are subsets $A_0 \subseteq A, B_0 \subseteq B$ such that $A_0 \cup B_0$ is a basis. We claim that for each $a \in A \setminus A_0$, there exists $b \in B_0$ such that $A_0 \cup \{a\} \cup (B_0 \setminus \{b\})$ is a basis. For, let $C \subseteq B_0$ be of minimum cardinality such that $a \in cl(A_0 \cup C)$; we know $C$ must be inhabited since $a \notin cl(A \setminus \{a\}) \supseteq cl(A_0)$; clearly $C \cap A_0 = \emptyset$. So let $b$ be an element of $C$. Since by minimality of $C$ we have

$$a \in cl(A_0 \cup (C \setminus \{b\}) \cup \{b\}) \cap \neg cl(A_0 \cup (C \setminus \{b\})),$$

it follows from the exchange axiom that $b \in cl(A_0 \cup (C \setminus \{b\}) \cup \{a\})$. Thus $b \in cl(A_0 \cup (B_0 \setminus \{b\}) \cup \{a\})$, whence

$$cl(A_0 \cup (B_0 \setminus \{b\}) \cup \{a\}) = cl(A_0 \cup B_0 \cup \{a\}) = X$$

so that $D \coloneqq A_0 \cup (B_0 \setminus \{b\}) \cup \{a\}$ "spans" $X$. Also $D$ is independent: if $x \in D$ and $x \neq a$, then

$$cl(D \setminus \{x\}) \subseteq cl((A_0 \cup B_0) \setminus \{x\})$$

with neither side containing $x$ since $A_0 \cup B_0$ is independent; whereas if $x = a$ and supposing to the contrary that $a \in cl(D \setminus \{a\}) = cl((A_0 \cup (B_0 \setminus \{b\}))$, we conclude $A_0 \cup (B \setminus \{b\})$ has the same span as $D$. Since $D$ already spans, $b \in cl(A_0 \cup (B_0 \setminus \{b\}))$, again impossible since $A_0 \cup B_0$ is independent. This proves the claim.

Again assuming $A$ independent and $B$ is a finite basis, now we show that $card(A) \leq card(B)$, which will finish the proof. Let $n = card(B)$, and suppose on the contrary that there are distinct elements $a_1, \ldots, a_{n+1} \in A$. Set $A_0 = \emptyset$ and $B_0 = B$. Applying the claim above inductively, we have that $\{a_1, \ldots, a_i\} \cup (B \setminus \{b_1, \ldots, b_i\})$ is a basis for $1 \leq i \leq n$, so in particular $\{a_1, \ldots, a_n\}$ spans $X$. Hence $a_{n+1} \in cl(\{a_1, \ldots, a_{n}\})$, contradicting the independence of $A$.

=--

### Model-theoretic geometry
 {#ModelTheoreticGeometry}

Essentially the very same notion arises in [[model theory]], except instead of being called a matroid it is called a "pregeometry" or "geometry", and in contrast to combinatorialists, model theorists usually mean _infinite_ matroids. The notion arises in the study of geometry of strongly minimal sets, with applications to stability theory (part of Shelah's classification theory).

\begin{definition}

A _pregeometry_ (in [[model theory]]) is a (possibly infinite) matroid (Def. \ref{Matroid}, given by a [[set]] $X$ equipped with a [[closure operator]] $cl$) that is _finitary_:

* for all $S \subseteq X$, if $x \in cl(S)$ then $x \in cl(S_0)$ for some [[finite set|finite]] [[subset]] $S_0 \subseteq S$.

A *geometry* is a pregeometry such that, in addition:

1. $cl(\varnothing) = \varnothing$ (the [[empty set|empty subset]] is closed),

1. $cl(\{x\}) = \{x\}$ for every $x \in X$ (all [[singleton]] subsets are closed).

\end{definition}

(See also *[[geometric stability theory]]*.)

The language of independence, spanning, and basis carry over as before. A maximal independent set spans (i.e., is a basis), and maximal independent sets exist according to [[axiom of choice|Zorn's lemma]]. Again we have a notion of dimension by the following proposition.

+-- {: .num_prop #welldefined}
###### Proposition
In a pregeometry $(X, cl)$, any two bases have the same cardinality.
=--

+-- {: .proof}
###### Proof
We already [proved this](#dim1) in the case where the pregeometry has a finite basis. Otherwise, if $A$ is independent and $B$ is an infinite basis, then

$$A = A \cap X = A \cap \bigcup_{B_0 \subseteq B\; finite} cl(B_0) = \bigcup_{B_0 \subseteq B\; finite} A \cap cl(B_0)$$

where the second equality follows from the finitary condition. Since each summand $A \cap cl(B_0)$ has cardinality less than that of $B_0$ by independence of $A$ (noting that $B_0$ is a basis of $cl(B_0)$), the union on the right has cardinality bounded above by $card(B)$. From $card(A) \leq card(B)$ it follows that any two bases have the same cardinality.
=--

## Examples
 {#Examples}

Vector spaces, algebraic closures, graphs, restrictions, localizations, ...

* A *graphic matroid* is a matroid $M$ derived from a [[simple graph]] by taking the underlying set of $M$ to be the set of edges $E$, and taking as independent sets of $M$ those $S \subseteq E$ such that the edges of $S$ (and their incident vertices) form a [[forest]], i.e., a graph without cycles.

* A *vector matroid* is a matroid $M$ derived from a a collection of vectors $E$ in a vector space. Independent sets of $M$ are those $S \subseteq E$ such that $S$ is linearly independent. Providing an equivalence to such a vector matroid is the problem of representability over a specified field.

* An *algebraic matroid* for a field extension $K/F$ is derived from a collection of elements $E \subset K$ and independent sets of $M$ are those such that $F(S)$ has transcendence degree equal to $\mid S \mid$ so all of $S$ are algebraically independent.

* If $M$ is a (finite) matroid, then the *matroid dual* $M^\ast$ of $M$ has the same underlying set as $M$, and where a basis in $M^\ast$ is precisely the complement of a basis of $M$. It follows that $M^{\ast\ast} \cong M$ (this is even an equality according to the definition).

* For a topological space $X$ the exchange condition is equivalent to
$$ \forall x,y \in X: x\in cl({y}) \Leftrightarrow y\in cl({x}) $$
In particular, every $T1$-space is a matroid.

## Properties

### Mnev's theorem

Mnëv's universality theorem says that any [[semialgebraic set]] in $\mathbb{R}^n$ defined over integers is stably equivalent to the realization space of some oriented matroid.

### Combinatorial optimization

(...)

## Related concepts

* [[oriented matroid]]

## References

The original article:

* {#Whitney35} [[Hassler Whitney]], _On the abstract properties of linear dependence_, American Journal of Mathematics (The Johns Hopkins University Press) **57** 3  (1935) 509-533 [[jstor:2371182](http://jstor.org/stable/2371182), [MR1507091](http://www.ams.org/mathscinet-getitem?mr=1507091)]

On the [[category]] of matroids with *strong maps* between them:

* [[Chris Heunen]], [[Vaia Patta]], *The category of matroids*, Applied Categorical Structures **26** (2018) 205–237 [[arXiv:1512.01390](https://arxiv.org/abs/1512.01390), [doi:10.1007/s10485-017-9490-2](https://doi.org/10.1007/s10485-017-9490-2)]

On [[Hodge theory]] and [[intersection cohomology]] of matroids:

* Tom Braden, [[June Huh]], Jacob P. Matherne, [[Nicholas Proudfoot]], Botong Wang, *Singular Hodge theory for combinatorial geometries* [[arXiv:2010.06088](https://arxiv.org/abs/2010.06088)]

* {#Huh22} [[June Huh]], _Combinatorics and Hodge theory_, Proc. Int. Cong. Math. **1** (2022)
 [[doi:10.4171/ICM2022/205](), [pdf](https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf), [[Huh-CombinatoricsAndHodgeTheory.pdf:file]]]

On the axiomatization of [[infinite set|infinite]] matroids:

* [[Henning Bruhn]], [[Reinhard Diestel]], [[Matthias Kriesell]], [[Rudi Pendavingh]], [[Paul Wollan]], *Axioms for infinite matroids*, Advances in Mathematics, **239** (2013) 18–46 [[arXiv:1003.3919](https://arxiv.org/abs/1003.3919), [doi:10.1016/j.aim.2013.01.011](https://doi.org/10.1016/j.aim.2013.01.011)]

Further references:

* Wikipedia [matroid](http://en.wikipedia.org/wiki/Matroid), [Mnëv's universality theorem](http://en.wikipedia.org/wiki/Mnev%27s_universality_theorem), [oriented matroid](http://en.wikipedia.org/wiki/Oriented_matroid)

* J. Oxley, _What is a matroid_, [pdf](http://www.math.lsu.edu/~oxley/survey4.pdf)
* James G. Oxley, _Matroid theory_, Oxford Grad. Texts in Math. 1992, 2010
* some papers on Coxeter matroids [html](http://www.math.ufl.edu/~white/cox.html)

* MathOverflow question [Mnëv's universality corollaries, quantitative versions](http://mathoverflow.net/questions/72154/mnevs-universality-corollaries-quantitative-versions)
* Eric Katz, Sam Payne, _Realization space for [[tropical geometry|tropical]] fans_, [pdf](http://users.math.yale.edu/~sp547/pdf/Realization-spaces.pdf)
* Nikolai E. Mnev, _The universality theorems on the classification problem of configuration varieties and convex polytopes varieties_, pp. 527-543, in "Topology and geometry: Rohlin Seminar." Edited by O. Ya. Viro. Lecture Notes in Mathematics, __1346__, Springer 1988; _A lecture on universality theorem_ (in Russian) [pdf](http://club.pdmi.ras.ru/~panina/9.pdf)
* Talal Ali Al-Hawary, _Free objects in the category of geometries_, [pdf](http://www.emis.de/journals/HOA/IJMMS/Volume26_12/770.pdf)
* Talal Ali Al-Hawary, D. George McRae, _Toward an elementary axiomatic theory of the category of LP-matroids_, Applied Categorical Structures __11__: 157–169, 2003, [doi](http://dx.doi.org/10.1023/A:1023557229668)
* Hirokazu Nishimura, Susumu Kuroda, _A lost mathematician, Takeo Nakasawa: the forgotten father of matroid theory_ 1996, 2009
* William H. Cunningham, _Matching, matroids, and extensions_, Math. Program., Ser. B __91__: 515–542 (2002) [doi](http://dx.doi.org/10.1007/s101070100256)
* L. Lovász, _Matroid matching and some applications_, J. Combinatorial Theory B __28__, 208–236 (1980)
* A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler,  _Oriented matroids_, Cambridge Univ. Press 1993, 2000, [view at reslib.com](http://reslib.com/book/Oriented_Matroids)
* Matthew Baker, _Matroids over hyperfields_, [arxiv/1601.01204](http://arxiv.org/abs/1601.01204)
* David Marker, _Model Theory: An Introduction_, Graduate Texts in Math. 217, Springer-Verlag New York, 2002.
* Tom Braden, Carl Mautner, _Matroidal Schur algebras_, [arxiv/1609.04507](http://arxiv.org/abs/1609.04507)
* P. S. Kung, _A Source Book in Matroid Theory_, Birkhäuser, Boston, 1986.

category: combinatorics, model theory

[[!redirects matroids]]
[[!redirects pregeometry]]
[[!redirects pregeometries]]


Matt Earnshaw

* [GitHub page](https://github.com/mattearnshaw/)

## related $n$Lab entries

* [[William Lawvere]]

category: people


Matt Insall

Associate Professor of Mathematics, Missouri University of Science and Technology in Rolla Missouri, USA


Matt Reece

* [webpage](http://users.physics.harvard.edu/~mreece/)

## Selected writings

On [[perturbative string theory]]-corrections (for small [['t Hooft coupling]] $\lambda = g_{YM}^2 N$) and/or [[M-theory]]-corrections (for [[small N limit|small N]]) to the [[supergravity]]-approximation of the [[AdS/CFT correspondence]]:

* Csaba Csaki, [[Matthew Reece]], John Terning, _The AdS/QCD Correspondence: Still Undelivered_, JHEP 0905:067, 2009 ([arXiv:0811.3001](https://arxiv.org/abs/0811.3001))

On the [[weak gravity conjecture]]:

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Sharpening the Weak Gravity Conjecture with Dimensional Reduction_, JHEP02(2016)140 ([arXiv:1509.06374](https://arxiv.org/abs/1509.06374))

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Repulsive Forces and the Weak Gravity Conjecture_ ([arXiv:1906.02206](https://arxiv.org/abs/1906.02206))

category: people

[[!redirects Matthew Reece]]


Matt Rosenzweig

* [web](https://matthewhr.wordpress.com/)

## related $n$Lab entries

* [[shrinking lemma]]

category: people


Matt Strassler > history

See _[[Matthew Strassler]]_


Matteo Baggioli

* [personal page](https://members.ift.uam-csic.es/matteo.baggioli/who-i-am/)

* [research group page](https://www.theorylab-sjtu.com/)

## Selected writings

On the [[AdS/CMT correspondence]]:

* [[Matteo Baggioli]], *Gravity, holography and applications to condensed matter* $[$[arXiv:1610.02681](https://arxiv.org/abs/1610.02681)$]$

category: people


Matteo Bertolini

* [webpage](http://people.sissa.it/~bertmat/)

## Related $n$Lab entries

* [[supersymmetry breaking]]

catgeory: people


Matteo Capucci

* [personal page](https://matteocapucci.wordpress.com/)

## Selected writings

On [[categorical systems theory]] ([[cybernetics]]):

* [[Matteo Capucci]], [[Bruno Gavranović]], [[Jules Hedges]], [[Eigil F. Rischel]], *Towards Foundations of Categorical Cybernetics*, EPTCS **372** (2022) 235-248 [[arXiv:2105.06332](https://arxiv.org/abs/2105.06332), [doi:10.4204/EPTCS.372.17](https://doi.org/10.4204/EPTCS.372.17)]

On [[actegories]]:

* {#CG22} [[Matteo Capucci]], [[Bruno Gavranović]], _Actegories for the working amthematician_ [[arXiv:2203.16351](https://arxiv.org/abs/2203.16351)]

On [external choice](axiom+of+choice#ExternalForm) in [[game theory]] as a [[quasi-limit]]:

* {#Capucci22} [[Matteo Capucci]], _The Universal Property of External Choice_ (2022) [[pdf](https://matteocapucci.files.wordpress.com/2022/03/main.pdf)]

On [[higher category theory|higher category theoretic]] generalizations of the [[para construction]] as [[category theory for dynamical systems]]:

* {#Capucci23} [[Matteo Capucci]], _Constructing triple categories of cybernetic processes_ [[slides](https://zenodo.org/record/8221550/files/main.pdf), [video](https://youtu.be/jYxnUy1vH7Q)]

* [[Matteo Capucci]], *Para Construction as a Wreath Product*, [talk at](CQTS#CapucciJan2024) *[[CQTS]]* (Jan 2024) [slides: [[Capucci-ParaAsWreathProduct.pdf:file]]]

category: people


Matteo Spadetto

* [personal page](https://spadetto.github.io/)

* [institute page](https://eps.leeds.ac.uk/maths/pgr/8476/matteo-spadetto)

## Selected writings

On [[weak type theory]] in relation to [[homotopy type theory]]:

* [[Matteo Spadetto]], *Weak type theories: a conservativity result for homotopy elementary types*, talk at *[[DutchCATS]]* (12 May 2023) [[slides pdf](https://dutchcats.github.io/Weak_type_theories/slides_spadetto.pdf)]

category: people


matter

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In particle [[physics]], _matter_ is a [[bound state]] of [[fermion|fermionic]] [[particles]].

This is in contrast to [[force]] [[gauge fields|fields]], whose [[quanta]] are [[bosons|bosonic]] [[particles]].

In the [[standard model of particle physics]], all matter fields are [[sections]] of a [[spinor bundle]] on [[spacetime]] which is [[associated bundle|associated]] of a $U(1) \times SU(2) \times SU(3)$-[[principal bundle]].

In the [[standard model of cosmology]] matter is characterized as part of the energy-momentum density with specific _parameter of state_ $w = p/\rho$, see at _[[FRW model]]_.

## Related concepts

* [[energy]]

  * **matter**, [[radiation]], [[cosmological constant]]

* [[force]], [[spacetime]]

* [[spinors in Yang-Mills theory]], [[minimal coupling]]

* [[dark matter]]

[[!redirects matter field]]
[[!redirects matter fields]]


Matthew Ando

* [website](http://www.math.uiuc.edu/~mando/)

## Selected writrings

On [[elliptic genera]], the [[Witten genus]] and the [[string orientation of tmf]]:

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850 ([doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf))

* {#AndoHopkinsStrickland02} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _The sigma orientation is an H-infinity map_, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 ([arXiv:math/0204053](http://arxiv.org/abs/math/0204053), [doi:10.1353/ajm.2004.0008](https://doi.org/10.1353/ajm.2004.0008))

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of $KO$-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

On [[equivariant elliptic genera]] in [[equivariant elliptic cohomology]]:

* [[Matthew Ando]], [[Maria Basterra]], _The Witten genus and equivariant elliptic cohomology_, Math Z 240, 787–822 (2002) ([arXiv:math/0008192](https://arxiv.org/abs/math/0008192), [doi:10.1007/s002090100399](https://doi.org/10.1007/s002090100399))

* [[Matthew Ando]], [[John Greenlees]], _Circle-equivariant classifying spaces and the rational equivariant sigma genus_, Math. Z. 269, 1021–1104 (2011) ([arXiv:0705.2687](https://arxiv.org/abs/0705.2687), [doi:10.1007/s00209-010-0773-7](https://doi.org/10.1007/s00209-010-0773-7))

On [[elliptic genera]] of [[Landau-Ginzburg models]]:

* [[Matthew Ando]], [[Eric Sharpe]], _Elliptic genera of Landau-Ginzburg models over nontrivial spaces_, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 ([arXiv:0905.1285](https://arxiv.org/abs/0905.1285))

On [[∞-groups of units]], [[Thom spectra]] and [[twisted generalized cohomology]]:

* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory_, Journal of Topology, Volume7, Issue 4, December 2014 ([arXiv:1403.4320](https://arxiv.org/abs/1403.4320), [arXiv:10.1112/jtopol/jtu009](https://doi.org/10.1112/jtopol/jtu009))

* {#ABGHR14} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology_, Journal of Topology Volume 7, Issue 3 2014 Pages 869–893  ([arXiv:1403.4325](http://arxiv.org/abs/1403.4325), [doi:10.1112/jtopol/jtt035](https://doi.org/10.1112/jtopol/jtt035))

Specifically on [[twisted K-theory]] and twisted [[tmf]]:

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics 81, American Mathematical Society 2010 ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004), [ISBN:978-0-8218-4887-6](https://bookstore.ams.org/pspum-81))

category: people


Matthew B. Young

* [personal page](https://sites.google.com/site/youngmatthewb/)

## Selected writings

On [[self-dual object|self-dual]] [[modules]] over [[Hall algebras]]:

* [[Matthew B. Young]], *Self-Dual Hall modules*, PhD thesis, Stony Brook (2013) [[pdf](https://www.math.stonybrook.edu/alumni/2013-Matthew-Young.pdf), [[Young-HallModules.pdf:file]]]

On [[higher Segal spaces]]:

* [[Matthew B. Young]], *Relative 2-Segal spaces*, Algebraic & Geometric Topology **18** (2018) 975-1039 [[doi:10.2140/agt.2018.18.975](https://doi.org/10.2140/agt.2018.18.975)]

On [[transgression in group cohomology]] generalized to Real $\mathbb{Z}/2$-equivariant cohomology (as appropriate for twists of [[KR-theory]]):

* [[Behrang Noohi]], [[Matthew B. Young]], *Twisted loop transgression and higher Jandl gerbes over finite groupoids*, Algebr. Geom. Topol. **22** (2022) 1663-1712 [[arXiv:1910.01422](https://arxiv.org/abs/1910.01422), [doi:10.2140/agt.2022.22.1663](https://doi.org/10.2140/agt.2022.22.1663)]

Generalization of [[Tate K-theory]] (quasi-[[elliptic cohomology]]) to [[twisted equivariant K-theory|twisted equivariant]] [[KR-theory]]:

* [[Zhen Huan]], [[Matthew B. Young]], *Twisted Real quasi-elliptic cohomology* [[arXiv:2210.07511](https://arxiv.org/abs/2210.07511)]

On [[super Chern-Simons theory]] and [[Rozansky-Witten theory]]:

* [[Nathan Geer]], [[Matthew B. Young]], *Three dimensional topological quantum field theory from $U_q(\mathfrak{gl}(1\vert1))$ and $U(1\vert 1)$ Chern--Simons theory* [[arXiv:2210.04286](https://arxiv.org/abs/2210.04286)]

* [[Matthew B. Young]], *Non-semisimple TFT and $U(1\vert 1)$ Chern-Simons theory*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023)[[web](Center+for+Quantum+and+Topological+Systems#YoungMar2023), video: [YT](https://www.youtube.com/watch?v=m73KQCwXOrY)]

## Related entries

* [[hyperbolic functor]]

category: people

[[!redirects Matthew Young]]


Matthew Burke

## Selected writings

Expressing [[Goodwillie calculus]] in terms of [[tangent (∞,1)-categories]]:

* {#BauerBurkeChing21} [[Kristine Bauer]], [[Matthew Burke]], [[Michael Ching]], _Tangent $\infty$-categories and Goodwillie calculus_ ([arXiv:2101.07819](https://arxiv.org/abs/2101.07819))

category: people


Matthew Daws

Lecturer at [UCLAN](http://www.star.uclan.ac.uk/) in the UK.

category: people


Matthew Di Meglio

Matthew Di Meglio is a PhD student in the Laboratory for Foundations of Computer Science at the University of Edinburgh working with [[Chris Heunen]] and [[Perdita Stevens]]. He previously completed a Masters of Research at Macquarie University with [[Michael Johnson]].

* [website](https://mdimeglio.github.io)

## Selected writings

On [[lenses in computer science]]:

* [[Matthew Di Meglio]], _The category of asymmetric lenses and its proxy pullbacks_, Master's Thesis, Macquarie University, (2021) [[doi:10.25949/20236449.v1](https://doi.org/10.25949/20236449.v1)]

* [[Matthew Di Meglio]], _Coequalisers under the Lens_, EPTCS **372**, (2022) 149-163. [[doi:10.4204/EPTCS.372.11](https://doi.org/10.4204/EPTCS.372.11)]

* [[Bryce Clarke]] and [[Matthew Di Meglio]], _An introduction to enriched cofunctors_, (2022) [[arXiv:2209.01144](https://arxiv.org/abs/2209.01144)]

* [[Matthew Di Meglio]], _Universal Properties of Lens Proxy Pullbacks_, EPTCS **380** (2023) 400-416 [[doi:10.4204/EPTCS.380.23](https://doi.org/10.4204/EPTCS.380.23)]

On [[rational dagger categories]]:

* [[Matthew Di Meglio]], *Rational dagger categories* [[arXiv:2312.02883](https://arxiv.org/abs/2312.02883)]

category: people


Matthew Emerton

* [personal page](http://www.math.uchicago.edu/~emerton/)

## Selected writings

Introducing the [[Emerton-Gee stack]]:

* {#EmertonGee19} [[Matthew Emerton]], [[Toby Gee]], *Moduli stacks of étale $(\phi, \Gamma)$ modules and the existence of crystalline lifts* ([arXiv:1908.07185](https://arxiv.org/abs/1908.07185))

On [[prismatic cohomology]]:

* [[Matthew Emerton]], _Prismatic cohomology_, lecture notes (2020) [[webpage](http://www.math.uchicago.edu/~emerton/prismatic/prismatic.html)]

category: people


Matthew Fisher

* [institute page](https://www.kitp.ucsb.edu/mpaf)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Matthew_P._A._Fisher)

## Selected writings

On ([[non-perturbative effect|strongly]]) [[interaction|interacting]] [[topological phases of matter]] (such as [[topological order]]):

* [[Jason Alicea]], [[Matthew Fisher]], [[Marcel Franz]], [[Yong-Baek Kim]],  *Strongly Interacting Topological Phases*, report on Banff workshop [15w5051](http://www.birs.ca/events/2015/5-day-workshops/15w5051) (2015) [[pdf](https://www.birs.ca/workshops/2015/15w5051/report15w5051.pdf), [[AliceaEtAl-InteractingTopPhases.pdf:file]]]

category: people


Matthew Greenberg

* [webpage](https://people.ucalgary.ca/~mgreenbe/)

## Selected writings

On [[elliptic cohomology]]:

* [[Matthew Greenberg]], _Constructing elliptic cohomology_, McGill University 2002 ([oclc:898194373](https://www.bac-lac.gc.ca/eng/services/theses/Pages/item.aspx?idNumber=898194373), [[GreenbergEllipticCohomology.pdf:file]])

category: people


Matthew Hastings

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

* [Institute page](https://www.microsoft.com/en-us/research/people/mahastin/)

## Selected writings

Providing rigorous theoretical foundations for the [[quantum Hall effect]]:

* [[Matthew Hastings]], [[Spyridon Michalakis]], _Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions_, Commun. Math. Phys., 334:433–471, (2015) ([arXiv:0911.4706](https://arxiv.org/abs/0911.4706), [doi:10.1007/s00220-014-2167-x](https://doi.org/10.1007/s00220-014-2167-x))

On [[quantum adiabatic theorem|adiabatic transformations]] of certain 2d [[topological insulators]]:

* David Aasen, [[Zhenghan Wang]], [[Matthew B. Hastings]], *Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes* [[arXiv:2203.11137](https://arxiv.org/abs/2203.11137)]

category: people

[[!redirects Matthew B. Hastings]]


Matthew Headrick

* [personal page](http://people.brandeis.edu/~headrick/)

## Selected writings

Solutions to [[Joseph Polchinski]]'s textbook: *[[String Theory]]*:

* [[Matthew Headrick]], *A solution manual for Polchinski's "String Theory"* [[arXiv:0812.4408](https://arxiv.org/abs/0812.4408)]

On [[holographic entanglement entropy]] via [[bit threads]]:

* {#FreedmanHeadrick16} [[Michael Freedman]], [[Matthew Headrick]], *Bit threads and holographic entanglement*, Comm. Math. Phys. 352, 407 (2017) ([arXiv:1604.00354](https://arxiv.org/abs/1604.00354))

category: people


Matthew Heydeman

* [webpage](https://www.ias.edu/scholars/matthew-te-heydeman)

## Selected writings

On the [[p-adic AdS/CFT correspondence]] and [[tensor networks]]:

* [[Matthew Heydeman]], [[Matilde Marcolli]], [[Ingmar Saberi]], [[Bogdan Stoica]], _Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence_ ([arXiv:1605.07639](https://arxiv.org/abs/1605.07639))

Including [[BTZ black holes]]:

* {#HMPS18} [[Matthew Heydeman]], [[Matilde Marcolli]], Sarthak Parikh, [[Ingmar Saberi]], _Nonarchimedean Holographic Entropy from Networks of Perfect Tensors_ ([arXiv:1812.04057](https://arxiv.org/abs/1812.04057))

category: people


Matthew Kennedy

* [website](https://www.math.uwaterloo.ca/~m3kenned/)

## Selected writings

* [[Matthew Kennedy]], [[Astra Kolomatskaia]], [[Daniel Spivak]] *An infinite quantum Ramsey theorem*, Journal of Operator Theory, Volume 84, Issue 1, Summer 2020  pp. 49-65. ([doi:10.7900/jot.2018dec11.2259](http://dx.doi.org/10.7900/jot.2018dec11.2259), [arXiv:1711.09526](https://arxiv.org/abs/1711.09526))

category:people


Matthew Kleban

* [webpage](http://physics.as.nyu.edu/object/MatthewKleban.html)

## Selected writings

Relating the [[Brownian loop soup]] to [[conformal field theory]]:

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *Scalar conformal primary fields in the Brownian loop soup* [[arXiv:2109.12116](https://arxiv.org/abs/2109.12116)]

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *The Brownian loop soup stress-energy tensor* [[arXiv:2112.00074](https://arxiv.org/abs/2112.00074), doi:10.1007/JHEP11(2022)009]

## Related entries

* [[string phenomenology]]

* [[landscape of string theory vacua]]

* [[cosmology]]

category: people

[[!redirects Matt Kleban]]


Matthew Morrow

* [personal page](https://webusers.imj-prg.fr/~matthew.morrow/)

## Selected writings

On [[Hodge theory]] over the [[p-adic numbers]]:

* {#BhattMorrowScholze16}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]] *Integral $p$-adic Hodge Theory* [[arXiv:1602.03148](https://arxiv.org/abs/1602.03148)]

and in connection to [[topological Hochschild homology]]:

* {#BhattMorrowScholze18}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]], *Topological Hochschild Homology and Integral $p$-adic Hodge Theory* [[arXiv:1802.03261](https://arxiv.org/abs/1802.03261)]

## Related entries

* [[prismatic cohomology]]

category: people


Matthew Rupert

* [personal page](https://sites.google.com/view/mrrupert)

## Selected writings

On [[braided monoidal category|braided]] [[tensor categories]] arising as [[representation categories]] of [[logarithmic CFTs|logarithmic]] [[vertex operator algebras]]:

* [[Thomas Creutzig]], [[Simon Lentner]], [[Matthew Rupert]], *Characterizing braided tensor categories associated to logarithmic vertex operator algebras* ([arXiv:2104.13262](https://arxiv.org/abs/2104.13262))

category: people


Matthew Schwartz

* [webpage](https://www.physics.harvard.edu/people/facpages/schwartz)

## Selected writings

On [[glueballs]] via [[AdS-QCD]]:

* [[Emanuel Katz]], [[Adam Lewandowski]], [[Matthew Schwartz]], _Tensor Mesons in AdS/QCD_, Phys. Rev. D**74** 086004 (2006) [[arXiv:hep-ph/0510388](https://arxiv.org/abs/hep-ph/0510388), [doi:10.1103/PhysRevD.74.086004](https://doi.org/10.1103/PhysRevD.74.086004)]

On [[perturbative quantum field theory|perturbative]] [[quantum field theory]] and the [[standard model of particle physics]]:

* [[Matthew Schwartz]], *Quantum Field Theory and the Standard Model*, Cambridge University Press (2014) [[ISBN:9781107034730](https://www.cambridge.org/de/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-field-theory-and-standard-model?format=HB&isbn=9781107034730), [doi:10.1017/9781139540940](https://doi.org/10.1017/9781139540940), [pdf](https://indico.ihep.ac.cn/event/11134/attachments/3783/4308/QFT-Schwartz.pdf)]

category: people


Matthew Spong

* [personal page](http://matthewjspong.net)

## Selected writings

On [[equivariant elliptic cohomology]] via [[orbifold K-theory]] of [[Huan's inertia orbifolds]]:

* [[Zhen Huan]], [[Matthew Spong]], *Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology* ([arXiv:2006.00554](https://arxiv.org/abs/2006.00554))

category: people


Matthew Strassler

* [wikipedia entry](http://www.phys.washington.edu/users/strasslr/)

## Related entries

* [[worldline formalism]]

category: people

[[!redirects Matt Strassler]]

[[!redirects Matthew J. Strassler]]


Matthew Szudzik

* [webpage](http://www.szudzik.com)

category: people


Matthew Watkins

* [webpage](http://empslocal.ex.ac.uk/people/staff/mrwatkin/)

## Related entries

* [[number theory and physics]]

category: people


Matthew Weaver

* [webpage](http://www.cs.princeton.edu/~mzweaver/)

## Selected writings

On [[synthetic (∞,1)-category theory]] in [[cubical type theory]] with [[bicubical sets]]:

* {#WeaverLicata20} [[Matthew Weaver]], [[Daniel Licata]], *A Constructive Model of Directed Univalence in Bicubical Sets*, in: *Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’20*, Association for Computing Machinery (2020) 915–928 [[doi:10.1145/3373718.3394794](https://doi.org/10.1145/3373718.3394794)]

Exposition in:

* [[Matthew Weaver]], *A Constructive Model of Directed Univalence in Bicubical Sets*, talk at HoTTEST (April 2020) [[pdf](https://www.cs.princeton.edu/~mzweaver/pdfs/HoTTEST.pdf), [video](https://youtu.be/kkfNjqSx4Nw)]

category: people


Matthew Yu

* [InSpire page](https://inspirehep.net/authors/1847740)

## Selected writings

On [[fiber functor|fiber]] [[2-functors]] for [[fusion 2-categories]]:

* {#DY23} [[Thibault Decoppet]], [[Matthew Yu]], *Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories* (2023) [[arXiv:2306.08117](https://arxiv.org/abs/2306.08117)]

## Related entries

* [[generalized global symmetry]]

category: people


Matthias Blau

* [website](http://www.blau.itp.unibe.ch/blau.html)

## Selected writings

On [[topological field theory]]:

* Danny Birmingham, [[Matthias Blau]], Mark Rakowski, [[George Thompson]], *Topological field theory*, Physics Reports **209** 4–5 (1991) 129-340 \[doi:10.1016/0370-1573(91)90117-5\]

On the [[supersymmetry|supersymmetric]] [[WZW model]] using [[group algebra]]:

* [[Matthias Blau]], [[George Thompson]], *Equivariant Kähler Geometry and Localization in the $G/G$ Model*, Nucl. Phys. B **439** (1995) 367-394 [doi:10.1016/0550-3213(95)00058-Z, [arXiv:hep-th/9407042](https://arxiv.org/abs/hep-th/9407042)]

On [[general relativity]] ([[gravity]]):

* [[Matthias Blau]], _Lecture notes on [[general relativity]]_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

On [[pp-wave spacetimes]] and [[Penrose limits]]:

* [[Matthias Blau]], _Plane waves and Penrose limits_, 2011 ([pdf](http://www.blau.itp.unibe.ch/lecturesPP.pdf), [[BlauPlaneWavesAndPenroseLimit.pdf:file]])

On relation between [[Rozansky-Witten invariants]] and [[Seiberg-Witten invariants]] of [[3-manifolds]]:

* [[Matthias Blau]], George Thompson, _On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants_, Adv. Theor. Math. Phys. 5 (2002) 483-498 ([arXiv:hep-th/0006244](https://arxiv.org/abs/hep-th/0006244))

category: people


Matthias Burkardt

* [institute page](https://phys.nmsu.edu/facultydirectory/burkardt_matthias.html)

* [InSpire page](https://inspirehep.net/authors/1014938)

## Selected writings

On [[light front quantization]], especially of [[QCD]]:

* [[Matthias Burkardt]], *Light Front Quantization*, Adv. Nucl. Phys. **23** (1996) 1-74 [[arXiv:hep-ph/9505259](https://arxiv.org/abs/hep-ph/9505259), [doi:10.1007/0-306-47067-5_1](https://doi.org/10.1007/0-306-47067-5_1)]

category: people


Matthias Christandl

* [home page](https://www.math.ku.dk/english/staff/?pure=en/persons/475476)

## Selected writings

On [[quantum state tomography]]:

* [[Matthias Christandl]], [[Renato Renner]], *Reliable Quantum State Tomography*, Physical Review Letters **109** 120403 (2012) [[arXiv:1108.5329](https://arxiv.org/abs/1108.5329), [doi:10.1103/PhysRevLett.109.120403](https://doi.org/10.1103/PhysRevLett.109.120403)]

On [[quantum error correction]] for quantum communication protocols:

* [[Matthias Christandl]], Alexander Müller-Hermes, *Fault-tolerant Coding for Quantum Communication*,
IEEE Transactions on Information Theory, [[arXiv:2009.07161](https://arxiv.org/abs/2009.07161)  [doi:10.1109/TIT.2022.3169438](https://ieeexplore.ieee.org/document/9761242)]

category: people


Matthias Flach

* [webpage](http://math.caltech.edu/people/flach.html)

category: people


Matthias Gaberdiel

* [Institute page](https://itp.phys.ethz.ch/people/person-detail.MTEyNjI4.TGlzdC84NDYsLTE5MDQ4MDM5ODI=.html)

* [GoogleScholar](https://scholar.google.com/citations?user=j7lpopIAAAAJ&hl=en)

## Selected writings

On [[anyons]]:

*  [[Klaus Fredenhagen]], [[Matthias Gaberdiel]] and Stefan M. Rüger, *Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory*, Communications in Mathematical Physics **175** (1996) 319–335 $[$[doi:10.1007/BF02102411](https://doi.org/10.1007/BF02102411)$]$

On [[duality between type I and heterotic string theory]]:

* [[Oren Bergman]], [[Matthias Gaberdiel]], G. Lifschytz, _String Creation and Heterotic-Type I' Duality_, Nucl. Phys. B524 (1998) 524-544 ([arXiv:hep-th/9711098](https://arxiv.org/abs/hep-th/9711098))

The triplet algebra of the $c = -2$ Virasoro model as a [[logarithmic CFT]]:

* [[Matthias R. Gaberdiel]], Horst G. Kausch, *A Rational Logarithmic Conformal Field Theory*, Phys.Lett. B **386** (1996) 131-137 $[$[arXiv:hep-th/9606050](https://arxiv.org/abs/hep-th/9606050), doi:10.1016/0370-2693(96)00949-5$]$

On [[fractional-level WZW models]] and [[logarithmic CFTs]]:

* [[Matthias R. Gaberdiel]], *Fusion rules and logarithmic representations of a WZW model at fractional level*, Nucl. Phys. B **618** (2001) 407-436 $[$[arXiv:hep-th/0105046](https://arxiv.org/abs/hep-th/0105046), doi:10.1016/S0550-3213(01)00490-4)$]$

* {#Gaberdiel01} [[Matthias R. Gaberdiel]], *An algebraic approach to logarithmic conformal field theory*, Int. J. Mod. Phys. A **18** (2003) 4593-4638 $[$[arXiv:hep-th/0111260](https://arxiv.org/abs/hep-th/0111260), [doi:10.1142/S0217751X03016860](https://doi.org/10.1142/S0217751X03016860)$]$

On [[super-orbifolds]] as [[target spaces]] in [[superstring theory]] ([[orbispace|orbi-]][[superspace]]):

* [[Matthias Gaberdiel]], [[Dan Israel]], [[Eliezer Rabinovici]], _D-branes at multicritical points_, JHEP 0804:086, 2008 [[arXiv:0803.0291](https://arxiv.org/abs/0803.0291)]

On [[higher spin gauge theory]]:

* [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _String Theory as a Higher Spin Theory_, J. High Energ. Phys. (2016) 2016: 85 ([arXiv:1512.07237](https://arxiv.org/abs/1512.07237), doi:10.1007/JHEP09(2016)085)

On [[AdS/CFT]] for $AdS_3/CFT_2$:

* Shouvik Datta, [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$_ JHEP 1801 (2018) 146 ([arXiv:1709.06393](https://arxiv.org/abs/1709.06393))

* {#EberhardtGaberdiel19a} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _String theory on $AdS_3$ and the symmetric orbifold of Liouville theory_ ([arXiv:1903.00421](https://arxiv.org/abs/1903.00421))

* {#EberhardtGaberdiel19b} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Strings on $AdS_3 \times S^3 \times S^3 \times S^1$_ ([arXiv:1904.01585](https://arxiv.org/abs/1904.01585))

* {#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

* [[duality between type I and heterotic string theory]]

* [[higher spin gauge theory]]

category: people

[[!redirects Matthias R. Gaberdiel]]


Matthias Hutzler

* [institute page](https://www.gu.se/en/about/find-staff/matthiashutzler)

## Selected writings

On [[synthetic algebraic geometry]]:

* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]

category: people


Matthias Kreck

* [webpage](http://www.him.uni-bonn.de/homepages/prof-dr-matthias-kreck/)

## Selected writings

Introducing the notion of [[cutting and pasting of manifolds]]:

* U. Karras, [[Matthias Kreck]], [[Walter D. Neumann]], E. Ossa, *Cutting and Pasting of Manifolds; SK-Groups*, Publish or Perish (1973) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/skbook.pdf), [[Karras-Kreck-Neumann-Ossa_CutNPaste.pdf:file]]]

On [[elliptic genera]]:

* {#KreckStolz93} [[Matthias Kreck]], [[Stephan Stolz]], _$HP^2$-bundles and elliptic homology_, Acta Math, 171 (1993) 231-261 ([[KreckStolzElliptic.pdf:file]], [euclid:acta/1485890737](https://projecteuclid.org/euclid.acta/1485890737))

On [[supermanifolds]] form the point of view of [[functorial geometry]]:

* [[Henning Hohnhold]], [[Matthias Kreck]], [[Stephan Stolz]], [[Peter Teichner]], Sections 2-3 of: _Differential forms and 0-dimensional supersymmetric field theories_, Quantum Topology Volume 2, Issue 1 (2011) pp. 1–41 ([doi:10.4171/QT/12](http://dx.doi.org/10.4171/QT/12))

category: people


Matthias Kriesell

* [Institute page (Ilmenau)](https://www.tu-ilmenau.de/universitaet/fakultaeten/fakultaet-mathematik-und-naturwissenschaften/profil/institute-und-fachgebiete/institut-fuer-mathematik/profil/fachgebiet-diskrete-mathematik-und-algebra/team)

* [home page (Hamburg)](https://www.math.uni-hamburg.de/home/kriesell/)

## Selected writings

On the [[axiomatization]] of [[infinite set|infinite]] [[matroids]]:

* [[Henning Bruhn]], [[Reinhard Diestel]], [[Matthias Kriesell]], [[Rudi Pendavingh]], [[Paul Wollan]], *Axioms for infinite matroids*, Advances in Mathematics, **239** (2013) 18–46 [[arXiv:1003.3919](https://arxiv.org/abs/1003.3919), [doi:10.1016/j.aim.2013.01.011](https://doi.org/10.1016/j.aim.2013.01.011)]

category: people


Matthias König

## Selected writings

On apparently observed [[flavour anomalies]] and their potential explanation by [[leptoquarks]]:

* Michael J. Baker, [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias König]], _High-pT Signatures in Vector-Leptoquark Models_ ([arXiv:1901.10480](https://arxiv.org/abs/1901.10480))

* [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias König]], Nudzeim Selimovic, _Vector Leptoquarks Beyond Tree Level_ ([arxiv:1910.13474](https://arxiv.org/abs/1910.13474))

category: people


Matthias Ludewig

* [Website](https://homepages.uni-regensburg.de/~lum63364/).

## Selected writings

On [[2-vector bundles]] for [[2-vector spaces]] regarded ([here](2-vector+space#ReferencesAsAlgebrasWithBimodules)) as algebras with bimodules between them:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *The insidious bicategory of algebra bundles* [[arXiv:2204.03900](https://arxiv.org/abs/2204.03900)]

On [[2-representations]] of the [[string 2-group]] on [[2-vector spaces]] and the construction of the [[stringor bundle]]:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]

Review:

* [[Matthias Ludewig]], *The spinor bundle on loop space and its fusion product*, talk at [[CQTS]] (Apr 2023) [[web](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#LudewigApr2023), video: [YT](https://www.youtube.com/watch?v=iiRm47HavRU)]

category: people


Matthias Neubert

## Selected writings

On observed [[flavour anomalies]] and their potential explanation by [[leptoquarks]]:

* {#BauerNeubert15} Martin Bauer, [[Matthias Neubert]], _One Leptoquark to Rule Them All: A Minimal Explanation for $R_{D^{(\ast)}}$, $R_K$ and $(g-2)_\mu$_, Phys. Rev. Lett. 116, 141802 (2016) ([arXiv:1511.01900](https://arxiv.org/abs/1511.01900))

* Claudia Cornella, Darius A. Faroughy, [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias Neubert]], _Reading the footprints of the B-meson flavor anomalies_ ([arXiv:2103.16558](https://arxiv.org/abs/2103.16558))

> With the recent result $[$of [[LHCb]]$]$, for the first time a single observable affected by negligible theoretical uncertainties exhibits a deviation from the SM exceeding the $3 \sigma$ level.  Equally striking is the overall coherence of the picture that emerges, especially in $b \to s \ell^+ \ell^-$ −transitions.  As we shall show in this paper, combining all the $b \to s \ell^+ \ell^-$−observables in a very conservative way, the significance of the New Physics (NP) hypothesis formulated in 2014–2015 of a purely left-handed LFU-violating contact interaction has now reached a [[statistical significance|significance]] of $4.6 \sigma$.

> $[\ldots]$  A more structural way of addressing the flavor structure of the model is the idea of implementing [[GUT|Pati-Salam unification]]

category: people


Matthias Schmidt

* [personal page](https://www.mschmidt.uni-bayreuth.de/)

## Selected writings

On [[differential geometry]] and [[mathematical physics]]

specifically on [[differential geometry]] and [[Lie groups]]:

* [[Gerd Rudolph]], [[Matthias Schmidt]], *Differential Geometry and Mathematical Physics Part I. Manifolds, Lie Groups and Hamiltonian Systems*, Springer 2013 ([doi:10.1007/978-94-007-5345-7](https://link.springer.com/book/10.1007/978-94-007-5345-7))

specifically on [[fiber bundles in physics]] ([[principal bundles]], [[principal connections]], [[classifying spaces]], [[Chern-Weil theory]], [[spin bundles]], [[Yang-Mills theory]]):

* [[Gerd Rudolph]], [[Matthias Schmidt]], *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

category: people


Matthias Staudacher

## Selected writings

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 opetator]], 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]]:

* {#BeisertStaudacher03} [[Niklas Beisert]], [[Matthias Staudacher]], _The $\mathcal{N}=4$ SYM Integrable Super Spin Chain_,
Nucl. Phys. B670:439-463, 2003 ([arXiv:hep-th/0307042](https://arxiv.org/abs/hep-th/0307042))

* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))

* {#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))

category: people


Matthias Wendt

* [website](http://home.mathematik.uni-freiburg.de/arithmetische-geometrie/wendt.html)

## Selected writings

On [[representable functor|representability]] in [[motivic homotopy theory]]:

and [[representable functor|representability]]:

* [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], *Affine representability results in $\mathbb{A}^1$-homotopy theory I: Vector bundles*, Duke Math. J. **166** 10 (2017) 1923-1953 [[arXiv:1506.07093](http://arxiv.org/abs/1506.07093), [doi:10.1215/00127094-0000014X](https://doi.org/10.1215/00127094-0000014X)]

* [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], _Affine representability results in $\mathbb{A} ^1$-homotopy theory II: Principal  bundles and homogeneous spaces_, Geom. Topol. **22** (2018) 1181-1225 [[arXiv:1507.08020](http://arxiv.org/abs/1507.08020), [doi:10.2140/gt.2018.22.1181](https://doi.org/10.2140/gt.2018.22.1181)]

category: people


Matthieu Sozeau

* [personal webpage](https://www.irif.fr/~sozeau/)

## Selected writings

On combining [[homotopy type theory]] with [[opetopic type theory]]:

* {#AlliouxFinsterSozeau21} [[Antoine Allioux]], [[Eric Finster]], [[Matthieu Sozeau]], _Types are internal infinity-groupoids_, 2021 ([hal:03133144](https://hal.inria.fr/hal-03133144), [pdf](https://hal.inria.fr/hal-03133144/document))

category: people


Matthijs Vákár

* [webpage](https://www.uu.nl/medewerkers/MILVakar)

## Selected writings

On [[dependent linear type theory]] (for [[quantum computation]] with classical control) with [[categorical semantics]] in [[indexed monoidal categories]]:

* {#Vakar14} [[Matthijs Vákár]], _Syntax and Semantics of Linear Dependent Types_ ([arXiv:1405.0033](http://arxiv.org/abs/1405.0033))

* {#VakarSlides14} [[Matthijs Vákár]], _Splitting the Atom of Dependent Types... or Linear and Operational Dependent Type Theory_, November 2014 ([[VakarLinearAndOperationalDTT.pdf:file]])

* [[Matthijs Vákár]], *A Categorical Semantics for Linear Logical Frameworks*, In: A. Pitts  (ed.), *Foundations of Software Science and Computation Structures* FoSSaCS 2015. Lecture Notes in Computer Science, vol 9034. Springer 2015  ([arXiv:1501.05016](https://arxiv.org/abs/1501.05016), [doi:10.1007/978-3-662-46678-0_7](https://doi.org/10.1007/978-3-662-46678-0_7))

On [[side effects]] in [[functional programming]] with [[dependent type theory]]:

* [[Matthijs Vákár]], *In Search of Effectful Dependent Types* ([arXiv:1706.07997](https://arxiv.org/abs/1706.07997), [uuid:e91e19b3-7e10-4fda-9433-f23b469e4049](https://ora.ox.ac.uk/objects/uuid:e91e19b3-7e10-4fda-9433-f23b469e4049))

## Related $n$Lab entries

* [[dependent linear type theory]]

* [[automatic differentiation]]

category: people


Matti Järvinen

* [Institute page](https://www.ipm.ens.fr/spip.php?article44&lang=fr)

* [Google Scholar page](https://scholar.google.com/citations?user=TxkPsa0AAAAJ&hl=en)

## Selected writings

On the [QCD phase diagram](QCD#PhaseDiagram) and the description of [[neutron stars]] via [[holographic QCD]]:

* Takaaki Ishii, [[Matti Järvinen]], Govert Nijs, _Cool baryon and quark matter in holographic QCD_ ([arXiv:1903.06169](https://arxiv.org/abs/1903.06169))

* Carlos Hoyos, Niko Jokela, [[Matti Järvinen]], Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, _Transport in strongly coupled quark matter_ ([arXiv:2005.14205](https://arxiv.org/abs/2005.14205))

* Carlos Hoyos, Niko Jokela, [[Matti Järvinen]], Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, *Holographic approach to transport in dense QCD matter* ([arXiv:2109.12122](https://arxiv.org/abs/2109.12122))

* [[Matti Järvinen]], *Holographic modeling of nuclear matter and neutron stars*,  Eur.Phys.J.C. 56 2021 ([arXiv:2110.08281](https://arxiv.org/abs/2110.08281))

category: people

[[!redirects Matti Jarvinen]]


Mattia Cafasso

__Mattia Cafasso__ is a mathematical physicist in Angers, France. His main interests are in [[random matrix]] theory, [[Fredholm determinant]]s, [[Riemann-Hilbert problem]], [[orthogonal polynomials]] and [[integrable systems]].

* [homepage](https://sites.google.com/site/mattiacafasso)
* Mattia Cafasso, Chao-Zhong Wu, _Tau functions and the limit of block Toeplitz determinants_, [arxiv/1404.5149](http://arxiv.org/abs/1404.5149)
* M. Adler, M. Cafasso, P. van Moerbeke, _Nonlinear PDEs for Fredholm determinants arising from string equations_, [arxiv/1207.6341](http://arxiv.org/abs/1207.6341)
* M. Bertola, M. Cafasso, _Fredholm determinants and pole-free solutions to the noncommutative Painlevé II equation_, Comm. Math. Phys. __309__ (2012), no. 3, 793–833 [MR2885610](http://www.ams.org/mathscinet-getitem?mr=2885610) [doi](http://dx.doi.org/10.1007/s00220-011-1383-x)

category: people
[[!redirects M. Cafasso]]


Matériaux pour l'histoire des mathématiques au XXe siècle

* Michele Audin (editor), _[[Matériaux pour l’histoire des mathématiques au XXe siècle]]_ - Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996) Séminaires & Congrès. Numèro 3. Paris (Socièté Mathématique de France). 1998. 283 pp.

Table of contents with links to full articles in this collection can be found at
<http://www.emis.de/journals/SC/1998/3/html>

The collection is reviewed in

* Jeremy Gray, Review: Matériaux pour l'Histoire des Mathématiques au XXe Siècle. Actes du colloque à la mémoire de Jean Dieudonné (Nice, 1996), Historia Mathematica
__28__:2, May 2001, pp. 123-125 [doi](https://doi.org/10.1006/hmat.2001.2302)

#### Contents

Pierre Deligne

Quelques idées maîtresses de l'œuvre de A. Grothendieck

Beno Eckmann

Naissance des fibrés et homotopie

Lars Gårding

Hyperbolic Equations in the Twentieth Century

Thomas Hawkins

From General Relativity to Group Representations

Christian Houzel

Histoire de la théorie des faisceaux

Jean-Pierre Kahane

Le mouvement brownien

Yu. I. Manin

Interrelations between Mathematics and Physics

Gilles Pisier

Problèmes de similarité pour les opérateurs sur l'espace de Hilbert

Reinhold Remmert

From Riemann Surfaces to Complex Spaces

Norbert Schappacher

On the History of Hilbert's Twelfth Problem

[[!redirects Matériaux pour l’histoire des mathématiques au XXe siècle]]


Matías Menni

* [webpage](https://sites.google.com/site/matiasmenni/)

## relate entries

* [[cohesive topos]]

category: people

[[!redirects Matias Menni]]


Maurer-Cartan equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### In dg-Lie algebras

For $(\mathfrak{g}, [-,-],\partial)$ a [[dg-Lie algebra]] (the [[differential]] of degree -1), a **Maurer-Cartan element** in $\mathfrak{g}$ is

* an element $a \in \mathfrak{g}_1$ of degree -1

* such that the **Maurer-Cartan equation** holds

  $$
    \partial a + \frac{1}{2}[a,a] = 0
     \,.
  $$

This is a special case of MC elements for [[L-∞ algebras]], which we discuss next.

### In $L_\infty$-algebras
 {#InLInfinityAlgebras}

For $\mathfrak{g}$ an [[L-∞ algebra]] with brackets $[-,\cdots,-]_k$, a Maurer-Cartan element is an element $a \in \mathfrak{g}$ such that

$$
  \sum_{k = 0}^\infty \frac{1}{k!} [a, \cdots, a]_k
  =
  0
  \,.
$$

## Properties
 {#Properties}

### As $L_\infty$-homomorphisms
  {#AsHomomorphisms}

For $\mathfrak{g}$ an [[L-∞ algebra]] and $A$ a [[dg-algebra]], also the [[tensor product]] $A \otimes \mathfrak{g}$ naturally inherits the structure of an [[L-∞ algebra]].

Let $\mathfrak{g}$ be of [[finite type]] and write $CE(\mathfrak{g})$ for the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$. Then MC-elements in $A \otimes \mathfrak{g}$ correspond bijectively to dg-algebra homomorphisms $A \leftarrow CE(\mathfrak{g})$:

$$
  MC(A \otimes \mathfrak{g})
  \simeq
  Hom_{dgAlg}(CE(\mathfrak{g}), A)
  \,.
$$

A reference for this is for instance around def. 3.1 in ([Hain 1983](#Hain)).

We unwind in steps how this comes about:

First, the space of [[graded algebra]] [[homomorphisms]] $A  \leftarrow CE(\mathfrak{g})$ is a subspace of the space of [[linear maps]] of [[graded vector spaces]], and since $\mathrm{CE}(\mathfrak{g})$ is freely
generated as a graded algebra and is of finite type by assumption, this is isomorphic to the
space of grading preserving homomorphisms

$$
  \mathrm{Hom}_{\mathrm{Vect}[{\mathbb{Z}]}}(\mathfrak{g}^*,A)
$$

of linear grading-preserving maps from the graded vector space $\mathfrak{g}^*$ of dual generators to $A$. By the usual relation in $\mathrm{Vect}[\mathbb{Z}]$ for $\mathfrak{g}$ of finite type, this is isomorphic to the space of elements of total degree degree 1 in elements of $A$ tensored with $\mathfrak{g}$:

$$
  (A \otimes \mathfrak{g})_1
  \simeq
  \mathrm{Hom}_{\mathrm{Vect}[{\mathbb{Z}]}}(\mathfrak{g}^*,A)
  \,.
$$

The dg-algebra homomorphism form the subspace of this space

$$
    \mathrm{Hom}_{dgAlg}(\mathrm{CE}(\mathfrak{g}),A)
    \hookrightarrow
    \mathrm{Hom}_{grAlg}(\mathrm{CE}(\mathfrak{g}),A)
    \simeq
    (A \otimes \mathfrak{g})_1
$$

on elements that respect the differential. Under the above equivalence this are elements $a$ in $(A \otimes \mathfrak{g})_1$ satisfying a certain condition. By inspection one finds that this condition is precisely the MC equation

$$
  d_A A + \partial a  + [a \cdot_A a]_2 + [a \cdot_A a \cdot_A a ]_3
      + \cdots = 0
      \,.
$$

For instance if $A = \Omega^\bullet(X)$ is the [[de Rham algebra]] of a [[smooth manifold]] $X$, then $MC(A \otimes \mathfrak{g})$ is the space of flat [[infinity-Lie algebroid-valued differential form|L-∞ algebra valued differential forms]] on $X$. See there for more details.

## Examples

For $\mathfrak{g}$ a [[Lie algebra]], $X$ a [[smooth manifold]], there is a canonical dg-Lie algebra structure on $\Omega^\bullet(X) \otimes \mathfrak{g}$.

A Maurer-Cartan element is then precisely a [[Lie algebra valued 1-form]] $A$ whose [[curvature]] 2-form vanishes

$$
  d_{dR} A + [A \wedge A] = 0
  \,.
$$

## Range of versions and applications

_Maurer--Cartan equation_ is a name for very many related equations in [[geometry]], [[algebra]], [[deformation theory]], [[category theory]] and [[deformation quantization]]. Such equations express for example certain conditions in theory of isometric embedding of submanifolds into a euclidean space ('structure equations', with relations to the Lie groups $O(n)$), invariance of invariant differential forms ([[Maurer-Cartan form|Maurer-Cartan forms]]) on Lie groups, flatness of connections on principal or associated fibre bundles, the solutions in some contexts parametrize infinitesimal deformations, or define [[twisting cochain|twisting cochains]].
In the context of BV-quantization, a Maurer--Cartan equation has the role of classical master equation.

A Maurer--Cartan equation for $A_\infty$-[[A-infinity-algebra|algebra]]s is usually referred to as a _generalized Maurer--Cartan equation_ as it has more summands than the one for [[dg-algebra]]s. In some contexts like $A_\infty$-[[A-infinity-category|categories]], some authors prefer the geometric terminology 'homological vector field' as a datum on a formal geometric space which satisfies a Maurer--Cartan equation. Solutions to Maurer-Cartan equation for a dg- or $A_\infty$ algebra are called Maurer-Cartan elements.

## Maurer-Cartan and Lie theory

[[Sophus Lie]] considered groups of transformations first and discovered [[Lie algebra]]s only later (letter to Mayer, 1874). He has shown that infinitesimally one can solve the Maurer--Cartan equations for a given set of structure constants of a finite-dimensional Lie algebra. This means that one can construct a neighborhood with either the invariant differential form, or dually the invariant vector fields whose commutator corresponds to the commutator of the Lie algebra. This amounts to integrating the Lie algebra to a [[local Lie group]]. Only much later, Elie Cartan succeeded in proving the global version of integration, that is the Cartan--Lie theorem. J-P. Serre in an influential textbook called the Cartan--Lie theorem the "[[Lie's three theorems|third Lie theorem]]", which became a rather popular term in recent years, though one should correctly call so just the theorem on local solvability of Maurer--Cartan equation.

## References

The original article:

* [[Ludwig Maurer]], _Über allgemeinere Invarianten-Systeme_, Münch. Ber. __18__ (1888), 103-150.

Textbook accounts:

* {#Nakahara03} [[Mikio Nakahara]], Section 5.6.4 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [[Martin Markl]], Def. 3.11 in: _Deformation theory of algebras and their diagrams_, 129 pp, CBMS __116__, AMS 2012 ([ISBN:978-0-8218-8979-4](http://www.ams.org/bookstore-getitem/item=CBMS-116), [toc pdf](https://www.gbv.de/dms/goettingen/721467989.pdf))

* [[Gerd Rudolph]], [[Matthias Schmidt]], Prop. 1.4.9 in: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

A MathOverflow entry about Maurer-Cartan forms for Lie groups: [maurer-cartan-form](http://mathoverflow.net/questions/34418/maurer-cartan-form)

On  [[Lie's third theorem]] from the point of view of Maurer--Cartan equations:

* Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces

* N. Bourbaki, Lie algebras and lie groups, historical appendix

* F. Engel, P. Heegaard, Sophus Lie Samlede Avhandliger (Collected works)

In the generality of [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Martin Doubek]], [[Martin Markl]], [[Petr Zima]], equation (31) in: *Deformation Theory (lecture notes)*, Archivum mathematicum 43(5), 2007, 333-371 ([arXiv:0705.3719](https://arxiv.org/abs/0705.3719))

* [[Andrey Lazarev]], Def. 5.1 in: *Maurer-Cartan moduli and models for function spaces*, Advances in Mathematics Volume 235, 1 March 2013, Pages 296-320 ([arxiv:1109.3715](http://arxiv.org/abs/1109.3715), [doi:10.1016/j.aim.2012.11.009](https://doi.org/10.1016/j.aim.2012.11.009))

* Joseph Chuang, [[Andrey Lazarev]], Def. 1.6 in: *Combinatorics and formal geometry of the master equation*,  Lett. Math. Phys. **103** (2013) 79–112  ([arXiv:1205.5970](https://arxiv.org/abs/1205.5970), [doi:10.1007/s11005-012-0586-1](https://doi.org/10.1007/s11005-012-0586-1))

Also around def. 3.1 in

* {#Hain} R. M. Hain, _Twisting cochains and duality between minimal algebras and minimal Lie algebras_, Trans. Amer. Math. Soc. **277** (1983), no. 1, 397--411.

In relation to [[equations of motion]] of [[Yang-Mills theory]] and [[gravity]] (by truncation of [[string field theory]]):

* [[Anton M. Zeitlin]], *Formal Maurer-Cartan Structures: from CFT to Classical Field Equations*, JHEP 0712:098, 2007 ([arXiv:0708.0955](https://arxiv.org/abs/0708.0955))

[[!redirects Maurer-Cartan equations]]

[[!redirects Maurer-Cartan equation]]

[[!redirects Maurer-Cartan element]]
[[!redirects Maurer-Cartan elements]]


Maurer-Cartan equation > history

< [[Maurer-Cartan equation]]
[[!redirects Maurer-Cartan equation -- history]]


Maurer-Cartan form

+-- {: .rightHandSide}
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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## On Lie groups

### Idea

For $G$ a [[Lie group]], the _Maurer-Cartan form_ on $G$ is a canonical [[Lie-algebra valued 1-form]] on $G$. One can generalize also to the Maurer-Cartan form on a principal bundle.

### Definition in synthetic differential geometry

Speaking in terms of [[synthetic differential geometry]] the Maurer-Cartan form has the following definition:

any two points $x,y \in G$ are related by a unique group element $\theta(x,y)$ such that $y = x \cdot \theta(x,y)$. If $x$ and $y$ are [[infinitesimal space|infinitesimally close]] points, defining a [[tangent vector]], then $\theta(x,y)$ is an element of the [[Lie algebra]] of $G$. So $\theta$ restricted to infinitesimally close points is a $\mathfrak{g}$-valued 1-form, and this is the Maurer-Cartan form.

### Analytic definition

In terms of [[analysis]] there is a direct analogue of this definition: a [[tangent vector]] on $G$ at $g \in G$ may be identified with an equivalence class of [[smooth function]] $\gamma : [0,1] \to G$ with $\gamma(0) = g$. The tangent vectors through the origin $x = e$ are canonically identified with the [[Lie algebra]] of $G$. By left-translating a path through $g$ back to the origin $g^{-1}\gamma : [0,1] \to G \stackrel{g^{-1} \cdot(-)}{\to} G$ it represents a Lie algebra element. This map

$$
  \theta := g^{-1}_* : [\gamma] \mapsto [g^{-1} \gamma]
$$

of tangent vectors to Lie algebra elements is the Maurer-Cartan form.

If we write $g : G \to G$ for the identity function on $G$, then $d g : T G \to T G$ is the identity function on the tangent vectors of $G$.  With this the Maurer-Cartan form may be written

$$
  g^{-1}_* d g : T G \to T_e G = \mathfrak{g}
  \,.
$$

If $G$ is a [[matrix Lie group]], then $g^{-1}_*$ is literally just left-multiplication of matrices and therefore the Maurer-Cartan form is often written just

$$
  \theta = g^{-1} d g
  \,.
$$

## Properties

### Curvature

The Maurer-Cartan form is a Lie-algebra valued form with vanishing [[curvature]].

$$
  d \theta + \frac{1}{2}[\theta \wedge \theta] = 0
$$

This is known as the [[Maurer-Cartan equation]].

Synthetically this is just a restatement of the fact that for $x,y \in G$ there is a _unique_ group element such that $y = x \cdot g$: therefore for three points $x,y,z$ we have

$$
  \array{
    && y
    \\
    & {}^{\mathllap{\theta}(x,y)}\nearrow && \searrow^{\mathrlap{\theta}(y,z)}
    \\
    x &&\stackrel{\theta(x,z)}{\to}&& z
  }
$$

i.e. $\theta(x,y) \theta(y,z) = \theta(x,z)$. This is what analytically becomes the statement of vanishing curvature.

### Pullback

If $X$ is a [[smooth manifold]] and $h : X \to G$ a [[smooth function]] with values in $G$, we have the [[pullback form]]

$$
  h^* \theta \in \Omega^1(X,\mathfrak{g})
$$

of the Maurer-Cartan form on $X$. Using the above notation, writing simply $h^{-1}$ for $h^{-1}_*$ this is

$$
  h^* \theta = h^{-1} d h
 \,.
$$

Now $d h : T X \to T G$ is no longer (necessarily) the identity map as $g$ was when we wrote $\theta = g^{-1} d g$ above, but the form of this equation shows why it can be useful to think of $\theta$ itself in terms of the identity map $d g : T G \to T G$.

### Gauge transformations

The Maurer-Cartan form crucially appears in the formula for the gauge transformation of [[Lie-algebra valued 1-form]]s.

For $u : \mathbb{R} \to G$ a smooth function and $A \in \Omega^1(\mathbb{R}, \mathfrak{g})$ a Lie-algebra valued form, the condition that $u$ is _flat_ with respect to $A$ is that it satisfies the [[differential equation]]

$$
  d u = -(R_u)_* \circ A
$$

(where $R$ denotes the right multiplication action of $G$ on itself). This is such that if $G$ happens to be a [[matrix Lie group]] it is equivalent to

$$
  (d + A) u = 0
  \,.
$$

We call the unique solution $u $ of this differential equation that satisfies $u(0) = e$ the [[parallel transport]] of $A$ and write it $u = P \exp(\int_0^{(-)} A)$.

Now for $g : \mathbb{R} \to G$ a function, the _gauge transformed_ parallel transport is

$$
  g^{-1} P \exp(\int_0^{(-)} A) g
  \,.
$$

This solves a differential equation as above, but for a different 1-form $A'$. The relation is

$$
  A' = Ad_{g^{-1}} A + g^* \theta
$$

or equivalently, with adopted notation

$$
  A' = g^{-1}A g + g^{-1} d g
  \,.
$$

## On smooth $\infty$-groups {#OnInftyLieGroup}

The theory of [[Lie groups]] embeds into the more general context of [[smooth ∞-groupoid]]s. In this context the Maurer-Cartan form has an (even) more general abstract definition that does not even presuppose the notion of differential form as such:

for every [[smooth ∞-group]] $G \in Smooth\infty Grpd$ with [[delooping]] $\mathbf{B}G$ there is canonically an [[smooth ∞-groupoid]] $\mathbf{\flat}_{dR} \mathbf{B}G$ as described here. Morphisms $X\to \mathbf{\flat}_{dR}\mathbf{B}G$ correspond to flat $\mathfrak{g}$-valued differential forms on $G$.

This fits into a double [[(∞,1)-pullback]] diagram

$$
  \array{
    G &\to& *
    \\
    {}^{\mathllap{\theta}}\downarrow && \downarrow
    \\
    \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat} \mathbf{B}G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}G
  }
  \,.
$$

The morphism

$$
  \theta : G \to \mathbf{\flat}_{dR}\mathbf{B}G
$$

in this diagram is the $\infty$-Maurer-Cartan form on $G$. For $G$ an ordinary Lie group, this reduces to the above definition. This statement and its proof is spelled out [here](smooth+infinity-groupoid+--+structures#CanonicalFormOnLieGroup).

## On cohesive and stable homotopy types
 {#OnCohesiveHomotopyTypes}

### Definition

Therefore generally for $\mathbf{H}$ a [[cohesive (∞,1)-topos]] and $G \in \mathbf{H}$ an [[∞-group]] object, one may think of

$$
  \theta \coloneqq fib(\flat \to \mathbf{B})
$$

as the Maurer-Cartan form on [[∞-group]] objects

$$
  \theta_G \;\colon\; G \stackrel{}{\longrightarrow} \flat_{dR}\mathbf{B}G
  \,.
$$

This is discussed at _[[cohesive infinity-topos -- structures]]_ in the section _[Maurer-Cartan forms and curvature characteristics](cohesive+infinity-topos+--+structures#CurvatureCharacteristics)_.

This includes then for instance Maurer-Cartan forms in higher [[supergeometry]] as discussed at _[[Super Gerbes]]_.

### Properties

#### Relation to the Chern character

Given a [[stable homotopy type]] $\hat E$ in [[cohesion]], then the [[shape modality|shape]] of the Maurer-Cartan form plays the role of the _[[Chern character]]_ on $E \coloneqq \Pi(\hat E)$-cohomology.

See at _[[Chern character]]_ for more on this, and see at _[[differential cohomology diagram]]_.

## Related concepts

* [[invariant differential form]]

## References

The [[synthetic differential geometry|synthetic]] view on the Maurer-Cartan form is discussed in

* [[Anders Kock]], _Synthetic geometry of manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

The synthetic Maurer-Cartan form itself appears in example 3.7.2. The synthetic vanishing of its curvature is corollary 6.7.2.

[[!redirects Maurer-Cartan forms]]


Maurice G. Kendall

[[!redirects M. G. Kendall]]

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

## Selected writings

Introducing the [[Kendall tau distance]]:

* [[Maurice Kendall]], *A new measure of rank correlation*, Biometrika, Volume 30, Issue 1-2, June 1938, Pages 81–93 ([doi:10.1093/biomet/30.1-2.81](https://doi.org/10.1093/biomet/30.1-2.81))

category: people

[[!redirects Maurice Kendall]]


Mauricio Romo

* [webpage](https://www.ias.edu/scholars/mauricio-romo)

## related $n$Lab entries

* [[discrete torsion]]

* [[ABJM model]]

* [[supergravity C-field]]

category: people


Mauro Jaskelioff

* [personal page](https://www.fceia.unr.edu.ar/~mauro/)

* [GoogleScholar page](https://scholar.google.com/citations?user=9icg_zQAAAAJ)

## Selected writings

On [[monad transformers]] for [[monadic effects]]:

* [[Mauro Jaskelioff]], [[Eugenio Moggi]], *Monad Transformers as Monoid Transformers*, Theoretical Computer Science **411** 51–52 (2010) 4441-4466 [[doi:10.1016/j.tcs.2010.09.011](https://doi.org/10.1016/j.tcs.2010.09.011), [pdf](http://www.disi.unige.it/person/MoggiE/ftp/tcs10.pdf)]

category: people


Mauro Porta

* [MO page](http://mathoverflow.net/users/36690/mauro-porta)

## Related $n$Lab entries

* [[formal moduli problem]]
* [[Hall algebra]]
* [[McKay correspondence]]
* [[derived analytic geometry]]

## Selected works

* Mauro Porta, Francesco Salla, _Two-dimensional categorified Hall algebras_, J. Eur. Math. Soc. __25__:3 (2023) 1113-1205 [doi](https://doi.org/10.4171/jems/1303)
* Mauro Porta, _Derived formal moduli problems_, master thesis 2013, [pdf](http://algant.eu/documents/theses/porta.pdf)
* [[Duiliu-Emanuel Diaconescu]], Mauro Porta, Francesco Sala, _McKay correspondence, cohomological Hall algebras and categorification_, [arXiv:2004.13685](https://arxiv.org/abs/2004.13685)
* Mauro Porta, Tony Yue Yu, _Higher analytic stacks and GAGA theorems_, Adv. Math. __302__ (2016) 351-409 [arXiv:1412.5166](https://arxiv.org/abs/1412.5166) [doi](https://doi.org/10.1016/j.aim.2016.07.017)

* Mauro Porta, Tony Yue Yu, _Derived non-archimedean analytic spaces_, Selecta Math. New Ser. 24, 609-665 (2018) [arXiv:1601.00859](https://arxiv.org/abs/1601.00859)  [doi](https://doi.org/10.1007/s00029-017-0310-1)

* Julian Holstein, Mauro Porta, _Analytification of mapping stacks_, [arXiv:1812.09300](https://arxiv.org/abs/1812.09300)

* [[Mauro Porta]], _Derived complex analytic geometry I: GAGA theorems_, [arXiv:1506.09042](https://arxiv.org/abs/1506.09042)

* [[Mauro Porta]], _Derived complex analytic geometry II: square-zero extensions_, [arXiv:1507.06602](https://arxiv.org/abs/1507.06602)

category: people


Mauro Spera

__Mauro Spera__ is an Italian mathematical phycisist. Much of his research is about differential geometry applied to quantum mechanics and applications.

* [web](https://docenti.unicatt.it/ppd2/en/docenti/21065/mauro-spera/ricerca)

## Selected works

* [[Mauro Spera]], _Geometric methods of quantum mechanics_, J.  Geometry and Symmetry in Physics __24__ 1-44 (2011) [euclid](https://projecteuclid.org/euclid.jgsp/1495677645)
* [[Emma Previato]], [[Mauro Spera]], _Isometric embeddings of infinite-dimensional Grassmannians_, Regul. Chaot. Dyn. __16__:356 (2011) [doi](https://doi.org/10.1134/S1560354711030099)

He related the generalized uncertainty principle for Perelomov [[coherent state]]s and the moment map in geometric quantization:

* [[Mauro Spera]], _On a generalized uncertainty principle, coherent states and the moment map_, J. of Geometry and Physics 12 (1993) 165-182.
* [[Mauro Spera]], _On Kählerian coherent states_, in Proc. Int. Conf. on Geometry, Integrability and Quantization 241-256, Institute of Biophysics and Biomedical Engineering, Bulg. Acad. Sci. 2000 ([djvu](http://www.bio21.bas.bg/proceedings/Proceedings_files/vol1/GEOM19.DJVU))

category: people


Max Born

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

## Selected writings

Early discussion of [[quantum mechanics]] (and apparently introducing that term):

* [[Max Born]], [[Pascual Jordan]], *Zur Quantenmechanik*, Zeitschrift für Physik **34** (1925) 858–888 [[doi:10.1007/BF01328531](https://doi.org/10.1007/BF01328531)]

Introducing what came to be called the *[[Born rule]]* of [[quantum mechanics]]:

* {#Born26a} [[Max Born]], *Zur Quantenmechanik der Stoßvorgänge*, Zeitschrift für Physik **37** (1926) 863–867 [[doi:10.1007/BF01397477](https://doi.org/10.1007/BF01397477)]

* {#Born26b} [[Max Born]], *Quantenmechanik der Stoßvorgänge*, Zeitschrift für Physik **38** (1926) 803–827 [[doi:10.1007/BF01397184](https://doi.org/10.1007/BF01397184)]

* {#Born72} [[Max Born]], *Das Adiabatenprinzip in der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 167–192 [[doi:10.1007/BF01400360](https://doi.org/10.1007/BF01400360)]

Introducing the [[adiabatic theorem]]:

* [[Max Born]], [[Vladimir Fock]],  *Beweis des Adiabatensatzes*, Zeitschrift für Physik **51**  (1928) 165–180 ([doi:10.1007/BF01343193](https://doi.org/10.1007/BF01343193))

Introducing the [[DBI action]]:

* {#BornInfeld34} [[Max Born]], [[Leopold Infeld]], _Foundations of the New Field Theory_,  Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 144, No. 852 (Mar. 29, 1934), pp. 425-451 ([jstor:2935568](https://www.jstor.org/stable/2935568))

category: people


Max Dehn

Max Dehn was a German-American topologist who made significant contributions to topology and geometry.

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

category:people


Max Guillen

## Selected writings

On the [[pure spinor]]-formulation of the [[superparticle]] in [[D=11 supergravity]]-backgrounds:

* [[Nathan Berkovits]], Eduardo Casali, [[Max Guillen]], Lionel Mason, _Notes on the $D=11$ pure spinor superparticle_ ([arXiv:1905.03737](https://arxiv.org/abs/1905.03737))

On [[pure spinor]]-formalism in [[D=11 supergravity]]:

* [[Max Guillen]], _Pure spinors and $D=11$ supergravity_ ([arXiv:2006.06014](https://arxiv.org/abs/2006.06014))

On [[scattering amplitudes]] in [[D=10 super Yang-Mills theory]] in [[pure spinor]]-formalism:

* Maor Ben-Shahar, [[Max Guillen]], *10D Super-Yang-Mills Scattering Amplitudes From Its Pure Spinor Action* ([arXiv:2108.11708](https://arxiv.org/abs/2108.11708))

On the chiral [[Green-Schwarz superstring]] and [[Berkovits superstring]]:

* [[Max Guillen]], *Green-Schwarz and Pure Spinor Formulations of Chiral Strings* ([arXiv:2108.11724](https://arxiv.org/abs/2108.11724))

category: people


Max Karoubi

**Max Karoubi** is a French topologist, with main interests in algebraic and topological K-theory, cyclic (co)homology and [[noncommutative geometry]] à là Connes. He introduced the noncommutative differential forms to generalize the de Rham complex to [[noncommutative geometry]].

* personal [webpage](http://people.math.jussieu.fr/~karoubi/);

* German [wikipedia](http://de.wikipedia.org/wiki/Max_Karoubi)

## Selected writings

Introducing what came to be known as [[Karoubi K-theory]]:

* [[Max Karoubi]], *Espaces Classifiants en K-Théorie*, Transactions of the American Mathematical Society **147** 1 (Jan., 1970) 75-115 $[$[doi:10.2307/1995218](https://doi.org/10.2307/1995218)$]$

Introducing [[Hermitian K-theory]]:

* [[Max Karoubi]], [[Orlando Villamayor]], *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]

* {#Karoubi10} [[Max Karoubi]], *Le théorème de périodicité en K-théorie hermitienne*, in *Quanta of Maths*, Clay Mathematics Proceesings **11**, AMS and Clay Math Institute Publications (2010) 257-282 [[arXiv:0810.4707](https://arxiv.org/abs/0810.4707), [pdf](https://www.claymath.org/library/proceedings/cmip011c.pdf)]

* [[Max Karoubi]], [[Mariusz Wodzicki]], *Algebraic and Hermitian K-theory of $\mathcal{K}$-rings*, The Quarterly Journal of Mathematics **64** 3 (2013) 903–940 [[arXiv:1310.4123](https://arxiv.org/abs/1310.4123), [doi:10.1093/qmath/hat030](https://doi.org/10.1093/qmath/hat030)]

and [[topological Hermitian K-theory]]:

* [[Max Karoubi]], *Périodicité de la K-théorie hermitienne*, in: [[Hyman Bass]] (ed.), *Algebraic K-Theory III -- Hermitian K-Theory and Geometric Applications*, Lecture Notes in Mathematics **343** (1973) 301-411 [[doi:10.1007/BFb0061366](https://doi.org/10.1007/BFb0061366)]

On [[topological K-theory]]:

* {#Karoubi78} [[Max Karoubi]], _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

* _Lectures on K-theory_, Proc. of the Trieste Conference in 2002, Contemporary Developments in Algebraic K-theory, ICTP Lecture notes __15__ (2003).

On [[cyclic homology]], [[topological K-theory]] and [[differential K-theory]]:

* [[Max Karoubi]], *Homologie cyclique et K-théorie*, Astérisque, no. 149 (1987) [[numdam:AST_1987__149__1_0](http://www.numdam.org/item/AST_1987__149__1_0)]

On [[algebraic topology]] and [[differential geometry]]:

* _Algebraic Topology via Differential Geometry_ (with Christian Leruste). Cambridge Univ. Press (1987), based on _Méthodes de géométrie différentielle en topologie algébrique_ (1982, Paris 7)

On a form of [[Hodge-filtered differential cohomology|Hodge-filtered differential]] [[topological K-theory]]:

* [[Max Karoubi]], *Théorie générale des classes caractéristiques secondaires*, K-Theory **4** 1 (1990) 55-87 [[doi:10.1007/BF00534193](http://dx.doi.org/10.1007/BF00534193), [pdf](https://webusers.imj-prg.fr/~max.karoubi/Publications/43.pdf)]

* [[Max Karoubi]], *Classes Caractéristiques de Fibrés Feuilletés, Holomorphes ou Algébriques*, in: *Proceedings of Conference on Algebraic Geometry and Ring Theory* in honor of Michael Artin, Part II (Antwerp 1992), K-Theory **8** 2 (1994) 153-211 [[doi:10.1007/BF00961455](http://dx.doi.org/10.1007/BF00961455)]

On [[equivariant K-theory]]:

* [[Max Karoubi]], _Equivariant K-theory of real vector spaces and real vector bundles_, Topology and its Applications, 122, (2002) 531-456 ([arXiv:math/0509497](https://arxiv.org/abs/math/0509497))

On [[KR-theory]] of [[algebraic varieties]]:

* [[Max Karoubi]], [[Charles Weibel]], *Algebraic and Real K-theory of Algebraic varieties*, Topology **42** (2003) 715-742 [[arXiv:math/0509412](https://arxiv.org/abs/math/0509412), doi:10.1016/S0040-9383(02)00069-1]

On [[Bott periodicity]]:

* {#Karoubi05} [[Max Karoubi]], _Bott Periodicity in Topological, Algebraic and Hermitian K-Theory_,  in: [[Eric Friedlander]], [[Daniel Grayson]] (eds.) *[[Handbook of K-Theory]]*, Springer (2005) [[doi:10.1007/978-3-540-27855-9_4](https://doi.org/10.1007/978-3-540-27855-9_4)]

On [[twisted K-theory]], [[Hermitian K-theory]] and [[KR-theory]]:

* [[Max Karoubi]], [[Marco Schlichting]], [[Charles Weibel]], *The Witt group of real algebraic varieties*, Journal of Topology **9** 4 (2016) 1257-1302 [[arXiv:1506.03862](https://arxiv.org/abs/1506.03862), [doi:10.1112/jtopol/jtw024]( https://doi.org/10.1112/jtopol/jtw024)]

* [[Max Karoubi]], [[Charles Weibel]], *Twisted $K$-theory, Real $A$-bundles and Grothendieck–Witt groups*, Journal of Pure and Applied Algebra **221** 7 (2017) 1629-1640 [[doi:10.1016/j.jpaa.2016.12.020](https://doi.org/10.1016/j.jpaa.2016.12.020)]

## Related entries

* [[Karoubian category]] ([[pseudo-abelian category]])

* [[Karoubian envelope]]

* [[hyperbolic functor]]

category: people