Markov chain

A __Markov chain__ (named for [[Andrey Markov]]) is a sequence of random variable taking values in the state space of the chain, with the property that the probability of moving to the next state depends only upon the current state:

$$
Pr(X_{n + 1} = x | X_1 = x_1, X_2 = x_2,..., X_n = x_n) = Pr(X_{n + 1} = x | X_n = x_n).
$$

A Markov chain can also be described as a [[coalgebra for an endofunctor|coalgebra]] for the [[endofunctor]] on [[Set]] which maps a set $X$ to the set of [[probability distribution|probability distributions]] on $X$.

+--{: .query}
[[Ian Durham]]: I assume that a Markov chain can be represented as a directed graph in some way and thus can be used to generate a free category of some sort.  Is this a correct assumption?

[[Eric]]: I'm pretty sure that is the case for _finite_ Markov chains.

[[Ian Durham]]: Hmmm.  I'll have to think about that.
=--

## Literature

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

Baez and his students recently defined a generalization, open Markov chains:

* [[John C. Baez]], [[Brendan Fong]], [[Blake S. Pollard]], _A compositional framework for Markov processes_, [arxiv/1508.06448](http://arxiv.org/abs/1508.06448)

category: probability

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Markov kernel

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### Context
#### Measure and probability theory
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#Contents#
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## Idea

A **Markov kernel** (also called **transition kernel**, **stochastic kernel**, or **probability kernel**) is a mathematical formalization of a "function with random outcomes", or "random transition".
(Sometimes the term *transition kernel* denotes a more general, non-normalized mapping.)

It can be thought of as a generalization of a [[stochastic map]] outside the finite discrete case.
(Sometimes the term "stochastic map" is itself used to denote a Markov kernel.)

Markov kernels, in the form of regular conditional distributions (see [below](#regular_conditional_distributions)) are also the standard setting for talking about [[conditional probability]] outside of the discrete setting.

Markov kernels and their categories are among the basic building blocks of [[categorical probability]].

## Definition

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be [[measurable spaces]], i.e. sets equipped each with a [[sigma-algebra]].
A **Markov kernel** $(X,\mathcal{A})\to(Y,\mathcal{B})$ is an assignment
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
X \times \mathcal{B} \ar{r}{k} & {[0,1]} \\
(x,B) \ar[mapsto]{r} & k(B|x)
\end{tikzcd}
which is

* [[measurable function|measurable]] as a function of $x$, for all $B\in\mathcal{B}$;
* a [[probability measure]] as a function of $B$, for all $x\in X$.

The quantity $k(B|x)$ can be thought of as the probability of transitioning to $B$ if we are in $x$, or of the [[conditional probability]] of the event $B$ given $x$.

## Basic structures and properties

### As Kleisli morphisms

A Markov kernel $k:(X,\mathcal{A})\to(Y,\mathcal{B})$ can be equivalently expressed as a measurable map
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
X \ar{r}{k^\sharp} & PY
\end{tikzcd}
where $P$ denotes the [[Giry monad]] on [[Meas]], with the Giry sigma-algebra.
In other words, a Markov kernel is exactly a [[Kleisli morphism]] for the Giry monad.
Particular Markov kernels are also Kleisli morphisms for other [[probability monads]].

### Almost sure equality

Consider two $k,h:(X,\mathcal{A})\to(Y,\mathcal{B})$ and a measure $p$ on $(X,\mathcal{A})$.
The kernels $k$ and $h$ are said to be **$p$-almost surely equal** or **$p$-almost everywhere equal** if for every $B\in\mathcal{B}$, the quantities $k(B|x)$ and $h(B|x)$ only differ on a set of measure zero.
Note that in general this set depends on the choice of $B$, but in some cases, such as when $(Y,\mathcal{B})$ is [[standard Borel]], it can be taken independently of $B$.

### Measure-preserving kernels

Let $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$ be [[probability spaces]], i.e. [[measurable spaces]] equipped each with a [[probability measure]].
A **measure-preserving kernel** $(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$ is a kernel between the underlying measurable spaces $k:(X,\mathcal{A})\to(Y,\mathcal{B})$
such that for every $B\in\mathcal{B}$,
$$
\int_X k(B|x)\,p(d x) = q(B) .
$$

### Kernels from deterministic functions

The [[identity]] function on a measurable set $(X,\mathcal{A})$ defines a kernel as follows,
$$
\delta(A|x) = \delta_x(A) = 1_A(x) = \begin{cases}
1 & x\in A ; \\
0 & x\notin A
\end{cases}
$$
for every $x\in X$ and $A\in\mathcal{A}$. This gives the identity morphisms in the categories of kernels [[Markov kernel#categories_of_markov_kernels|below]].

More generally, let $f:(X,\mathcal{A})\to (Y,\mathcal{B})$ be a [[measurable function]]. We can define the kernel $\delta_f:(X,\mathcal{A})\to (Y,\mathcal{B})$ as follows,
$$
\delta_f(B|x) = \delta_f(x)(B) = 1_{f^{-1}(B)}(x) = \begin{cases}
1 & f(x)\in B ; \\
0 & f(x)\notin B
\end{cases}
$$
for every $x\in X$ and $B\in\mathcal{B}$.
Intuitively, this kernel represents the deterministic transition, from $x$ to $f(x)$ with probability one.
This construction induces a [[functor]] from the category [[Meas]] to the categories of kernels [[Markov kernel#categories_of_markov_kernels|below]].
(Compare with the [[stochastic map#stochastic_maps_from_deterministic_functions|analogous construction for stochastic maps]].)

Note that if $f:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$ is a [[measure-preserving function]], then $\delta_f$ is a measure-preserving kernel in the sense defined above.

### Bayesian inversion

Given a measure-preserving Markov kernel $k:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$, a **[[Bayesian inverse]]** of $k$ is a Markov kernel $k^\dagger:(Y,\mathcal{B},q)\to(X,\mathcal{A},p)$ such that for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$,
$$
\int_A k(B|x)\,p(d x) = \int_B k^\dagger(A|y)\,q(d y) .
$$
The last formula can be seen as an instance of [[Bayes' formula]], as it generalize the usual discrete formula
$$
P(b|a)\,P(a) = P(a|b)\,P(b) .
$$
We have that

* Every kernel between [[standard Borel]] spaces admits a Bayesian inverse;

* If a kernel $k:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$ admits a Bayesian inverse $k^\dagger:(Y,\mathcal{B},q)\to(X,\mathcal{A},p)$, then any kernel $h:(Y,\mathcal{B},q)\to(X,\mathcal{A},p)$ is a Bayesian inverse of $k$ if and only if it is $q$-almost surely equal to $k^\dagger$.

* Bayesian inverses, when they exist, depend crucially on the measure $p$.

* Bayesian inversion makes the category [[Krn]] (see [[Markov kernel#categories_of_markov_kernels|below]]) a [[dagger category]].

## Regular conditional distributions

A Markov kernel can be seen as a [[probability measure]] which depends [[measurable function|measurably]] on a parameter.
Therefore, together with the idea of [[conditional expectation]], they a very useful tool to talk about [[conditional probability]] without incurring into paradoxes. Let's see how.

Let $(X,\mathcal{A},p)$ be a [[probability space]], and consider a sub-[[sigma-algebra]] $\mathcal{B}\subseteq\mathcal{A}$.
Recall that

* the [[conditional expectation]] of an integrable function $f:(X,\mathcal{A},p)\to\mathbb{R}$ given $\mathcal{B}$ is a function $\mathbb{E}[f|\mathcal{B}]:X\to\mathbb{R}$ which is $\mathcal{B}$-measurable ("coarser" than $f$), and such that for every $B\in\mathcal{B}$,
$$
\int_B f \, d p = \int_B \mathbb{E}[f|\mathcal{B}] \, d p .
$$
(Such a function always exists by the [[Radon-Nikodym theorem]].)

* the [[conditional probability]] of an event (measurable subset) $A\in\mathcal{A}$ is the conditional expectation of its indicator function:
$$
\mathbb{P}[A|\mathcal{B}] = \mathbb{E}[1_A|\mathcal{B}] .
$$

Note that $\mathbb{P}[A|\mathcal{B}]$ is a function,
$$
x\mapsto \mathbb{P}[A|\mathcal{B}](x)
$$
which is $\mathcal{B}$-measurable for every $A\in\mathcal{A}$. Therefore, in order for the assignment
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
X \times \mathcal{A} \ar{r} & {[0,1]} \\
(x,A) \ar[mapsto]{r} & \mathbb{P}{[A|\mathcal{B}]}(x)
\end{tikzcd}
to be a Markov kernel $(X,\mathcal{B})\to(X,\mathcal{A})$ (see the definition [above](#definition)), we moreover need that for every $x\in X$ the assignment
$$
A \mapsto \mathbb{P}[A|\mathcal{B}](x)
$$
be a probability measure on $(X,\mathcal{A})$.
This is not always the case. When this happens, we call the resulting kernel $(X,\mathcal{B})\to(X,\mathcal{A})$ a **regular conditional distribution** given $\mathcal{B}$.
We can view a regular conditional distribution as a probability measure on $(X,\mathcal{A})$ which is parametrized in a measurable way by $(X,\mathcal{B})$.

The **disintegration theorem** says that when $(X,\mathcal{A})$ is standard Borel, regular conditional distributions in the form above always exist.

### Regular conditionals from a joint distribution

Using the disintegration theorem we can obtain conditional probability distributions in a way that generalizes the discrete formula
$$
p(b|a) = \frac{p(a,b)}{p(a)}
$$
to the non-discrete setting.

Given standard Borel spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, their product $(X\times Y,\mathcal{A}\otimes\mathcal{B})$ is again standard Borel. The projection map $\pi_1:(X\times Y,\mathcal{A}\otimes\mathcal{B})\to (X,\mathcal{A})$ induces a sub-sigma-algebra $\pi_1^{-1}(\mathcal{A})$ on $X\times Y$ (which makes it isomorphic to $(X,\mathcal{A})$ in both [[Stoch]] and [[Krn]], see [Section 2.1 of this paper](#ergodic_dagger)).

Given a joint distribution $r$ on $(X\times Y,\mathcal{A}\otimes\mathcal{B})$, we can then form the regular conditional $(X\times Y,\pi_1^{-1}(\mathcal{A}))\to(X\times Y,\mathcal{A}\otimes\mathcal{B})$ (or equivalently, up to isomorphism, $(X,\mathcal{A})\to(X\times Y,\mathcal{A}\otimes\mathcal{B})$).
Further composing with the projection $\pi_2:(X\times Y,\mathcal{A}\otimes\mathcal{B})\to(Y,\mathcal{B})$ (or equivalently restricting the kernel to the sub-sigma-algebra $\pi_2^{-1}(\mathcal{B})$) we then get a kernel
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
X\times\mathcal{B} \ar{r} & {[0,1]} \\
(x,B) \ar[mapsto]{r} & \mathbb{P}{[\pi_2^{-1}(B)|\pi_1^{-1}(\mathcal{A})]}(x) ,
\end{tikzcd}
sometimes denoted simply $r(B|x)$ or $r'(B|x)$.

This kernel satisfies the property that for every $A\in\mathcal{A}$ and $B\in\mathcal{B}$,
$$
\int_A r'(B|x) \, p(d x) = r(A\times B) ,
$$
and is therefore a generalization of conditional probability to the non-discrete setting: compare it to the formula
$$
P(b|a)\,P(a) = P(a,b) .
$$

## Categories of Markov kernels

* [[Stoch]] is the category of [[measurable spaces]] and Markov kernels between them.

* One can also form the category of [[probability spaces]] and equivalence classes of Markov kernels between them (under [[Markov kernel#almost_sure_equality|almost sure equality]]).

* One can restrict the category above to [[standard Borel]] probability spaces, obtaining a [[dagger category]] sometimes denoted by [[Krn]], which is equivalent to the [[category of couplings]].

## Related concepts

* [[Markov category]]

* [[Stoch]]

* [[category of couplings]]

* [[stochastic map]]

* [[kernel method]]

* [[Mallows kernel]]

* [[Kendall kernel]]

* [[Cayley distance kernel]]

## References

See also:

* Wikipedia, *[Markov kernel](https://en.m.wikipedia.org/wiki/Markov_kernel)*

Discussion via [[category theory]]:

* [[F. W. Lawvere]], _The Category of Probabilistic Mappings_, 1962, [PDF](https://ncatlab.org/nlab/files/lawvereprobability1962.pdf).

* Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Alexandra Silva, _Borel kernels and
their approximation, categorically_. In MFPS 34: Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics, volume 341, 91–119, 2018.  [arXiv](https://arxiv.org/abs/1803.02651).

* [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_. Adv. Math., 370:107239, 2020. [arXiv:1908.07021](https://arxiv.org/abs/1908.07021).

* {#ergodic_dagger} Noé Ensarguet, [[Paolo Perrone]], _Categorical probability spaces, ergodic decompositions, and transitions to equilibrium_.  [arXiv](https://arxiv.org/abs/2310.04267).

category: probability

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Markov's principle

# Markov\'s Principle
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{: toc}

## Definition

In [[constructive mathematics]], Markov\'s principle is the (classically trivial) statement that any infinite [[sequence]] of $0$ and $1$ that is not all $1$s must have a $0$ somewhere.  Stated in a more [[logic]]al form, if $P$ is a predicate on [[natural number]]s, then
$$ \forall{n}\, \big(P(n) \vee \neg{P(n)}\big) \Rightarrow \neg\forall{n}\, P(n) \Rightarrow \exists{n}\, \neg{P(n)} .$$
Compare this to
$$ \neg\exists{n}\, P(n) \Rightarrow \forall{n}\, \neg{P(n)} ,$$
which is a theorem of [[intuitionistic logic]].
More generally, a [[set]] $S$ may be called _Markovian_ if this principle holds for all predicates on $S$.  Every Kuratowski-[[finite set]] is Markovian, for example.

Markov\'s principle is often stated in terms of [[infinite sequences]] of natural numbers, using Greek letters for sequences and Latin letters for individual numbers:
$$ \forall{\alpha}\, \big(\neg\forall{n}\, (\alpha_n = 0) \Rightarrow \exists{n}\, (\alpha_n \ne 0)\big) .$$
Here, $P(n)$ is the statement that $\alpha_n = 0$, and the hypothesis $\forall{n} (P(n) \vee \neg{P(n)})$ is true for any sequence of natural numbers.  Conversely, given any predicate $P$ satisfying this hypothesis, we can define $\alpha$ by $\alpha_n = 0$ if $P(n)$ is true and $\alpha_n = 1$ if $P(n)$ is false, so the two versions of Markov\'s principle are equivalent.

## Discussion

In standard constructive mathematics (such as in the [[internal logic]] of a [[topos]]), it is possible that the only Markovian sets are the Kuratowski-[[finite sets]].  Thus, Markov\'s principle, stating that the set of natural numbers is Markovian, is nontrivial.  (It is true, of course, in a [[Boolean topos]]; that is, Markov\'s principle follows from the principle of [[excluded middle]].)

[[Andrey Markov Jr]] (the one who proved undecidability theorems, and son of the great stochastician) belonged to the [[Russian constructivism|Russian school of constructivism]], which saw mathematics as about computability.  From this perspective, Markov\'s principle is justified as follows:  We are justified in concluding $\exists{n}\, \neg{P(n)}$ if we can actually compute a value of $n$ such that $\neg{P(n)}$ can be proved; since $P$ is decidable, it\'s enough to compute $n$ such that $\neg{P(n)}$ is true.  And to compute this, you just set a computer working, deciding $P(0), P(1), P(2), \ldots$, until it finds $n$.  Other constructivists find this argument unconvincing, since they\'re not convinced that the computer will ever stop, even though it\'s impossible that it continue forever.

Equivalent forms:

*  If a [[Turing machine]] does not run forever, then it halts.
*  If an [[extended natural number system|extended natural number]] is not infinite, then it is finite.
*  If a [[modulated Cantor real number]] does not equal zero, then it is [[apartness relation|apart]] from zero in that it has a multiplicative inverse.

Note that the contrapositives of these are all valid regardless of Markov\'s principle.

The other major historical school of constructivism, Brouwer\'s [[intuitionism]], rejects Markov\'s principle. Brouwer\'s viewpoint has since his time been formalized, and via this formalization Markov\'s principle can be proved false. Namely, [[Kripke's schema]] with MP proves Excluded Middle, and Excluded Middle is incompatible with [[Brouwer continuity principle|continuity]]. Several models have been built satisfying Kripke\'s schema and continuity, thereby falsifying MP. These include topological models (e.g. M. Krol, ``A Topological model for intuitionistic analysis with Kripke's
Scheme," Zeitschr. f. math. Logic und Grundlagen d. Math. 24, p. 427-436, 1978), Beth models (e.g. D. van Dalen, ``An interpretation of intuitionistic analysis," Annals of Mathematical Logic 13(1), p. 1-43), realizability models (e.g. J. van Oosten, Realizability, Elsevier, 2008), and a [[Kripke model]] [BridgesRichman, p138](#BridgesRichman).  Note however that some intuitionists have advocated in favour of Markov\'s principle (and presumably then against Kripke\'s schema).

## Weak Markov\'s Principle

More recently, a weakened form of Markov\'s principle has been identified (first in ([Mandelkern 1988](#Mandelkern))) and seen to be of interest, aptly named Weak Markov\'s Principle. It states that if a binary sequence is pseudo-positive then it is positive:

$$
\forall \alpha \, \Big(\forall \beta \, \Big(\neg\neg\exists n \, (\beta(n)=1)\vee \neg\neg\exists n \, \big((\alpha(n)=1)\wedge(\beta(n)=0)\big)\Big)\rightarrow\exists n \, (\alpha(n)=1)\Big).
$$

## Analytic Markov\'s Principle

Markov\'s principle is equivalent to the assertion that for every [[modulated Cantor real number]] $a$, if $a \leq 0$ is false, then $a \gt 0$.  Another way to say this is that the standard [[tight apartness relation]] on modulated Cantor reals is a [[stable relation]].

The analogous statement for [[Dedekind real numbers]] might be called the **[[analytic Markov's principle]]**, by analogy with the [analytic LPO](principle+of+omniscience#analytic).  The [[Russian constructivism|Russian constructivists]], who used Markov\'s principle most, accepted [[countable choice]], or at least $AC_{0,0}$, which implies that these two principles are equivalent.  However, in other varieties of constructive mathematics, the analytic Markov\'s principle is stronger.

## References

* {#Mandelkern} Mark Mandelkern, _Constructively Complete Finite Sets_, Mathematical Logic Quarterly **34**, issue 2 (1988) 97–103, doi:[10.1002/malq.19880340202](https://doi.org/10.1002/malq.19880340202).

For a recent comparison see:

* Matt Hendtlass and Robert Lubarsky, _Separating fragments of WLEM, LPO, and MP_, [PDF](http://math.fau.edu/lubarsky/Separating%20LLPO.pdf).
  {#HendtlassLubarsky}

* Douglas Bridges and Fred Richman, _Varieties of Constructive Mathematics_.
  {#BridgesRichman}

category: foundational axiom

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Markus Banagl

* [webpage](http://www.mathi.uni-heidelberg.de/~banagl/)

## Selected writings

On [[topological quantum field theory]]:

* [[Markus Banagl]], _Positive topological quantum field theories_ [[arxiv/1303.4276](http://arxiv.org/abs/1303.4276)]

On [[topology|topological]] [[invariants]] of [[stratified spaces]]:

* [[Markus Banagl]], *Topological invariants of stratified spaces*, Springer Monographs in Mathematics, Springer (2000) [[doi:10.1007/3-540-38587-8](https://doi.org/10.1007/3-540-38587-8)]

On [[intersection homology]]:

* [[Markus Banagl]], *Rational generalized intersection homology theories*, Homology, Homotopy and Applications, **12** 1 (2010) 157-185  [[pdf](http://www.mathi.uni-heidelberg.de/~banagl/pdfdocs/spectrumqih.pdf), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-1/Rational-generalized-intersection-homology-theories/hha/1296223826.full)]

## Related entries

* [[Pólya enumeration theorem]]

category: people


Markus Dierigl

* [Institute page](https://live-sas-physics.pantheon.sas.upenn.edu/people/markus-dierigl)

## Selected writings

On the [[swampland cobordism conjecture]]:

* [[Markus Dierigl]], [[Jonathan Heckman]], _On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups_, Physical Review D 2021 ([arXiv:2012.00013](https://arxiv.org/abs/2012.00013))

category: people


Markus Fierz

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

## related $n$Lab entries

* [[spin-statistics theorem]]

* [[Fierz identity]]

category: people

[[!redirects Markus Eduard Fierz]]


Markus Fröb

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=211948)

## Selected writings

* {#Froeb18} [[Markus Fröb]], _Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories_ ([arXiv:1803.10235](https://arxiv.org/abs/1803.10235))

  (on [[L-infinity algebra]]-structure in [[perturbative quantum field theory]])

[[!redirects Markus Froeb]]

category: people


Markus Hausmann

* [webpage](http://www.hcm.uni-bonn.de/people/profile/markus-hausmann/)

## Selected writings

On [[proper equivariant homotopy theory]]:

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_ ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))

On [[enhanced triangulated categories]] ([[stable (infinity,1)-categories|stable $(\infty,1)$-categories]]):

* [[Stefan Schwede]], [[Markus Hausmann]], _Enhancements of triangulated categories_, Graduate Seminar Topology, Bonn 2019-2020 ([[Schwede_EnhancedSeminar.pdf:file]])

## Related $n$Lab entries

* [[global equivariant stable homotopy theory]]

category: people


Markus Land

* [webpage](http://www.math.uni-bonn.de/people/land/)

## Selected writings

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

* [[Markus Land]], *Introduction to Infinity-Categories*, Birkhäuser 2021 ([doi:10.1007/978-3-030-61524-6](https://link.springer.com/book/10.1007/978-3-030-61524-6))

On ([[equivariant K-theory|equivariant]]) [[KK-theory]] as the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (infinity,1)-category|stable $(\infty,1)$-category]]:

* {#BunkeEngelLand21} [[Ulrich Bunke]], [[Alexander Engel]], [[Markus Land]], *A stable $\infty$-category for equivariant KK-theory* $[$[arXiv:2102.13372](https://arxiv.org/abs/2102.13372)$]$

category: people


Markus Müller

* [homepage](https://markus-mueller.website/)

## Selected writings

On [[experiment|experimental]] realization of [[quantum error correction]]:

* [[Markus Müller]], [[Andreas Wallraff]] et al., *Realizing Repeated Quantum Error Correction in a Distance-Three Surface Code*, Nature **605** (2022) 669–674 [[arXiv:2112.03708](https://arxiv.org/abs/2112.03708), [doi:10.1038/s41586-022-04566-8](https://doi.org/10.1038/s41586-022-04566-8)]

* [[Markus Müller]] et al., *Demonstration of fault-tolerant universal quantum gate operations*, Nature **605** (2022) 675-680 [[doi:10.1038/s41586-022-04721-1](https://doi.org/10.1038/s41586-022-04721-1), [arXiv:2111.12654](https://arxiv.org/abs/2111.12654)]

category: people

[[!redirects Markus Mueller]]
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Markus Pflaum

* [webpage](http://euclid.colorado.edu/~pflaum/)

## Selected writings

On [[equivariant differential topology]] of [[Lie groupoids]]:

* {#PflaumPosthumaTang11} [[Markus Pflaum]], [[Hessel Posthuma]], X. Tang, _Geometry of orbit spaces of proper Lie groupoids_, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 ([arXiv:1101.0180](https://arxiv.org/abs/1101.0180), [doi:10.1515/crelle-2012-0092](https://doi.org/10.1515/crelle-2012-0092))

On [[formally integrable differential equations]]:

* Batu Güneysu, [[Markus Pflaum]], _The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE's_ ([arXiv:1308.1005](https://arxiv.org/abs/1308.1005))

See also:

* {#PflaumWilkin17} [[Markus Pflaum]], Graeme Wilkin, _Equivariant control data and neighborhood deformation retractions_, Methods and Applications of Analysis, 2019 ([arXiv:1706.09539](https://arxiv.org/abs/1706.09539))

On the [[K-theory classification of topological phases of matter]] (and more general classification in [[Whitehead generalized cohomology theories]]):

* {#BHMP23} [[Agnès Beaudry]], Michael Hermele, Juan Moreno, [[Markus Pflaum]], Marvin Qi, Daniel Spiegel, *Homotopical Foundations of Parametrized Quantum Spin Systems* [[arXiv:2303.07431](https://arxiv.org/abs/2303.07431)]

## Related entries

* [[equivariant differential topology]]

* [[formally integrable differential equation]]

category: people


Markus Reineke

* [webpage](https://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Algebra/reineke.html.de)

## Selected writings

Introducing the [[knots-quivers correspondence]]:

* {#KRSS19} [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *Knots-quivers correspondence*, Adv. Theor. Math. Phys. **23** 7 (2019) 1849-1902 [[arXiv:1707.04017](https://arxiv.org/abs/1707.04017), [doi:10.4310/ATMP.2019.v23.n7.a4](https://doi.org/10.4310/ATMP.2019.v23.n7.a4)]

* [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *BPS states, knots and quivers*, Phys. Rev. D **96** 121902 (2017) [[arXiv:1707.02991](https://arxiv.org/abs/1707.02991), [doi:10.1103/PhysRevD.96.121902](https://doi.org/10.1103/PhysRevD.96.121902)]

* Markus Reineke, _Moduli of representations of quivers_ [arXiv:0802.2147](https://arxiv.org/abs/0802.2147)

* Markus Reineke, _Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants_, Compositio Mathematica __147__:3 (2011) 943--964 [doi](https://doi.org/10.1112/S0010437X1000521X) [arXiv:0903.0261](https://arXiv.org/abs/0903.0261)

* A B Buan, R Marsh, M Reineke, I Reiten, G. Todorov, _Tilting theory and cluster combinatorics_, Adv. Math. __204__:2 (2006) 572--618 [doi](https://doi.org/10.1016/j.aim.2005.06.003)

category: people


Markus Spitzweck

* [webpage](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/preprints/Spitzweck.htm)

category: people


Markus Szymik

* [webpage](http://www.math.ntnu.no/~markussz)

## Selected writings

On [[Brauer ∞-groups]]

* {#Szymik11} [[Markus Szymik]], _Brauer spaces for commutative rings and structured ring spectra_, Manifolds and K-theory. Contemp. Math. 682 (2017) 189-208 ([arXiv:1110.2956](http://arxiv.org/abs/1110.2956))

* {#BakerRichterSzymik12} [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], _Brauer groups for commutative $\mathbb{S}$-algebras_, J. Pure Appl. Algebra 216 (2012) 2361–2376 ([arXiv:1005.5370](http://arxiv.org/abs/1005.5370))

On [[K3-spectra]]:

* {#Szymik10} [[Markus Szymik]], _K3 spectra_, Bull. Lond. Math. Soc. 42 (2010) 137-148 ([pdf](http://www.math.ku.dk/~xvd217/szymik.K3spectra.pdf), [publisher](http://blms.oxfordjournals.org/content/early/2010/01/12/blms.bdp106.abstract))

* [[Markus Szymik]], _Crystals and derived local moduli for ordinary K3 surfaces_, Adv. Math. 228 (2011) 1-21

On [[equivariant stable cohomotopy|stable]] [[equivariant cohomotopy]] and the [[equivariant Hopf degree theorem]]:

* [[Markus Szymik]], _A stable approach to the equivariant Hopf theorem_, Topology Appl. 154 (2007) 2323-2332 ([pdf](https://folk.ntnu.no/markussz/papers/szymik.hopf.pdf), [[StableApproachToEquivariantHopfTheorem.pdf:file]])

On [[characteristic]] of [[E-∞ rings]]:

* [[Markus Szymik]], _Commutative S-algebras of prime characteristics and applications to unoriented bordism_, Algebr. Geom. Topol. 14 (2014) 3717-3743 ([arXiv:1211.3239](http://arxiv.org/abs/1211.3239))

* [[Markus Szymik]],  _String bordism and chromatic characteristics_, Homotopy Theory: Tools and Applications. Contemp. Math. 729 (2019) 239-254 ([arXiv:1312.4658](http://arxiv.org/abs/1312.4658))

category: people


Marshall Stone

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

On Stone's influence on [[category theory]]:

* {#MacLane70} [[Saunders MacLane]], *The Influence of M. H. Stone on the Origins of Category Theory*, in *Functional Analysis and Related Fields*, Springer (1970) [[doi:10.1007/978-3-642-48272-4_12](https://doi.org/10.1007/978-3-642-48272-4_12)]

and via his original observation of the notion of [[adjoint operators]]

* {#MacLane88} [[Saunders Mac Lane]]: §5 in: *Concepts and Categories in Perspective*, in: P. Duren, *A century of mathematics in America* Part 1, AMS (1988) 323-365. [[pdf](http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf), [ISBN:0-8218-0124-4](https://www.ams.org/publicoutreach/math-history/hmath1-index)]

## Related entries

* [[Stone duality]]
* [[Stone-von Neumann theorem]]

[[!redirects Marshall Harvey Stone]]
[[!redirects Marshall H. Stone]]
[[!redirects M. H. Stone]]
[[!redirects M. Stone]]


Marston Morse

> Not to be confused with [[Anthony P. Morse]], known for the [[Morse–Sard lemma]].

Marston Morse (24 March 1892 – 22 June 1977) was an American mathematician, best known for [[Morse theory]].

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

## Related entries

* [[Morse theory]]

category: people


Marta Bunge

**Marta Bunge** (1938-2022) was a Argentinian-Canadian mathematician. Her research included [[fibered categories]], [[stacks]] (with values in [[categories]], hence [[2-sheaves]]) and [[groupoids]], [[topos theory]], geometric [[Galois theory]] of [[covering spaces]], [[synthetic differential topology]], amongst others.

* [personal page](https://www.math.mcgill.ca/bunge)

* [institute obituary](https://www.mcgill.ca/mathstat/people-0/memoriam/marta-c-bunge)

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

## Selected writings

On [[categories of presheaves]]:

* {#Bunge66} [[Marta Bunge]], *Categories of Set-Valued Functors*, PhD thesis, University of Pennsylvania (1966) [[[Bunge-CategoriesOfSetValuedFunctors.pdf:file]]]

On [[internal presheaves]] in [[elementary toposes]]:

* [[Marta Bunge]], *Internal Presheaf Toposes*, [[Cahiers|Cahiers de topologie et géométrie différentielle catégoriques]] **18** 3 (1977) 291-330 [[numdam:CTGDC_1977__18_3_291_0](http://www.numdam.org/item/?id=CTGDC_1977__18_3_291_0)]

On [[category]]-valued [[stacks]] ([[2-sheaves]]) as [[internal categories]] in a [[sheaf topos]], and on [[weak equivalences of internal categories]]:

* {#BungePare} [[Marta Bunge]], [[Robert Paré]], _Stacks and equivalence of indexed categories_, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4 (1979) 373-399 [[numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0)]

* {#Bunge} [[Marta Bunge]], *Stack completions and Morita equivalence for categories in a topos*, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4, (1979) 401-436 [[numdam](http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0), [MR558106](http://www.ams.org/mathscinet-getitem?mr=558106)]

See also:

* _Relative functor categories and categories of algebras_, J. of Algebra __11__,1 (1969) 64--101, [MR236238](https://mathscinet.ams.org/mathscinet-getitem?mr=236238) doi

* [[Marta Bunge]], [[Eduardo Dubuc]], _Archimedian local $C^\infty$-rings and models of synthetic differential geometry_ Cahiers de Topologie et Géométrie Différentielle Catégoriques __27__, no. 3 (1986), p. 3-22, [numdam](http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0)

* [[Marta Bunge]], [[Aurelio Carboni]], _The symmetric topos_, J. Pure Appl. Algebra __105__ (1995), no. 3, 233–249, [MR96i:18004](http://www.ams.org/mathscinet-getitem?mr=1367869), doi

* [[Marta Bunge]], [[Steve Lack]], _van Kampen theorem for toposes_, [ps](http://www.maths.usyd.edu.au/u/stevel/papers/vkt.ps.gz)

* Marta Bunge, Jonathan Funk, _Singular coverings of toposes_, Springer Lect. Notes in Math. __1890__, (2006); _Quasi locally connected toposes_, Theory Appl. Categ. __18__ (2007), No. 8, 209–239, [pdf](http://www.tac.mta.ca/tac/volumes/18/8/18-08.pdf)

* _Galois groupoids and covering morphisms in topos theory_,  Galois theory, Hopf algebras, and semiabelian categories,  131--161, Fields Inst. Commun. __43__, Amer. Math. Soc. 2004, [links](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.7071).

* _Classifying toposes and fundamental localic groupoids_, Category theory 1991 (Montreal, PQ, 1991), 75--96, CMS Conf. Proc. __13__, Amer. Math. Soc. 1992.

* Marta Bunge, [[Claudio Hermida]], _Pseudomonodacity and 2-stack completions_, in Models, Logics, and Higher-Dimensional Categories, CRM Proceedings and Lecture Notes __53__, pp. 29-54, Amer. Math. Soc. 2011.

* [[Marta Bunge]], Felipe Gago, Ana Maria San Luis, _Synthetic Differential Topology_, 2018, ([CUP](http://www.cambridge.org/gb/academic/subjects/mathematics/logic-categories-and-sets/synthetic-differential-topology))

* Marta Bunge reported at the category conference in Calais in June 2008 about her joint work with [[Claudio Hermida]] about some aspects of [[Diaconescu's theorem|Diaconescu-type]] results in [[2-category|categorical dimension 2]]. Her slides from Calais can be found [here](http://saxo.univ-littoral.fr/CT08/slides/Bunge.pdf).  There are also some conjectural statements about higher n, with an interesting definition when an n-functor between strict n-categories should be called a fibered n-category.

category: people

[[!redirects M. Bunge]]


Martijn den Besten

## Selected writings

On [[objective type theory]]:

* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

category: people


Martin A. Guest

* [personal page](http://www.f.waseda.jp/martin/)

## Selected wrirings

Identification of the higher [[homotopy groups]] of the [[homotopy of rational maps]] on the [[Riemann sphere]]:

*  [[Martin A. Guest]], [[Andrzej Kozlowski]], M. Murayama, [[Kohhei Yamaguchi]], *The homotopy type of the space of rational functions*, J. Math. Kyoto Univ. 35(4): 631-638 (1995) ([doi:10.1215/kjm/1250518652](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-35/issue-4/The-homotopy-type-of-the-space-of-rational-functions/10.1215/kjm/1250518652.full))

On [[homotopy of rational maps]] from a compact [[Riemann surface]] to a [[toric variety]]:

* [[Martin A. Guest]], *Configuration spaces and the space of rational curves on a toric variety*, Bull. Amer. Math. Soc. 31 (1994), 191-196 ([doi:10.1090/S0273-0979-1994-00515-4](https://doi.org/10.1090/S0273-0979-1994-00515-4))

* [[Martin A. Guest]], *The topology of the space of rational curves on a toric variety*, Acta Math. 174(1): 119-145 (1995) ([doi:10.1007/BF02392803](https://projecteuclid.org/journals/acta-mathematica/volume-174/issue-1/The-topology-of-the-space-of-rational-curves-on-a/10.1007/BF02392803.full), [arXiv:alg-geom/9301005](https://arxiv.org/abs/alg-geom/9301005))

On [[quantum cohomology rings]]:

* [[Martin A. Guest]], *Introduction to Quantum Cohomology*, Vietnam Journal of Mathematics **33** SI (2005) 29–59 [[pdf](http://www.math.ac.vn/publications/vjm/VJM_33/Pdf_files_DB_2005/Bai3_Guest.pdf)]

* [[Martin A. Guest]], *Quantum cohomology: is it still relevant?* [[arXiv:2210.05413](https://arxiv.org/abs/2210.05413)]

category: people

[[!redirects Martin Guest]]


Martin Arkowitz

* [Mathematics Genealogy Page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=36985)

## Selected writings

On [[Whitehead products]] in relation to [[Pontrjagin products]]:

* [[Martin Arkowitz]], *Whitehead Products as Images of Pontrjagin Products*, Transactions of the American Mathematical Society, **158** 2 (1971) 453-463 [[doi:10.2307/1995917](https://doi.org/10.2307/1995917)]

On [[Whitehead products]] in [[rational homotopy theory]], and the statement that *[[the co-binary Sullivan differential is the dual of the Whitehead product]]*:

* {#AndrewsArkowitz78} Peter Andrews, [[Martin Arkowitz]], _Sullivan's Minimal Models and Higher Order Whitehead Products_, Canadian Journal of Mathematics, 30(5), 961-982, 1978 ([doi:10.4153/CJM-1978-083-6](https://doi.org/10.4153/CJM-1978-083-6))

On [[H-spaces]] and [[co-H-spaces]]:

* [[Martin Arkowitz]], _Co-H-spaces_, chapter 23 of [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_

* [Arkowitz 2011, Ch. 2](#Arkowitz11)

On [[homotopy theory]]:

* {#Arkowitz11} [[Martin Arkowitz]], *Introduction to Homotopy Theory* Springer (2011) $[$[doi:10.1007/978-1-4419-7329-0](https://doi.org/10.1007/978-1-4419-7329-0)$]$

## related $n$Lab entries

* [[H-space]], [[co-H-space]]

* [[Whitehead product]]

* [[rational homotopy theory]]

* [[co-binary Sullivan differential is Whitehead product]]

category: people


Martin Bendersky

Martin Bendersky is an algebraic topologist.

* [website](http://math.hunter.cuny.edu/mbenders/)

### Selected writings

On the [[Hopf invariant]] and [[e-invariant]] in [[BP-theory]]:

* [[Martin Bendersky]], _The BP Hopf Invariant_, American Journal of Mathematics, Vol. 108, No. 5 (Oct., 1986) ([jstor:2374595](https://www.jstor.org/stable/2374595))

See also:

* [[Anthony Bahri]], [[Martin Bendersky]], [[Fred Cohen]], _Polyhedral products and features of their homotopy theory_, in
_[[Handbook of Homotopy Theory]]_
([arXiv:1903.04114](https://arxiv.org/abs/1903.04114))

category: people


Martin Beneke

* [Institute page 1](https://www.professoren.tum.de/en/beneke-martin)

* [Institute page 2](https://www.ph.tum.de/about/people/vcard/2FCC11BEE5200CA2/?language=en)

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

## Selected writings

On the [[soft graviton theorem]]:

* [[Martin Beneke]], Patrick Hager, Robert Szafron, *Soft-Collinear Gravity and Soft Theorems*, in section *Effective Quantum Gravity* of *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09336](https://arxiv.org/abs/2210.09336)]

category: people


Martin Bojowald

* [Institute page](https://science.psu.edu/physics/people/mart)

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

## Selected writings

On [[structure formation]] during [[cosmic inflation]] via [[inhomogeneous cosmology|inhomogeneous]] [[quantum cosmology]]:

* [[Martin Bojowald]], Ding Ding, *Canonical description of cosmological backreaction*, JCAP **03** (2021) 083 ([arXiv:2011.03018](https://arxiv.org/abs/2011.03018))

category: people


Martin Bordemann

* [Mathematics Genealogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=107591)

## Selected writings

On [[geometric quantization]] of [[compact topological space|compact]] [[Kähler manifolds]]:

* {#BordemannMeinrenkenSchlichenmaier93} [[Martin Bordemann]], [[Eckhard Meinrenken]], [[Martin Schlichenmaier]], _Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$_,  	Commun. Math. Phys. **165** (1994) 281-296 [[arXiv:hep-th/9309134](http://arxiv.org/abs/hep-th/9309134), [doi:10.1007/BF02099772](https://doi.org/10.1007/BF02099772)]

## Related entries

* [[GNS-construction]]

* [[deformation quantization of the 2-sphere]]

category: people


Martin Brandenburg

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=207180)

## Selected writings

**Publications**

* [Algebraic games – playing with groups and rings](https://link.springer.com/article/10.1007/s00182-017-0577-7), International Journal of Game Theory, 47, 417–450(2018), [arXiv](https://arxiv.org/abs/1205.2884)

* [Operations on categories of modules are given by Schur functors](https://link.springer.com/article/10.1007/s10485-017-9494-y),
Applied Categorical Structures, 26, 287–308(2018), [arXiv](https://arxiv.org/abs/1610.02180)

* [Rosenberg's reconstruction theorem (after Gabber)](https://www.sciencedirect.com/science/article/pii/S0723086917300579), Expositiones Mathematicae, Volume 36, Issue 1, March 2018, Pages 98-117, [arXiv](http://arxiv.org/abs/1310.5978)

* with [[Alexandru Chirvasitu|A. Chirvasitu]] and [[Theo Johnson-Freyd|T. Johnson-Freyd]], [Reflexivity and dualizability in categorified linear algebra](http://www.tac.mta.ca/tac/volumes/30/23/30-23.pdf), Theory and Applications of Categories 30 (2015), No. 23, 808-835, [arXiv](https://arxiv.org/abs/1409.5934)

* with [[Alexandru Chirvasitu|A. Chirvasitu]], [Tensor functors between categories of quasi-coherent sheaves](http://www.sciencedirect.com/science/article/pii/S0021869313006030), Journal of Algebra 399 (2014), 675-692, [arXiv](https://arxiv.org/abs/1202.5147)

**Books**

On [[category theory]]:

* [Einführung in die Kategorientheorie: Mit ausführlichen Erklärungen und zahlreichen Beispielen](http://www.springer.com/de/book/9783662535202), Second edition, Springer, 2016.

**Preprints**

On [[ring theory]]:

* Equational proofs of Jacobson's Theorem, 2023, [arxiv](https://arxiv.org/abs/2310.05301)

On [[Galois theory]]:

* A simple proof of the fundamental theorem of Galois theory, 2022, [arxiv](https://arxiv.org/abs/2211.07508)

On [[limit sketches]]:

* Large limit sketches and topological space objects, 2021, [arXiv](https://arxiv.org/abs/2106.11115)

On [[localization]] of [[tensor categories]]

* Localizations of tensor categories and fiber products of schemes, 2020, [arXiv](https://arxiv.org/abs/2002.00383)

On [[2-limits]] of [[tensor categories]]:

* Bicategorical colimits of tensor categories, 2020, [arXiv](https://arxiv.org/abs/2001.10123)

On [[tensor category]]-theory for [[algebraic geometry]]:

* Tensor categorical foundations of algebraic geometry, PhD thesis, University of Münster, 2014, [arXiv](http://arxiv.org/abs/1410.1716)

  > Errata for PhD thesis: The reflectors in 5.8.17, 5.8.18, 5.8.20, 5.8.21, 5.10.6 are not correct in general and should be removed.

category: people


Martin Bridson

[[!redirects M. Bridson]]
Martin Bridson is a Manx mathematician who holds the Whitehead chair of Pure mathematics at the University of Oxford. He works on geometric group theory, low-dimensional topology, and the study of metric spaces of non-positive curvature.

[home page](http://people.maths.ox.ac.uk/bridson/)

*  His slides for ICM 2006 look very good (see his home page).

category: people


Martin Cederwall

* [webpage](http://www.chalmers.se/en/research/professors/Pages/descriptions/martin-cederwall-theoretical-physics.aspx)

## Selected writings

On [[DBI-action]] and [[Green-Schwarz action functional]] for [[D-branes]]:

* [[Martin Cederwall]], Alexander von Gussich, [[Aleksandar Mikovic]], [[Bengt Nilsson]], Anders Westerberg, _On the Dirac-Born-Infeld Action for D-branes_,  Phys.Lett.B390:148-152, 1997 ([arXiv:hep-th/9606173](https://arxiv.org/abs/hep-th/9606173))

* {#CGNSW96} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))

On [[U-duality]] and [[diffeomorphisms]] in [[exceptional generalized geometry]]:

* [[David Berman]], [[Martin Cederwall]], [[Axel Kleinschmidt]], [[Daniel C. Thompson]], _The gauge structure of generalised diffeomorphisms_, High Energ. Phys. 2013, 64 (2013) [[arXiv:1208.5884](http://arxiv.org/abs/1208.5884)]

On [[doubled supergeometry]]:

* {#Cederwall16} [[Martin Cederwall]], _Double supergeometry_, J. High Energ. Phys. (2016) 2016: 155 ([arXiv:1603.04684](https://arxiv.org/abs/1603.04684))

Relating [[supermultiplets]] to [[L-infinity algebras|$L_\infty$-algebras]] via [[pure spinor]]-formalism:

* [[Martin Cederwall]], Simon Jonsson, [[Jakob Palmkvist]], [[Ingmar Saberi]], *Canonical supermultiplets and their Koszul duals* [[arXiv:2304.01258](https://arxiv.org/abs/2304.01258)]

category: people


Martin Doubek

* [GoogleScholar page](https://scholar.google.co.uk/citations?user=x5XNZwIAAAAJ&hl=en)

## Selected writings

On [[deformation theory]] in terms of [[L-infinity algebra|$L_\infty$-algebras]] and their [[Maurer-Cartan equations]]:

* [[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))

On [[string field theory]] in terms of [[L-infinity algebras|$L_\infty$-algebras]]:

* {#DoubekJurcoMuenster13} [[Martin Doubek]], [[Branislav Jurco]], Korbinian Muenster, _Modular operads and the quantum open-closed homotopy algebra_, J. High Energ. Phys. 2015, 1–55 (2015) ([arXiv:1308.3223](https://arxiv.org/abs/1308.3223), doi:10.1007/JHEP12(2015)158)

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Martin Doubek]], [[Branislav Jurčo]], [[Ján Pulmann]], *Quantum L∞ Algebras and the Homological Perturbation Lemma*  Comm. Math. Phys. **367** (2019) 215–240 [[arXiv:1712.02696](https://arxiv.org/abs/1712.02696), [doi:10.1007/s00220-019-03375-x](https://doi.org/10.1007/s00220-019-03375-x)]

category: people


Martin Fraas

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

## Selected writings

On the [[quantum adiabatic theorem]] in [[condensed matter theory]] with an eye towards [[topological phases of matter]]:

* [[Sven Bachmann]], Spyridon Michalakis, [[Bruno Nachtergaele]], Robert Sims, *Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems*, Commun. Math. Phys. **309** (2012) 835-871 ([arXiv:1102.0842](https://arxiv.org/abs/1102.0842), [doi:10.1007/s00220-011-1380-0](https://doi.org/10.1007/s00220-011-1380-0))

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *Adiabatic Theorem for Quantum Spin Systems*, Phys. Rev. Lett. **119** (2017) 060201 ([doi:10.1103/PhysRevLett.119.060201](https://doi.org/10.1103/PhysRevLett.119.060201))

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *The adiabatic theorem in a quantum many-body setting*, in: *Analytic Trends in Mathematical Physics*, Contemporary Mathematics **741** (2020) 43-58 ([arXiv:1808.09985](https://arxiv.org/abs/1808.09985), [ISBN:978-1-4704-5388-6](https://bookstore.ams.org/conm-741))

category: people


Martin Frankland

Hello. Here is a link to [my website](http://uregina.ca/~franklam/).


Martin Gallauer

* [personal page](https://homepages.warwick.ac.uk/staff/Martin.Gallauer/)

## Selected writings

On [[Grothendieck's yoga of six operations]]:

* [[Martin Gallauer]], *An introduction to six-functor formalism*, lecture at *[The Six-Functor Formalism and Motivic Homotopy Theory](https://sites.google.com/view/summer-school-motivic/home)*, Università degli Studi di Milano (Sept. 2021) [[arXiv:2112.10456](https://arxiv.org/abs/2112.10456), [pdf](https://homepages.warwick.ac.uk/staff/Martin.Gallauer/docs/m6ff.pdf)]

On [[Grothendieck's yoga of six operations]] for [[motives]] in [[rigid analytic geometry]]:

* [[Joseph Ayoub]], [[Martin Gallauer]], [[Alberto Vezzani]], *The six-functor formalism for rigid analytic motives*, Forum of Mathematics, Sigma **10** (2022) E61 [[doi:10.1017/fms.2022.55](https://doi.org/10.1017/fms.2022.55), [arXiv:2010.15004](https://arxiv.org/abs/2010.15004)]

category: people


Martin Gallauer Alves de Souza

**Martin Gallauer Alves de Souza** is a mathematician interested in algebraic geometry (especially motives) and abstract homotopy theory.

* [web](http://www.math.ucla.edu/~gallauer/)

* [Traces, homotopy theory and motivic Galois groups, PhD Thesis](http://www.math.ucla.edu/~gallauer/thesis/thesis.pdf)

* [[motives]]
  * [[motivic Galois groups]]


Martin Greiter

* [institute page](https://www.physik.uni-wuerzburg.de/tp1/team/professoren/prof-dr-martin-greiter/)

* [GoogleScholar page](https://scholar.google.de/citations?user=MIQT5sMAAAAJ&hl=en)

## Selected writings

On [[braid group statistics]]:

* [[Martin Greiter]], [[Frank Wilczek]], *Fractional Statistics* [[arXiv:2210.02530](https://arxiv.org/abs/2210.02530)]

category: people


Martin Heidegger

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

* [[Heidegger's topology]]

category: people

[[!redirects Heidegger]]


Martin Hofmann

* [webpage](http://www2.tcs.ifi.lmu.de/~mhofmann/)

* {#Altenkirch21} [[Thorsten Altenkirch]], *Martin Hofmann’s contributions to type theory: Groupoids and univalence*, Mathematical Structures in Computer Science **31** 9 (2021) 953-957 [[doi:10.1017/S0960129520000316](https://doi.org/10.1017/S0960129520000316)]

## Selected writings

On [[identity types]] in [[extensional type theory|extensional]]/[[intensional type theory|intensional]] [[dependent type theory]] ([[Martin-Löf dependent type theory]]):

* {#Hofmann95} [[Martin Hofmann]], _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

whose chapter 2 on [[syntax]] and [[semantics]] of [[dependent type theory]] is also published as:

* [[Martin Hofmann]], *Syntax and semantics of dependent types*, in *Semantics and logics of computation*, Publ. Newton Inst. **14**, Cambridge Univ. Press (1997) 79-130 [[doi:10.1017/CBO9780511526619.004](https://doi.org/10.1017/CBO9780511526619.004)]

On the [[categorical semantics]] of [[dependent type theory]] with [[function types]] in [[locally cartesian closed categories]] (see at *[[relation between category theory and type theory]]*):

* {#Hofmann} [[Martin Hofmann]], *On the interpretation of type theory in locally cartesian closed categories*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** (1994) 427–441 [[doi:10.1007/BFb0022273](https://doi.org/10.1007/BFb0022273)]

* {#CurienGarnerHofmann} [[Pierre-Louis Curien]], [[Richard Garner]], [[Martin Hofmann]], *Revisiting the categorical interpretation of dependent type theory*, Theoretical Computer Science **546** 21 (2014) 99-119 [[doi:10.1016/j.tcs.2014.03.003](https://doi.org/10.1016/j.tcs.2014.03.003), [[CurienGarnerHofmann.pdf:file]]]

Introducing the [[homotopy type theory]]-interpretation of [[identity types]] (the "groupoid interpretation")

* [[Martin Hofmann]], [[Thomas Streicher]], *The groupoid model refutes uniqueness of identity proofs*, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science (1994) [[doi:10.1109/LICS.1994.316071](https://doi.org/10.1109/LICS.1994.316071)]

and introducing what came to be known the [[univalence axiom]] (under the name "universe extensionality"):

* {#HofmannStreicher98} [[Martin Hofmann]], [[Thomas Streicher]]  _The groupoid interpretation of type theory_, in: [[Giovanni Sambin]] et al. (eds.), *Twenty-five years of constructive type theory*, Proceedings of a congress, Venice, Italy, October 19-21, 1995, Oxf. Logic Guides. **36** Clarendon Press (1998) 83-111  [[ISBN:9780198501275](https://global.oup.com/academic/product/twenty-five-years-of-constructive-type-theory-9780198501275), [ps.gz](http://www.mathematik.tu-darmstadt.de/~streicher/venedig.ps.gz), [[HofmannStreicherGroupoidInterpretation.pdf:file]]]

  see also:

  Ethan Lewis, Max Bohnet, *The groupoid model of type theory*, seminar notes (2017) [[pdf](https://staff.fnwi.uva.nl/b.vandenberg3/Onderwijs/Homotopy_Type_Theory_2017/HoTT-bohnet-lewis-handout.pdf), [[LewisBohnet.GroupoidModelOfTT.pdf:file]]]

* [[Martin Hofmann]], *The groupoid interpretation of type theory, a personal retrospective*, talk at *[[HoTT at DMV2015]]* (2015) [[[HofmannDMV.pdf|slides:file]]]

On [[subtypes]] (with an early discussion of what came to be called *[[lenses (in computer science)]]*, motivated by [[object-oriented programming]]):

* {#HofmannPierce96} [[Martin Hofmann]], [[Benjamin Pierce]], *Positive Subtyping*, Information and Computation **126** 1 (1996) 11-33 [[doi:10.1006/inco.1996.0031](https://doi.org/10.1006/inco.1996.0031)]

## Related entries

* [[relation between category theory and type theory]]

* [[judgement]]

* [[homotopy type theory]]

category: people


Martin Hyland

Martin Hyland is Professor in Mathematical Logic in the University of Cambridge, and is a Fellow of King's College, Cambridge. Hyland is  known for work on the effective topos (within topos theory), on the lambda-calculus and on game semantics.

His  main research interests are in the following.

*  Mathematical Logic: Lambda Calculus, Recursion Theory, Realizability, Proof Theory, Linear Logic.

*  Category Theory: Topos Theory, Categorical algebra, Operads, Higher-dimensional Categories.

*  Theoretical Computer Science: Applications of Category Theory, Domain Theory, Polymorphism, Game Semantics.

His webpages are:

* [DPMMS website](http://www.dpmms.cam.ac.uk/~martin/)

and a [research page](http://www.dpmms.cam.ac.uk/~martin/Research/research.html)

## Selected writings

On [[categorical semantics]] for the [[exponential modality]] in [[linear logic]]:

* {#BBPH92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], [[Martin Hyland]], *Linear $\lambda$-Calculus and Categorical Models Revisited*, in *Computer Science Logic. CSL 1992*, Lecture Notes in Computer Science **702**, Springer (1993) [[doi:10.1007/3-540-56992-8_6](https://doi.org/10.1007/3-540-56992-8_6)]

On [[realizability]] and the [[propositional axiom of choice]]:

* [[Martin Hyland]], _Variations on realizability: realizing the propositional axiom of choice_,  Mathematical Structures in Computer Science, Volume 12 , Issue 3 , June 2002 , pp. 295 - 317  ([doi:10.1017/S0960129502003651](https://doi.org/10.1017/S0960129502003651), [pdf](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2002/vor02.pdf))

On [[categorical semantics]] for [[linear logic]]:

* [[Martin Hyland]], Andrea Schalk, *Glueing and orthogonality for models of linear logic*, Theoretical Computer Science **294** 1–2 (2003) 183-231 [doi:10.1016/S0304-3975(01)00241-9]

On the history of [[categorical algebra|categorical]] [[universal algebra]] via [[Lawvere theories]] and [[monads]]:

* [[Martin Hyland]], [[John Power]], *The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads*, Electronic Notes in Theor. Comp. Sci. **172** (2007) 437-458 [[doi:10.1016/j.entcs.2007.02.019](https://doi.org/10.1016/j.entcs.2007.02.019), [preprint](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf)]

category: people


Martin Klaus

* [personal page](https://math.vt.edu/people/faculty/klaus-martin.html)

## Selected writings

On [[vacua]]/[[ground states]] of the [[free field|free]] [[Dirac field]] in a [[electromagnetic field|Coulomb]] [[background field]]:

* {#KlausScharf77} [[Martin Klaus]], [[Günter Scharf]], *The regular external field problem in quantum electrodynamics*, Helv. Phys. Acta **50** (1977) $[$[doi:10.5169/seals-114890](http://doi.org/10.5169/seals-114890)$]$

category: people


Martin Kliesch

* [personal page](http://www.mkliesch.eu/)

* [GoogleScholar page](https://scholar.google.com/citations?user=q8N6ibAAAAAJ&hl=en)

## Selected writings

On [[hardware verification]] of  [[quantum computing]]-[[quantum systems|systems]]:

* [[Martin Kliesch]], [[Ingo Roth]], *Theory of Quantum System Certification*, PRX Quantum **2** (2021) 010201 [[doi:10.1103/PRXQuantum.2.010201](https://doi.org/10.1103/PRXQuantum.2.010201), [arXiv:2010.05925](https://arxiv.org/abs/2010.05925)]

category: people


Martin Kruczenski

## Selected writings

On [[D-brane polarization]] into [[supertubes]]:

* [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], David J. Winters, _Aspects of supertubes_, JHEP 0205:017, 2002 ([arXiv:hep-th/0204103](https://arxiv.org/abs/hep-th/0204103))

On [[S-branes]]:

* {#KruczenskiMyersPeet02} [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], _Supergravity S-Branes_, JHEP 0205 (2002) 039 ([arXiv:hep-th/0204144](https://arxiv.org/abs/hep-th/0204144))

On [[single trace operators]] as spiky [[strings]] seen under the [[AdS-CFT correspondence]]:

* {#Kruczenski04} [[Martin Kruczenski]], _Spiky strings and single trace operators in gauge theories_, JHEP 0508:014, 2005 ([arXiv:hep-th/0410226](https://arxiv.org/abs/hep-th/0410226))

category: people


Martin Lundfall

* [CypherHunter page](https://www.cypherhunter.com/en/p/martin-lundfall/)

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

## Selected writings

On [[constructive analysis]] with [[real numbers]] formalized in the [[Agda]] [[proof assistant]]:

* [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [pdf, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

On [[dependent linear type theory]]:

* {#Lundfall17} [[Martin Lundfall]], *Models of linear dependent type theory*, 2017 ([[LundfallModelsOfLinerDTT.pdf:file]])

* {#Lundfall18} [[Martin Lundfall]], _A diagram model of linear dependent type theory_, ([arXiv:1806.09593](https://arxiv.org/abs/1806.09593))

category: people


Martin Markl

* [personal page](https://users.math.cas.cz/~markl/)

* [institute page](https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=3857)

## Selected writings

On [[deformation theory]] in terms of [[L-infinity algebra|$L_\infty$-algebras]] and their [[Maurer-Cartan equations]]:

* [[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))

* [[Martin Markl]], _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))

On [[monoidal enriched categories]]:

* {#BataninMarkl12} [[Michael Batanin]], [[Martin Markl]], Section 2 of *Centers and homotopy centers in enriched monoidal categories*, Advances in Mathematics **230** 4–6 (2012) 1811-1858 [[doi:10.1016/j.aim.2012.04.011](https://doi.org/10.1016/j.aim.2012.04.011), [arXiv:1109.4084](https://arxiv.org/abs/1109.4084)]

* (...)

## Related $n$Lab entries

* [[operad]]

* [[L-infinity algebra]]

* [[associahedron]]

category: people


Martin Neuchl

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=27412)

## Selected writings

On [[monoidal 2-categories]] and [[braided monoidal 2-categories]]:

* [[John C. Baez]], [[Martin Neuchl]], *Higher-Dimensional Algebra I: Braided Monoidal 2-Categories*, Adv. Math. 121 (1996), 196-244 ([arXiv:q-alg/9511013](https://arxiv.org/abs/q-alg/9511013))

category: people


Martin Olbrich

* [webpage](http://www.uni-math.gwdg.de/olbrich/)

## related $n$Lab entries

* [[Selberg zeta function]]

category: people


Martin Olsson

_Martin C. Olsson_ is an [[algebraic geometer]] at University of Berkeley (aritmetic geometry, crystalline site, [[algebraic stack]]s, [[logarithmic geometry]]).

* [homepage](http://math.berkeley.edu/~molsson)

## Selected writings

On [[algebraic spaces]] and [[algebraic stacks]]:

* [[Martin Olsson]], *Algebraic Spaces and Stacks*, Colloquium Publications **62** (2016) [[doi:10.1090/coll/062](https://doi.org/10.1090/coll/062), [ISBN:978-1-4704-2798-6](https://bookstore.ams.org/coll-62)]

category: people

[[!redirects Martin C. Olsson]]


Martin Otto

* [personal page](https://www2.mathematik.tu-darmstadt.de/~otto/)

Wikipedia entry (de)

## Selected writings

On [[semantics]]/[[model theory]] of [[modal logic]] ([[possible worlds semantics]], [[Kripke frames]]):

* [[Valentin Goranko]], [[Martin Otto]], *Model Theory of Modal Logic*, in Section 5 in: *Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 249-329 [[pdf](https://www2.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf),  [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

category: people


Martin Palmer

* [personal page](https://mdp.ac/)

## Selected writings

On the [[ordinary homology]] of [[configuration spaces of points]] in a [[surface]] as a [[group representation]] of the [[mapping class group]]:

* {#Palmer12} [[Martin Palmer]], *Configuration spaces and homological stability*, Oxford University Research Archive (2012) [[pdf](https://mdp.ac/thesis/thesis.pdf), [web](https://mdp.ac/thesis/index.html), [[Palmer-ConfigurationSpaces.pdf:file]]]

* [[Christian Blanchet]], [[Martin Palmer]], [[Awais Shaukat]], *Heisenberg homology on surface configurations* [[arXiv:2109.00515](https://arxiv.org/abs/2109.00515)]

On [[mapping class group]]-[[linear representation|representations]]:

* [[Martin Palmer]], *Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann* (2022) $[$[talk](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#MartinPalmerMarch22)$]$

category: people

[[!redirects Martin David Palmer]]


Martin Raussen

Martin Raussen is a mathematician in the Department of Mathematics
Aalborg University, Denmark. He has published papers in Algebraic Topology and most recently in [[directed homotopy theory]].

His webpage is [here](http://people.math.aau.dk/~raussen/).

category:people

[[!redirects M. Raussen]]


Martin Rees

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

## Related $n$Lab entries

* [[cosmology]]

* [[hydrogen line]]

category: people


Martin Reuter

## Selected writings

* {#Reuter96} [[Martin Reuter]], _Nonperturbative Evolution Equation for Quantum Gravity_, Phys.Rev. D57 (1998) 971-985 ([arXiv:hep-th/9605030](https://arxiv.org/abs/hep-th/9605030))

* {#NinkReuter12} Andreas Nink, [[Martin Reuter]], _On quantum gravity, Asymptotic Safety, and paramagnetic dominance_, Int. J. Mod. Phys. D22 (2013) 1330008 ([arXiv:1212.4325](https://arxiv.org/abs/1212.4325))

* O. Lauscher, [[Martin Reuter]], _Asymptotic Safety in Quantum Einstein Gravity: nonperturbative renormalizability and fractal spacetime structure_,  In: Fauser B., Tolksdorf J., Zeidler E. (eds.) )Quantum Gravity_ Birkhäuser Basel 2006 ([arXiv:hep-th/0511260](https://arxiv.org/abs/hep-th/0511260), [doi:10.1007/978-3-7643-7978-0_15](https://doi.org/10.1007/978-3-7643-7978-0_15))

* [[Max Niedermaier]], [[Martin Reuter]], _The Asymptotic Safety Scenario in Quantum Gravity_, Living Reviews in Relativity December 2006, 9:5 [doi:10.12942/lrr-2006-5](http://link.springer.com/article/10.12942/lrr-2006-5)

## Related $n$Lab entries

* [[asymptotic safety]]

category: people


Martin Roček

[[!redirects Martin Rocek]]

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

* {#GatesGrisaruRocekSiegel83} [[Jim Gates]] Jr, [[Marcus Grisaru]], [[Martin Roček]], [[Warren Siegel]], _Superspace, or One thousand and one lessons in supersymmetry_, Front.Phys. 58 (1983) 1-548  (1983) ([arXiv:hep-th/0108200](https://arxiv.org/abs/hep-th/0108200), [spire:195126](inspirehep.net/record/195126))

  (on [[superspace]])

## related $n$Lab entries

* [[string theory]]

* [[supersymmetry]]

* [[generalized complex geometry]]

* [[T-duality]]

category: people


Martin Schlichenmaier

* [personal page](https://math.uni.lu/schlichenmaier/)

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

## Selected writings

On [[geometric quantization]] of [[compact topological space|compact]] [[Kähler manifolds]]:

* {#BordemannMeinrenkenSchlichenmaier93} [[Martin Bordemann]], [[Eckhard Meinrenken]], [[Martin Schlichenmaier]], _Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$_,  	Commun. Math. Phys. **165** (1994) 281-296 [[arXiv:hep-th/9309134](http://arxiv.org/abs/hep-th/9309134), [doi:10.1007/BF02099772](https://doi.org/10.1007/BF02099772)]

On [[Fedosov deformation quantization]] for [[almost Kähler structures]]:

* {#KarabegovSchlichenmaier01} [[Alexander Karabegov]], [[Martin Schlichenmaier]],  _Almost-Kähler deformation quantization_, Letters in Mathematical Physics, **57** 2 (2001) 135–148  [[doi:10.1023/A:1017993513935](https://doi.org/10.1023/A:1017993513935), [arXiv:0102169](https://arxiv.org/abs/math/0102169)]

with a survey in

* [[Martin Schlichenmaier]], _Deformation quantization for almost-Kähler manifolds_, J. of Nonlinear Math. Phys.
**11**, Supplement (2004) 49–54 [[doi:10.2991/jnmp.2004.11.s1.6](https://doi.org/10.2991/jnmp.2004.11.s1.6), [pdf](http://www.atlantis-press.com/php/download_paper.php?id=540)]

## Related pages

* [[deformation quantization of the 2-sphere]]

category: people


Martin Schnabl

* [webpage](http://www-hep2.fzu.cz/~schnabl/)

category: people


Martin Schottenloher

* [webpage](http://www.mathematik.uni-muenchen.de/~schotten/)

## Selected writings

On [[conformal field theory]]:

* [[Martin Schottenloher]], *A Mathematical Introduction to Conformal Field Theory*, Lecture Notes in Physics **759**, Springer 2008 ([doi:10.1007/978-3-540-68628-6](https://link.springer.com/book/10.1007/978-3-540-68628-6), [web](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/))

On the [[topology]] of the infinite [[unitary group]] [[U(ℋ)]] on a [[separable Hilbert space]] ([[operator topology]] and [[compact-open topology]]):

* {#Schottenloher} [[Martin Schottenloher]], _The Unitary Group In Its Strong Topology_ ([arXiv:1309.5891](https://arxiv.org/abs/1309.5891)), Advances in Pure Mathematics **08** 05 (2018) ([doi:10.4236/apm.2018.85029](https://www.scirp.org/html/4-5301487_84967.htm))

category: people


Martin Sohnius

## Selected writings

Introducing the [[Haag-Łopuszański-Sohnius theorem]]:

* {#HaagLopuszanskiSohnius75} [[Rudolf Haag]], [[Jan Łopuszański]], [[Martin Sohnius]], _All possible generators of supersymmetries of the S-matrix_, Nuclear Physics B, 88: 257–274 (1975) (doi:10.1016/0550-3213(75)90279-5, [spire:90850](https://inspirehep.net/literature/90850))

category: people


Martin Speirs

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

## Selected writings

On [[model categories]] and application to [[rational homotopy theory]]:

* [[Martin Speirs]], *Model Categories with a view towards rational homotopy theory*, MSc thesis, Copenhagen (2015) [[pdf](https://speirs.sites.ku.dk/files/2017/10/Master.pdf), [[Speirs-ModelCategories.pdf:file]]]

## Related pages

* [[model structure on simplicial groups]]

category: people


Martin Tangora

* [webpage](http://homepages.math.uic.edu/~tangora/)

## Selected writings

On [[cohomology operations]]:

* [[Robert Mosher]], [[Martin Tangora]], _Cohomology Operations and Application in Homotopy Theory_, Harper and Row (1968), Dover (2008) ([ISBN10:0486466647](https://store.doverpublications.com/0486466647.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf))

## Related $n$Lab entries

* [[cohomology operation]]

* [[Steenrod algebra]]

* [[May spectral sequence]]

* [[Hopf construction]]

* [[Hopf invariant]]

[[!redirects Martin C. Tangora]]

category: people


Martin Wijnholt

* [webpage](http://www.theorie.physik.uni-muenchen.de/lsluest/members/asc/sci_mem/wijnholt_martijn/index.html)

## Writings

* _String compactification_, [PITP 2014](https://pitp2014.ias.edu) lecture notes ([[WijnholtCompactification14.pdf:file]])

on [[F-theory]] [[string phenomenology]]

category: people


Martin Wolf

* [webpage](http://www.surrey.ac.uk/maths/people/wolf_martin/)

## Selected writings

On [[higher structures]] in [[string theory]] and [[M-theory]]:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _Higher Structures in M-Theory_, Introduction to _[[Higher Structures in M-Theory 2018]]_, Fortsch. d. Phys. **67** (2019) 8-9 [[arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001](https://doi.org/10.1002/prop.201910001)]

On [[higher structures]] in [[string theory]] and [[M-theory]]:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _Higher Structures in M-Theory_, Introduction to _[[Higher Structures in M-Theory 2018]]_, Fortsch. d. Phys. **67** (2019) 8-9 [[arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001](https://doi.org/10.1002/prop.201910001)]

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Branislav Jurčo]], [[Tommaso Macrelli]], [[Christian Sämann]], [[Martin Wolf]], *Loop Amplitudes and Quantum Homotopy Algebras*, Journal of High Energy Physics **2020** 3 (2020) [doi:10.1007/JHEP07(2020)003, [arXiv:1912.06695](https://arxiv.org/abs/1912.06695)]

* [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], *Perturbative Quantum Field Theory and Homotopy Algebras*, PoS (CORFU2019) 199 [[arXiv:2002.11168](https://arxiv.org/abs/2002.11168), [doi:10.22323/1.376.0199](https://doi.org/10.22323/1.376.0199)]

On the [[classical double copy]] ([[KLT relations]], relating [[gravity]] to [[Yang-Mills theory]]) via [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8-9 (2021) 2100075 $[$[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)$]$

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

## Related entries

* [[twistor correspondence]]

category: people


Martin Ziegler

* [personal webpage](http://m.zie.de)

* [institute page](http://home.mathematik.uni-freiburg.de/ziegler/Ziegler.html)

## related $n$Lab entries

* [[model theory]]

category: people


Martin Čadek

* [webpage](http://www.math.muni.cz/~cadek/)

## Selected writings

On [[Sp(2).Sp(1)]]-[[G-structure|structures]] and related issues:

* {#CadekVanzura92} [[Martin Čadek]], [[Jiří Vanžura]], _On oriented vector bundles over CW-complexes of dimension 6 and 7_, Comment. Math. Univ. Carolin. 33,4(1992)727–736 ([pdf](http://emis.impa.br/EMIS/journals/CMUC/pdf/cmuc9204/cadek.pdf))

* {#CadekVanzura93} [[Martin Čadek]], [[Jiří Vanžura]], _On the classification of oriented vector bundles over 5-complexes_, Czechoslovak Mathematical Journal, Vol. 43 (1993), No. 4, 753–764 ([dml:128427](https://dml.cz/handle/10338.dmlcz/128427))

* {#CadekVanzura95} [[Martin Čadek]], [[Jiří Vanžura]], _On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes_, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 ([dml-cz:118764](https://dml.cz/handle/10338.dmlcz/118764))

* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], _On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures  in 8-dimensional vector bundles_, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 ([jstor:43737249](https://www.jstor.org/stable/43737249))

* {#CadekVanzura98a} [[Martin Čadek]], [[Jiří Vanžura]], _On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes_,  Colloquium Mathematicum (1998), 76 (2), pp 213-228 ([web](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-cmv76z2p213bwm))

* {#CadekVanzura98b} [[Martin Čadek]], [[Jiří Vanžura]], _Almost quaternionic structures on eight-manifolds_, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 ([euclid:1200787905](https://projecteuclid.org/euclid.ojm/1200787905))

* {#CadekVanzura98c} [[Martin Čadek]], [[Jiří Vanžura]], _Various structures in 8-dimensional vector bundles over 8-manifolds_, Banach Center Publications (1998) Volume: 45, Issue: 1, page 183-197 ([dml:208903](https://eudml.org/doc/208903))

* {#CadekCrabbVanzura08} [[Martin Čadek]], Michael Crabb, [[Jiří Vanžura]],  _Obstruction theory on 8-manifolds_, manuscripta math. 127 (2008), 167-186 ([arXiv:0710.0734](https://arxiv.org/abs/0710.0734))

* {#CadekCrabbVanzura10} [[Martin Čadek]], Michael Crabb, [[Jiří Vanžura]], _Quaternionic structures_, Topology and its Applications Volume 157, Issue 18, 1 December (2010), Pages 2850-2863 ([doi:10.1016/j.topol.2010.09.005](https://doi.org/10.1016/j.topol.2010.09.005))

Discussion of [[homotopy groups]] in [[computational topology]]:

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension_, SIAM J. Comput., 43(5), 1728–1780  ([arXiv:1211.3093](https://arxiv.org/abs/1211.3093), [doi:10.1137/120899029](https://doi.org/10.1137/120899029))

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Extendability of continuous maps is undecidable_, Discrete Comput Geom (2014) 51: 24 ([arXiv:1302.2370](https://arxiv.org/abs/1302.2370), [doi:10.1007/s00454-013-9551-8](https://doi.org/10.1007/s00454-013-9551-8))

On [[cohomotopy]]-sets in [[computational topology]]:

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Francis Sergeraert]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Computing all maps into a sphere_, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 ([arxiv:1105.6257](https://arxiv.org/abs/1105.6257))

## related $n$Lab entries

* [[5-manifold]], [[8-manifold]]

* [[G-structure]]

* [[quaternion-Kähler manifold]]

* [[Sp(n).Sp(1)]]

* [[spin group]]

* [[Spin(4)]], [[Spin(5)]],  [[Spin(8)]]

* [[triality]]

* [[Euler class]]

category: people

[[!redirects Martin Cadek]]