math institutions

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### Context
#### Mathematics
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See also [[math blogs]], [[math archives]] and the top page [[math resources]]. This page is for *quick* links to the **main** institutes and to the *lists* of most societies and departments.

#Contents#
* table of contents
{:toc}

## Main math societies and lists

* [ams](http://www.ams.org) American Mathematical Society
* [imu](http://www.mathunion.org) International Mathematical Union
* [ems](http://www.emis.de) European Mathematical Society
* [siam](https://www.siam.org) Society for industrial and applied mathematics
* long [list of math societies](http://www.mathontheweb.org/mathweb/mi-sao.html) at (former metric conversion)
* [list of mathematics departments](http://www.mathontheweb.org/mathweb/mi-depts.html) at Math On the Web

## Main research institutes

Please do not list small unimportant institutions, they will be included in existing long lists which we will link to. This list should be for quick access and short reminder to resources.

### Europe including Russia

* [IHÉS](http://www.ihes.fr) Bures-sur-Yvette, France
* [mpim-bonn](http://www.mpim-bonn.mpg.de) Max Planck for Mathematics, Bonn
* [mpi mis](http://www.mis.mpg.de) Max Planck for Mathematics in Sciences, Leipzig
* [Oberwolfach](http://www.mfo.de)
* [MSRI](http://www.msri.org) Berkeley
* [ICTP](http://www.ictp.it) Trieste
* [Mittag-Leffler](http://www.mittag-leffler.se) Djursholm, Sweden
* [Isaac Newton Institute](http://www.newton.ac.uk) Cambridge, UK
* [IHP](http://www.ihp.jussieu.fr) l'Institute Henri Poincaré, Paris
* [ESI](http://www.esi.ac.at) Erwin Schrödinger International Institute for Mathematical Physics, Vienna
* [Steklov](http://www.mi.ras.ru/index.php?l=1) Mathematical Institute of Russian Academy of Sciences
* [Warwick](http://www2.warwick.ac.uk/fac/sci/maths) Mathematics Institute
* [IMPAN](http://www.impan.pl/BC) Warsaw with [Banach Center](http://www.impan.pl/BC)
* [IPMU](http://www.ipmu.jp) Institute for Physics and Mathematics of the Universe, Tokyo
* [Courant Research Centre "Higher Order Structures"](http://www.crcg.de/wiki/Main_Page) Göttingen
* [Hausdorff Institute](http://www.him.uni-bonn.de) and [Hausdorff Center](http://www.hausdorff-center.uni-bonn.de) at Uni Bonn
* University of [Oxford, Mathematical Institute](http://www.maths.ox.ac.uk)
* [CRM Barcelona](http://www.crm.es)
* [CIRM Marseilles](http://www.cirm-math.com)

### Asia

* [RIMS](http://www.kurims.kyoto-u.ac.jp/en/index.html) Kyōto
* [TIFR](http://www.tifr.res.in) Tata Institute, Mumbai
* [Harish-Chandra](http://www.mri.ernet.in) Allahabad
* [IMSC](http://www.imsc.res.in) Institute for Mathematical Sciences, Chennai
* [IMS](http://www2.ims.nus.edu.sg) Singapore
* [Chern Institute](http://www.nim.nankai.edu.cn/nim_e/index.htm) Nankai
* [NIMS](http://www.nims.re.kr/eng) South Korea
* [Razmadze](http://www.rmi.acnet.ge) Tbilisi
* [Weizman Institute, Faculty of Mathematics](http://www.wisdom.weizmann.ac.il)
* [Einstein Institute of Mathematics](http://www.ma.huji.ac.il) at Hebrew University of Jerusalem
* [IPM](http://www.ipm.ac.ir) Tehran, Iran

### Americas

* [Clay](http://www.claymath.org) Mathematical Institute
* [IAS](http://www.ias.edu) Institute for Advanced Studies, Princeton (and [Park City](http://pcmi.ias.edu))
* [AIM](http://www.aimath.org) American Institute of Mathematics
* [KITP](http://www.itp.ucsb.edu) Santa Barbara (Theoretical Physics)
* [Courant](http://www.cims.nyu.edu) Institute, New York
* [CRM Montreal](http://www.crm.umontreal.ca)
* [Fields](http://www.fields.utoronto.ca) Institute, Toronto, Canada
* [Banff](http://www.birs.ca) Research Institute Station (BIRS), Canada
* [PIMS](https://www.pims.math.ca) Pacific Institute for the Mathematical Sciences
* [Perimeter Institute](http://www.perimeterinstitute.ca) in Theoretical Physics
* [IMPA](http://www.impa.br) Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

### Global and lists

* [list](http://www.mathontheweb.org/mathweb/mi-inst.html) of math institutes at mathontheweb.org
* [IMSI](http://www.fields.utoronto.ca/aboutus/IMSI.html) consortium of math institutes (long list included!)
* Math Research Institute [list](http://www.dmoz.org/Science/Math/Research/Institutes) at Open Directory
* [physics institutes](http://greybook.cern.ch/institutes) -- long list maintained at CERN

[[!redirects mathematical institutions]]
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[[!redirects mathematical societies]]
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math resources by individuals

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The page level above is [[math resources]].

Most of interesting people for nlab will have their own pages in $n$lab; many  of them keep their own articles and books there, but some do more. Thus **this will not be a general list of homepages of mathematicians**, but rather of those pages which have *particularly important or large collections* of articles, books and especially advanced **lecture notes** of major importance to $n$lab and _in the areas of prime importance to $n$lab_ ([[sheaf|sheaves]], [[stack]]s, (higher) categories, modern geometry and [[homotopy theory]], [[algebraic topology]], foundations, [[mathematical physics]]). Still let us keep the alphabetic order (by family name)

* [[David Ben-Zvi]] has a nice [geometric Langlands page](http://www.math.utexas.edu/users/benzvi/Langlands.html) and of many other [lecture notes](http://www.math.utexas.edu/users/benzvi/notes.html)

* Dror Bar Natan keeps an [archive of his notebooks](http://drorbn.net/AcademicPensieve)

* [groupoid](http://wolfweb.unr.edu/homepage/ramazan/groupoid) homepage has various info on research on groupoids including list of addresses

* [[Jacob Lurie]]'s [website](http://www.math.harvard.edu/~lurie) has the latest editions of his papers (most of which are on arXiv but not all) of large importance to the subject of $n$lab

* [[Peter May]]: [books](http://www.math.uchicago.edu/~may/BOOKSMaster.html), [notes](http://www.math.uchicago.edu/~may/MISCMaster.html), [homepage, including talks](http://www.math.uchicago.edu/~may)

* [[James Milne]]'s [math page](http://www.jmilne.org/math/index.html) has several books and lecture notes, mainly on [[algebra]] and algebraic geometry

* Daniel Murfet's [The Rising Sea](http://therisingsea.org/?page_id=3) is classified under [[math blogs]] but has large amount of neat and modern lecture notes on [[algebraic geometry]] and [[homological algebra]]

* [[Andrew Ranicki]]'s [homepage](http://www.maths.ed.ac.uk/~aar) has several online books, slides of a number of talks and links to some papers and some other material like the [Maslov index](http://www.maths.ed.ac.uk/~aar/maslov.htm) seminar page with nice bibliography

* [[Bodo Pareigis]] has [here](http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html) several lecture notes on category theory, [[quantum group]]s, noncommutative geometry and [[Hopf algebra]]s

* Paul Smith's [noncommutative algebra and geometry page](http://www.math.washington.edu/~smith/Research/research.html) has a large list of links to homepages of geometers and algebraists

* [[Charles Weibel]] has [here](http://www.math.rutgers.edu/~weibel) lots of writings including a growing book on [[algebraic K-theory]] and lecture notes on motivic cohomology with Mazza and Voevodsky

* [David Mumford](http://www.dam.brown.edu/people/mumford/index.php), keeps lots of his online material

* Pointers to several similar collections in answers to MathOverflow question [free-high-quality-mathematical-writing-online](http://mathoverflow.net/questions/1722/free-high-quality-mathematical-writing-online)

* [The Assayer](http://www.theassayer.org/cgi-bin/asbrowsesubject.cgi?class=Q#freeclassQAm), a catalogue of freely available mathematics books in electronic formats (also computer science and other fields). Some books are even reviewed.

* [Tim Dokchitser](https://people.maths.bris.ac.uk/~matyd/GroupNames/index.html) GroupNames.org is a database, under construction, of names, extensions, properties and character tables of finite groups of small order.

Category: meta


Math-Net.Ru

[Math-Net.Ru](http://www.mathnet.ru/?option_lang=eng) is a database of mathematical publications in Russian (and Soviet) journals
operated by the [[Steklov Mathematical Institute]] in Moscow, Russia.

It does not perform any reviews,
but otherwise its functionality is similar to that of [[MathSciNet]] and [[zbMATH]],
with a much more comprehensive
coverage of Russian (and Soviet) journals,
as well as freely available full texts of almost all journal articles,
with the exception of a moving wall for the last few years for some journals.

## Related entries

* [[MathSciNet]]

* [[zbMATH]]

## References

* [Math-Net.Ru](http://www.mathnet.ru/?option_lang=eng).


Math-Phys-Cat

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**[Math-Phys-Cat](https://math-phys.group)** is a Brazilian research group concerned with [[mathematical physics|Mathematical-Physics]], [[categorification]] and [[higher category theory and physics]]. We are mainly interested in mathematical aspects of [[classical field theory|classical]] and [[quantum field theories]], especially those involving [[category theory]] and [[gauge theory]], also in the axiomatization problem of astrophysics and in using [[category theory|category-theoretic]] language in order to unify and abstract concepts in classical [[algebra]] and in [[differential geometry]].

* [Website](https://math-phys.group)
* [Members](https://math-phys.group/About/Members). We recently lost a member: Fábio Dadam. We leave here our homage to him.
* [Research Subgroups](https://math-phys.group/About/Research)
* [Writings](https://math-phys.group/Writings/PapersAndPreprints)

* [Twitter](https://twitter.com/math_phys_cat)

## Activies

* [Math-Phys-Cat Seminars](https://math-phys.group/Activities/Math-Phys-CatSeminars), more devoted to graduate topics and research. External speakers are welcome. Recordings are in our [Youtube Channel](https://www.youtube.com/channel/UCCjnnTiyS19M89olqLQQWOw).
*  [Student Seminars](https://math-phys.group/Activities/StudentSeminars). These are weekly seminars presented by our students, typically to their advisors our to a small group of people interested in the topics.
* [General Seminars](https://math-phys.group/Activities/GeneralSeminars). Seminars on general topics in the interface between mathematics and physics. Used to encourage the interaction between our students and also to find potential new members.
* [Courses](https://math-phys.group/Activities/Courses). Normally graduate courses and minicourses given by members of our group on our research topics.

## Math-Phys-Cat *Wiki*

Recently we created the [Math-Phys-Cat *Wiki*](wiki.math-phys.group), serving two purposes:

* share our research in the format we find most accessible and perhaps find new collaborations.
* be a repository for reference lists on our research topics. Contributions are welcome.

[[!redirects Math-Phys-Cat Group]]

category: people


Mathai Varghese

* [website](http://www.adelaide.edu.au/directory/mathai.varghese)

## Selected writings

On [[equivariant de Rham cohomology]]

* {#MathaiQuillen86} [[Varghese Mathai]], [[Daniel Quillen]], _Superconnections, Thom classes and equivariant differential forms_, Topology 25, 85 (1986)  (doi:10.1016/0040-9383(86)90007-8)

On [[non-commutative geometry]]-deformation of the [[self-dual higher gauge theory]] as it appears on the [[M5-brane]] [[worldvolume]], and its [[S-duality]]:

* {#MathaiSati14} [[Varghese Mathai]], [[Hisham Sati]], _Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality_, J. Geom. Phys. 92:240-251, 2015 ([arXiv:1404.2257](https://arxiv.org/abs/1404.2257))

On [[topological T-duality]]:

* {#BouwknegtEvslinMathai04} [[Peter Bouwknegt]], [[Jarah Evslin]], [[Varghese Mathai]], _T-Duality: Topology Change from H-flux_, Commun. Math. Phys. 249:383-415, 2004 ([hep-th/0306062](http://arxiv.org/abs/hep-th/0306062))

* {#BouwknegtHannabusMathai04} [[Peter Bouwknegt]], [[Keith Hannabus]], [[Varghese Mathai]], _T-duality for principal torus bundles_, JHEP 0403 (2004) 018 ([hep-th/0312284](http://arxiv.org/abs/hep-th/0312284))

* [[Varghese Mathai]], [[Jonathan Rosenberg]], _T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology_ ([arXiv:hep-th/0401168](http://arxiv.org/abs/hep-th/0401168))

On [[analytic torsion]] and [[twisted de Rham cohomology]]:

* [[Varghese Mathai]], [[Siye Wu]], _Analytic torsion for twisted de Rham complexes_, J. Diff. Geom. 88:297-332, 2011 ([arXiv:0810.4204](https://arxiv.org/abs/0810.4204))

On a twisted equivariant version of the [[Bismut-Chern character]]:

* [[Fei Han]], [[Varghese Mathai]], *Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality*,  Commun. Math. Phys. **337** (2015) 127-150 [[arXiv:1405.1320](https://arxiv.org/abs/1405.1320),  [doi:10.1007/s00220-014-2270-z](https://doi.org/10.1007/s00220-014-2270-z)]

reviewed in:

* [[Varghese Mathai]], *T-duality for loop spaces, or equivalently for the 1D sigma model*,  talk  at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#Mathai2023)*, NYU Abu Dhabi (Jan  2023) [[web](/nlab/show/M-Theory+and+Mathematics#Mathai2023)]

Aspects of the classification of [[semi-metals]] seen in [[ordinary cohomology]]:

* [[Varghese Mathai]], [[Guo Chuan Thiang]], *Global topology of Weyl semimetals and Fermi arcs*, J. Phys. A: Math. Theor. **50** (2017) 11LT01 ([arXiv:1607.02242](https://arxiv.org/abs/1607.02242),  [doi:10.1088/1751-8121/aa59b2](https://doi.org/10.1088/1751-8121/aa59b2))

* [[Varghese Mathai]], [[Guo Chuan Thiang]], *Differential Topology of Semimetals*,
Commun. Math. Phys. **355** (2017) 561–602, ([arXiv:1611.08961](https://arxiv.org/abs/1611.08961), [doi:10.1007/s00220-017-2965-z](https://doi.org/10.1007/s00220-017-2965-z))

On the [[Witten genus]]:

* [[Fei Han]], [[Varghese Mathai]], _Projective elliptic genera and elliptic pseudodifferential genera_, Adv. Math. 358 (2019) 106860  ([arXiv:1903.07035](https://arxiv.org/abs/1903.07035))

On [[spherical T-duality]]:

* Lachlan Macdonald, [[Varghese Mathai]], Hemanth Saratchandran, _On the Chern character in Higher Twisted K-theory and spherical T-duality_ ([arXiv:2007.02507](https://arxiv.org/abs/2007.02507))

On [[bundle gerbe modules]] with [[higher structure|higher]] [[G-structures]] (including [[String structure]], etc.) and the corresponding [[twisted Chern character]]:

* [[Fei Han]], [[Ruizhi Huang]], [[Varghese Mathai]], *Fractional structures on bundle gerbe modules and fractional classifying spaces* [[arXiv:2203.14439](https://arxiv.org/abs/2203.14439)]

category: people

[[!redirects Varghese Mathai]]


Mathematica

#Contents#
* table of contents
{:toc}

## Idea

_Mathematica_ is a computer algebra software package.

## Related entries

* [[list of mathematics software]]

* [[proof assistants]]:

  * [[Coq]], [[Agda]]

  * [[ELF]]

  * [[Sage]]

* [[GAP]]

## References and links

Mathematica is the brain child of [[Stephen Wolfram]].

* [www.wolfram.com/mathematica](https://www.wolfram.com/mathematica)

* Wikipedia, _[Mathematica](Mathematica)_

A Mathematica package for calculations with ([[super Lie algebra|super]])[[Lie algebras]]:

* [SuperLie](http://equaonline.com/math/SuperLie)

category: software


mathematical economics

## Scope

Mathematical economics studies mathematical methods and models which are applicable or otherwise directly relevant in economics.

Applications of mathematics to economics are a wide field mainly out of scope of $n$Lab;
the parts of mathematics applied typically involved are also those of minor presence in $n$Lab (numerical analysis and simulation, optimization, [[control theory]], [[stochastic process]]es, regression analysis, time-series analysis,
integral and [[differential equation]]s, [[game theory]], complex networks,
fixed point theory, [[statistics]]) or moderate presence ([[variational calculus]], [[dynamical system]]s) in $n$Lab.

We shall however include pointers in some cases which are relevant to
the $n$Lab, for example (nontrivial) connections to [[category theory]].

## References

* wikipedia: [Nash equilibrium](http://en.wikipedia.org/wiki/Nash_equilibrium)

* Ngoc Mai Tran, Josephine Yu, _Product-mix auctions and tropical geometry_ [arxiv/1505.05737](http://arxiv.org/abs/1505.05737)

category: applications


< [[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]


mathematical linguistics

See also [[linguistics]], [[categorial grammar]], [[context-free grammar]], [[linear logic]]

* J. Lambek, _The mathematics of sentence structure_, American Mathematical Monthly __65__, 154–170 (1958) [MR106170](http://www.ams.org/mathscinet-getitem?mr=106170) [jstor](http://www.jstor.org/stable/2310058)
* J. Y. Girard, _Linear logic: its syntax and semantics_, In: Girard, J.Y., Lafont, Y., Regnier, L.
(eds.) Advances in Linear Logic. London Math. Soc. Lec. Notes __222__, pp. 1–42. Cambridge Univ. Press 1995
* András Kornai, _Mathematical linguistics_, Springer 2008
* _Statistical linguistics_, an entry in Oxford International Encyclopedia of Linguistics reproduced [here](http://www.grsampson.net/aslj.html)
* M. Marcolli, Mathematical and computational linguistics, course materials and bibliography, [html](http://www.its.caltech.edu/~matilde/CS101Winter2015.html)
* joshu tan's blog: [formal-concepts-and-natural-languages/#more-136](http://www.joshuatan.com/formal-concepts-and-natural-languages/#more-136)


mathematical literature

##Databases aiming for complete coverage of the mathematical literature

* [The Mathematics Literature Project](http://tqft.net/mlp/wiki/The_Mathematics_Literature_Project)
* [AMS blog post](http://blogs.ams.org/blogonmathblogs/2013/05/13/building-the-world-digital-mathematical-library/) on the World Digital Mathematical Library
* MathSciNet
* Zentralblatt

##Other databases

* [The European Digital Mathematics Library](https://eudml.org)
* More sites listed at [Motivic Stuff](http://homotopical.wordpress.com/links/journals/)
* Some sites listed in this [MathOverflow question](http://mathoverflow.net/questions/87053/papers-archives-especially-not-indexed-by-google)

category:reference

[[!redirects test143]]
[[!redirects The Mathematical Literature]]
[[!redirects The mathematical literature]]
[[!redirects the mathematical literature]]
[[!redirects Mathematical Literature]]
[[!redirects Mathematical literature]]
[[!redirects mathematical literature]]


mathematical logic

see at _[logic - Mathematical logic](logic#MathematicalLogic)_.

and at *[[formal logic]]*

## References

* {#HilbertAckermann59} [[David Hilbert]], [[Wilhelm Ackermann]], _Grundzüge der Theoretischen Logik_ , 4th ed. Springer Heidelberg 1959 [1928]

* [[Georg Kreisel]], *Mathematical Logic*,  in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

* {#BellMach77} [[John Lane Bell]], M. Machover, _A Course in Mathematical Logic_, North-Holland Amsterdam 1977. (ch. 10,§5) ([ISBN:9780720428445](https://www.elsevier.com/books/a-course-in-mathematical-logic/bell/978-0-7204-2844-5))


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#### Symplectic geometry
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This entry collects links related to the textbook

* [[Vladimir Arnold]]:

  \linebreak

  **Mathematical methods of classical mechanics**

  \linebreak

  Graduate Texts in Mathematics **60**

  Springer (1989)

  [doi:10.1007/978-1-4757-1693-1](https://doi.org/10.1007/978-1-4757-1693-1)

on [[classical mechanics|classical]] [[Hamiltonian mechanics]] formulated via [[symplectic geometry]] (the [[phase space]] regarded as a [[symplectic manifold]]).

Later textbooks and lecture notes that pick up this perspective and further develop it in the context of [[geometric quantization]]:

*  {#Weinstein83} [[Alan Weinstein]], _Lectures on Symplectic Manifolds_, volume 29 of CBMS Regional Conf. Series in Math. Amer. Math. Soc., 1983. third  printing.

* Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_

category: reference


mathematical object

[[!redirects mathematical objects]]

#Contents#
* table of contents
{:toc}

## Idea

A mathematical object is an object studied by [[mathematics]]. This notion gets involved as soon as one asks the question what mathematical objects consist of, or---more generally---what it does mean for such an object to "exist". A further question, depending on the answer to the first, is how we can know about mathematical objects.

## Formalist definition

From a [[formalist]] viewpoint, a __mathematical object__
is a [[term]] in whatever system of [[formal logic]]
we happen to be using at the moment.
We can also consider [[types]] as mathematical
objects by interpreting them as terms in some [[universe]].

## Platonic views

Using the Platonic viewpoint,
one can identify mathematical objects
that do not fit the formalist paradigm.

A good example is the notion of an [[(∞,1)-category]],
which exists at the Platonic level,
and whose “shadows” (using Plato's terminology)
in the world of rigorous mathematics
have been exhibited as various models of [[(∞,1)-categories]]:
[[relative categories]], [[quasicategories]], [[Segal categories]],
[[complete Segal spaces]], [[simplicial categories]], etc.

However, the concept of [[(∞,1)-categories]] per se
resists formalization in a satisfactory way.
[[Homotopy type theory]] may succeed one day on this path,
but so far no working definition has been exhibited yet.

## Related entries

* [[homotopy theory FAQ]]

* [[Platonism]]

* [[formalism]]

* [[type theory]]

## References
 {#References}

For an overview of Platonism in the philosophy of mathematics see

* Linnebo, Øystein, _[Platonism in the Philosophy of Mathematics](https://plato.stanford.edu/entries/platonism-mathematics/)_, The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.).

[[!redirects Platonic form]]
[[!redirects Platonic forms]]


mathematical physics

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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

**Mathematical physics** is a discipline at the interface of [[mathematics]] and [[physics]], concerned with developing mathematical [[theory (physics)|theories]] and [[model (in theoretical physics)|models]] of physical phenomena and mathematical apparatus arising or needed in such models.

Mathematical physics intersects with **theoretical physics** which deals with theoretical arguments in consideration of physical phenomena and the development of models of known and of conjectured physics. Theoretical physics is commonly more widely conceived as it deals also with interpretations, non-rigorous and sometimes speculative argument from experiments or from rough comparisons of different models and various experimental data, not necessarily being mathematically satisfactoy.

For example, the calculations of fitting parameters and adjusting models to complicated experimental data, called *[[phenomenology]]* is part of the work of a theoretical physicist, but most such work is not nowadays considered to belong to mathematical physics, unless one is developing a really new mathematical model or tool for such work.

On the other hand, ever since [Galilei 1623](Galileo+Galilei#TheBookOfnature) ($\sim$ "The book of nature is written in the language of mathematics."), [[Naturerkennen und Logik|Hilbert 1930]] ("The instrument that mediates between theory and practice, between thought and observation, is mathematics.") and [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|Wigner 1959]] ("unreasonable effectiveness of mathematics in the natural sciences") it may be understood that *ultimately*, when the dust has settled, all physics is mathematical physics (cf. *[[theory of everything]]*).

## History

Historically, there has been some variance as to what exactly is comprised by "mathematical physics", see ([Fadeev 00](#Fadeev00)) for some history.

In the beginning of the 20th century the term was understood very broadly:

> Not only [[Henri Poincaré ]], but also [[Albert Einstein]], were called mathematical physicists. Newly established theoretical chairs were called chairs of mathematical physics. It follows from the documents in the archives of the Nobel Committee that MP had a right to appear both in the nominations and discussion of the candidates for the Nobel Prize in physics. Roughly speaking, the concept of Mathematical Physics covered theoretical papers where mathematical formulae were used. ([Fadeev 00, p. 1](#Fadeev00))

Also, in the list of [[Hilbert's problems]] of mathematics, the [[Hilbert's sixth problem|sixth problem]] regards the mathematical formulation of physics as one of the core problems in mathematics:

> **6. Mathematical Treatment of the Axioms of Physics.** The investigations on the [[foundations]] of [[geometry]] suggest the problem: To treat in the same manner, by means of [[axioms]], those [[physics|physical sciences]] in which already today [[mathematics]] plays an important part; in the first rank are the [[probability theory|theory of probabilities]] and [[mechanics]].

However, the 1920 and 1930s  the term "mathematical physics" began to be reserved more exclusively for the activity of making precise arguments that are already more or less understood by theoretical physicists informally. The term began to be referred to mathematical tools used in physics or yet more specifically in [[classical physics]] as the theory of [[partial differential equations]] and [[variational calculus]] and in [[quantum physics]] as [[functional analysis]] and [[representation theory]].

> One sees the quest for the rigorous mathematical theorems about results which are understood by physicists in their own way. ([Fadeev 00, p. 2](#Fadeev00))

More recently this narrow understanding has been called into question: [Fadeev 00, p. 3](#Fadeev00) writes:

> I consider as the main goal of Mathematical Physics the use of mathematical intuition for the derivation of really new results in the fundamental physics. In this sense, Mathematical Physics and Theoretical Physics are competitors.  Their goals in unraveling the laws of the structure of matter coincide. However, the methods and even the estimates of the importance of the results of work may differ quite significally.

In a similar spirit of trying to break out of an overly restrictive understanding of the term "mathematical physics" [[Gregory Moore]] has been advocating the alternative term **physical mathematics** for the study of mathematical constructions inspired by models of theoretical physics ([Moore 14](#Moore14)).

Some of the greatest names of mathematical physics have sought a more profound role for it in the belief that very deep mathematical ideas are required to describe nature. For instance, [[Paul Dirac]] expresses such a view here:

> It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. (Paul Dirac, *The Evolution of the Physicist's Picture of Nature*, Scientific American 1963)

This view led Dirac to advocate the following as a methodology:

>The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject, a method which has not yet been applied successfully, but which I feel confident will prove its value in the future. The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One should be influenced very much in this choice by considerations of mathematical beauty. It would probably be a good thing also to give a preference to those branches of mathematics that have an interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations are of more fundamental importance than equations. Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which it appears to lend itself naturally to physical interpretation. ([The Relation between Mathematics and Physics](http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html))

[[Hermann Weyl]] helped develop many of the ingredients of modern mathematical physics, as [[Michael Atiyah]] explains:

>The past 25 years have seen the rise of gauge theories–Kaluza-Klein models of high dimensions, string theories, and now M-theory, as physicists grapple with the challenge of combining all the basic forces of nature into one all embracing theory. This requires sophisticated mathematics involving Lie groups, manifolds, differential operators, all of which are part of Weyl's inheritance. There is no doubt that he would have been an enthusiastic supporter and admirer of this fusion of mathematics and physics. No other mathematician could claim to have initiated more of the theories that are now being explored. His vision has stood the test of time. (Michael Atiyah, *Hermann Weyl: 1885-1955*)

>[Weyl's] contemporaries are long since gone and only a few personal reminiscences survive. On the other hand the passage of time makes it easier to assess the long-term significance of Weyl's work, to see how his ideas have influenced his successors and helped to shape mathematics and physics in the second half of the twentieth century. In fact, the last fifty years have seen a remarkable blossoming of just those areas that Weyl initiated. In retrospect one might almost say that he defined the agenda and provided the proper framework for what followed. He made fundamental contributions to most branches of mathematics, and he also took a serious interest in theoretical physics. (Michael Atiyah, *Hermann Weyl: 1885-1955*)

The version of this belief in the necessity of deep mathematics for physics to be found at the $n$Lab is, naturally enough, that higher categories, and in particular cohesive $(\infty, 1)$-toposes, are required. In this regard, see [[higher category theory and physics]], [[geometry of physics]] and [differential cohomology in a cohesive topos](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos#Subprojects).

## Related entries

* [[theory (physics)]], [[model (in theoretical physics)]]

* [[high energy physics]]

* [[solid state physics]]

* [[mesoscopic physics]]

* [[nuclear physics]]

* [[quantum gravity]]

* [[theory of everything]]

* [[Hilbert's sixth problem]]

* [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]

* [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]

* See also the related, but disputable, notion of [[applied mathematics]].

## Literature and links
 {#Literature}

* {#Dirac78} [[Paul Dirac]], _The mathematical foundations of quantum theory_, pages 1-8 in A. R. Marlow (ed.) _Mathematical Foundations of Quantum Theory_, Ac. Press 1978

  > "One should keep the need for a sound mathematical basis dominating one’s search  for a new theory. Any physical or philosophical ideas that one has must be adjusted to  fit the mathematics. Not the way round.

> Too many physicists are inclined to start from some preconceived physical ideas  and then to try to develop them and a mathematical scheme that incorporates them.  Such a line of attack is unlikely to lead to success. One runs into difficulties and no  reasonable way out of them. One ought then to realize that one's whole line of approach  is wrong and to seek a new starting point with a sound mathematical basis."

* [[Michael Reed]], [[Barry Simon]]: *[[Methods of Modern Mathematical Physics]]*, Academic Press (1978)

* {#Geroch85} [[Robert Geroch]], _Mathematical Physics_, Chicago 1985 ([ISBN:9780226223063](https://press.uchicago.edu/ucp/books/book/chicago/M/bo4158035.html))

  > (introduces concepts via their [[categories]])

* {#Matolcsi86} [[Tamás Matolcsi]], _A Concept of Mathematical Physics_

  _Models for spacetimes_, Akadémiai Kiadó, 1984 ([djvu]( http://szofi.elte.hu/~szaboa/MatolcsiKonyvek/djvu/konyv/spacetime_ocr.djvu))

  _Models in Mechanics_, Akadémiai Kiadó, 1986 ([djvu](http://szofi.elte.hu/~szaboa/MatolcsiKonyvek/djvu/konyv/mechanics_ocr.djvu))

* [[Walter Thirring]], *A Course in Mathematical Physics* -- 1 *Classical Dynamical Systems* and 2 *Classical Field Theory*, Springer (1988, 1992) [[doi:10.1007/978-1-4684-0517-0](https://doi.org/10.1007/978-1-4684-0517-0)]

* {#Hassani99} [[Sadri Hassani]], *Mathematical Physics -- A Modern Introduction to Its Foundations* Springer 1999, 2002, 20013 ([ISBN:9783319011943](https://www.springer.com/de/book/9783319011943), [pdf](http://202.38.64.11/~jmy/documents/ebooks/Hassani%20Mathematical%20Physics%20A%20Modem%20Introduction%20to%20Its%20Foundations%20-%20S.%20Hassani%20%5B0-387-98579-4%5D.pdf))

* {#Fadeev00} [[Ludwig Faddeev]], _Modern Mathematical Physics: what it should be?_, in Fokos, Grigoryan, Kibble, Zegarlinski (eds) _Mathematical Physics 2000_, World Scientific (2000) ([arXiv:math-ph/0002018](https://arxiv.org/abs/math-ph/0002018))

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

  (with emphasis on [[differential geometry]] and [[topology]]/[[homotopy theory]] such as in [[fiber bundles in physics]])

* [[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))

* [[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))

  > (with emphasis on [[fiber bundles in physics]] and [[Yang-Mills theory]])

* [[Adam Marsh]],

  _Mathematics for Physics: An Illustrated Handbook_,

  World Scientific 2018

  ([doi:10.1142/10816](https://doi.org/10.1142/10816), [book webpage](https://www.mathphysicsbook.com/))

  chapter 10: _Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results_ ([arXiv:1607.03089](https://arxiv.org/abs/1607.03089))

  (with emphasis on [[differential geometry]] and [[topology]]/[[homotopy theory]] such as in [[fiber bundles in physics]])

On [[monoidal category]]-theory with an eye towards [[finite quantum mechanics in terms of dagger-compact categories]] and [[quantum computation]]:

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], *Categories for Quantum Theory*, Oxford University Press 2019 ([ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616))

A historically inclined article is in

* Wikipedia,  _[Mathematical physics](http://en.wikipedia.org/wiki/Mathematical_physics)_

An exposition from the point of view of modern [[gauge theory]] is in

* {#Freedman93} [[Daniel Freedman]], _Some beautiful equations of mathematical physics_, Dirac Prize lecture, November 1993 ([arXiv:hep-th/9408175](http://arxiv.org/abs/hep-th/9408175))

Some observations from the point of view of [[string theory]] are in

* {#Moore14} [[Gregory Moore]], _[[Physical Mathematics and the Future]]_ (2014)

For more on _physical mathematics_ see

* Piotr Sułkowski, _Quantum fields, strings, and physical mathematics_, ([arXiv:2104.03350](https://arxiv.org/abs/2104.03350))

For more pointers to the literature see also:

* [[books and reviews in mathematical physics]]

Further related entries include

* [[higher category theory and physics]], [[geometry of physics]]

[[!redirects physical mathematics]]
[[!redirects mathematical physicist]]
[[!redirects mathematical physicists]]


Mathematical Reviews

**Mathematical Reviews** is an indexing service provided by the [American Mathematical Society](http://www.ams.org), giving the publication data of mathematics publications which are most often (but not always) accompanied with a short or a featured review "article" or with the original abstract.
Its online version is called **MathSciNet**:

* web: [https://mathscinet.ams.org/mathscinet/](https://mathscinet.ams.org/mathscinet/)

It requires subscription, but some of its services, most notably the so called **Relay Station** does not. Relay Station gives the full reference (and sometimes a link to the journal) called by the MR number of the review, but does not provide the abstract or the review unless one has a subscription.

MR numbers have two formats. Relay station accepts both formats. To link from $n$lab, the URL is of the following form:

**https://mathscinet.ams.org/mathscinet-getitem?mr=FORMAT

where FORMAT includes the code after the MR. So MR92a:81191 has

**https://mathscinet.ams.org/mathscinet-getitem?mr=92a:81191

and MR1289330 has the URL

**https://mathscinet.ams.org/mathscinet-getitem?mr=1289330

MR numbers can be discovered using the free [**MR Lookup** service](https://mathscinet.ams.org/mrlookup).

This can also provide citations in BibTeX format.  The [MRef](https://mathscinet.ams.org/mref/) tool can also be used to parse citations copied and pasted from elsewhere.

## See also

* [[Zentralblatt MATH]]

* [[Math-Net.Ru]]

[[!redirects MathSciNet]]


mathematical statements --- contents

**mathematical statements**

* [[theory]]

* [[judgement]], [[assertion]]

* [[hypothesis]], [[consequence]]

* [[axiom]]

* [[assignment]]

* [[definition]] ([[inductive definition|inductive]], [[coinductive definition|coinductive]])

* [[lemma]]

* [[proposition]]/[[type]] ([[propositions as types]])

* [[theorem]]

* [[proof]]/[[program]] ([[proofs as programs]])

* [[example]], [[counterexample]]

* [[conjecture]], [[folklore]]

* [[exercise]]

* [[analogy]]

* [[tautology]]

* [[paradox]]


Mathematical Theory of Quantum Fields

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This entry is about the book

* [[Huzihiro Araki]], _Mathematical Theory of Quantum Fields_ Oxford University Press (1999) ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0998.81501&format=complete))

about [[quantum field theory]] from the [[AQFT]] perspective: formalized by [[local nets of observables]] according to the [[Haag-Kastler axioms]].

category: reference


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* [[Klaas Landsman]],

  \linebreak

  **Mathematical Topics Between Classical and Quantum Mechanics**

  \linebreak

  Springer Monographs in Mathematics

  Springer 1998

  [[doi:10.1007/978-1-4612-1680-3](https://doi.org/10.1007/978-1-4612-1680-3)]

on [[classical mechanics]] and [[quantum mechanics]].

category: reference


mathematics

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* [[philosophy of mathematics]]

* [[foundations of mathematics]]

  * [[set theory]], [[type theory]], [[(infinity,1)-category theory]]

  * [[mathematical logic]]

* mathematical tools

  * [[proof assistants]]

* [[topology and geometry|Topology and Geometry]]

  * [[geometry]] (general list), [[topology]] (general list)

  * [[general topology]]

  * [[algebraic topology]]

  * [[homotopy theory]]

  * [[rational homotopy theory]]

  * [[differential geometry]]

  * [[differential topology]]

  * [[algebraic geometry]]

  * [[noncommutative algebraic geometry]]

  * [[noncommutative geometry]] (general flavour)

  * [[higher geometry]]

  * [[Lie theory]]

* [[algebra|Algebra]]:

  * [[universal algebra]]

  * [[higher algebra]]

  * [[group theory]] (including generalizations)

  * [[representation theory]]

  * [[algebraic approaches to differential calculus]]

  * [[homological algebra]]

  * [[gebras]] (co(al)gebras, bi/Hopf-algebr-as/oids...)

* mathematical [[analysis]]

  * [[nonstandard analysis]], [[functional analysis]], [[operator algebras]]

[[!redirects mathematics]]
[[!redirects Mathematics]]
[[!redirects math]]
[[!redirects Math]]
[[!redirects maths]]
[[!redirects Maths]]


mathematics education

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#Contents#
* table of contents
{:toc}

## Idea

>$\mu\alpha\theta\eta\sigma\iota\varsigma$ - Learning (Gr.)

This page will discuss approaches to **teaching of mathematics** (as it is usual in the subject, with emphasis on the pre-university level, where the care in didactical aspects is the most relevant) and
the outstanding sources of the relevant materials about teaching.

#### Scope

Pedagogically well written introductory books in mathematics, rather than about pedagogical matter, are also of our concern, but they will preferably be posted under
[[elementary mathematics]], [[introductions to mathematics]], [[elementary geometry]] and related pages.

## Overview

Most traditionally, teaching methods were improvized adaptations of communication of subject matter from the knowledgeable teacher to a learner.

Modern educational theory is greatly influenced by the works on child psychology. In particular, it has been investigated which cognitive aspects can be achieved at certain age, or within certain educational or other cognitive experiences.

It is now commonly accepted that the advanced and long term knowledge is better achieved if the learner is also a discoverer. This means in mathematics that the emphasis on procedural knowledge should be replaced by wider experience in which a student discovers her own ways to approach the problems which make up the subject. The teacher and the learning environment hence have to anticipate and foster also specific _processes_
in learning the subject rather then only the goals and supposed content matter. Many authors however acknowledge the importance of balance with more traditional coaching and somewhat standardized procedural techniques (micromanagement being counterproductive). While in most educational taxonomies application of knowledge comes only at very hi stages in taxonomy, applying and experiencing concepts in practice, applications and in interaction with technology is considered necessary even at initial steps, and lower degrees of learning the subject. This should be therefore taken into account when creating the goals of mathematics curricula.

## References and online materials

* wikipedia: [mathematics education](https://en.wikipedia.org/wiki/Mathematics_education), [Common Core State Standards Initiative](https://en.wikipedia.org/wiki/Common_Core_State_Standards_Initiative), [New Math](https://en.wikipedia.org/wiki/New_Math), [Moore method](https://en.wikipedia.org/wiki/Moore_method), [modern elementary mathematics](https://en.wikipedia.org/wiki/Modern_elementary_mathematics), [Van Hiele model](https://en.wikipedia.org/wiki/Van_Hiele_model)
* [13th International Congress on Mathematical Education](http://icme13.org), Hamburg Aug 24-31, 2016
* Douglas A. Grouws ed., _Handbook of research on mathematics teaching and learning: a project of the National Council of Teachers of Mathematics_, Macmillan Library Reference 1992
* [GreenApples](http://GreenApples.wikispaces.com) -- resources for new math teachers

### Related $n$Lab pages

* [[New Math]]
* [[elementary mathematics]]
* [[foundations]]

### Software

* [geogebra.org](https://www.geogebra.org), wikipedia: [geogebra](https://en.wikipedia.org/wiki/GeoGebra)

### Organizations and initiatives in math education

* [NCTM](http://www.nctm.org) National Organization of Teachers of Mathematics
* [Common Core](http://www.corestandards.org/Math) math standards
* Next Generation Science Standards <http://www.nextgenscience.org>
* [ICMI](http://www.mathunion.org/icmi) International Commision on Mathematical Instruction (founded in 1908,  now under IMU)
* [websites of math education and math teacher associations](http://www.mathunion.org/icmi/links/math-ed-associations) (a list maintained at ICMI)
* [International Mathematical Olympiad](http://www.imo-official.org), official site

### Individual educators and their work

* Liping Ma, _Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States_ (Studies in Mathematical Thinking and Learning Series)
   * Liping Ma, _A critique of the structure of U.S. elementary school mathematics_, Notices AMS, Oct 2013 [pdf](http://www.ams.org/notices/201310/fea-ma.pdf)
   * Liping Ma, [webpage](http://lipingma.net/math/math.html)
   * blog opinion [What Does Liping Ma REALLY say?](http://rationalmathed.blogspot.hr/2007/06/what-does-liping-ma-really-say.html)
* [Guy Brousseau](http://www.ardm.eu/contenu/guy-brousseau-english) and at French [wikipedia](https://pt.wikipedia.org/wiki/Guy_Brousseau)
* Harold W. Stevenson (educational psychologist), _Mathematics learning in early childhood_, NCTM, 1985
* John A. Van de Walle et al. _Elementary and middle school mathematics. Teaching developmentally_, Pearson 2004, 2007, 2010, 2013
* J. Mayberry, _The Van Hiele levels of geometric thought in undergraduate preservice teachers_, Journal for Research in Mathematics Education 14 (1): 58–69 (1983) [doi](http://dx.doi.org/10.2307/748797) [jstor](http://www.jstor.org/stable/748797 )
* [[Hans Freudenthal]], _Why to teach mathematics so as to be useful_, 1968, [pdf](https://promathmedia.files.wordpress.com/2013/09/freudenthal-1968.pdf)
* (on [[Hans Freudenthal]]'s school) Marja van den Heuvel-Panhuizen, Paul Drijvers, _Realistic Mathematics Education_, [pdf](https://promathmedia.files.wordpress.com/2013/09/encyclopedia-realistic-mathematics-education_ref.pdf)

### Opinion articles

* Elizabeth Green, [Why Do Americans Stink at Math?](http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html), New York Times opinion story (moral: good ideas in the reforms never materialized in practice)
* Ralph A. Raimi, [Whatever Happened to the New Math?](http://www.math.rochester.edu/people/faculty/rarm/smsg.html)
* [[Richard Askey]], _Good intentions are not enough_, [pdf](http://www.math.wisc.edu/~askey/ask-gian.pdf)
* [Mariya Boyko](https://www.facebook.com/public/%D0%9C%D0%B0%D1%80%D1%96%D1%8F-%D0%91%D0%BE%D0%B9%D0%BA%D0%BE), _The "New Math" Movement in the U.S. vs Kolmogorov's Math Curriculum Reform in the U.S.S.R._, [html](https://mariyaboyko12.wordpress.com/2013/08/03/the-new-math-movement-in-the-u-s-vs-kolmogorovs-math-curriculum-reform-in-the-u-s-s-r)
* R. Balian, A. Connes, Bismut, Lafforgue, Serre, _Les savoirs fondamentaux au service de l'avenir scientifique et technique_, [pdf](https://www.laurentlafforgue.org/textes/SavoirsFondamentaux.pdf), a text lamenting the current state of the scientific part of education in France

### Other references

* wikipedia: [mathematics education](https://en.wikipedia.org/wiki/Mathematics_education), [Van Hiele model](https://en.wikipedia.org/wiki/Van_Hiele_model)
* G. Ziegler, _Teaching and learning "What is mathematics?"_, in Proc.ICM 2014, Seoul, vol. 4
* Alexander Karp, Bruce R Vogeli (eds.), _Russian mathematics education_, 2 vols, World Sci. Publ.
* Lingguo Bu, Robert Schoen (eds.), _Model centered learning_, Pathways to mathematical understanding using geogebra, vol. 6 of Modeling and simulations for learning and instruction
* Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckle, _The advantage of abstract examples in learning math_, [Science Magayibe 2012 pdf](https://promathmedia.files.wordpress.com/2013/02/2-1-the-advantage-of-abstract-examples-in-learning-math2.pdf)

The following epistemiology of math article offers also discussion on educational issues and has related bibliography

* Zbigniew Semadeni, _The triple nature of mathematics:
deep ideas, surface representations, formal models_, [article](https://www.researchgate.net/publication/228925441_The_triple_nature_of_mathematics_deep_ideas_surface_representations_formal_models)

Cognitive, linguistic and cultural aspects of mathematics (which are of relevance for learning) are emphasized in

* [[George Lakoff]], Rafael E. Núñez, _Where mathematics comes from_, Basic Books 2000, xviii+493 pp. [MR2001i:00013](http://www.ams.org/mathscinet-getitem?mr=1794854) (cf. also wikipedia: [George Lakoff#Mathematics](https://en.wikipedia.org/wiki/George_Lakoff#Mathematics))

category: education
[[!redirects mathematical education]]


mathematics presented in homotopy type theory

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# Contents
* table of contents
{: toc}

## Idea

[[Homotopy type theory]] is a formal language in which it is possible to carry out [[synthetic mathematics]] using proof assistants, such as [[Coq]] and [[Agda]]. This is also possible in extensions to [[modal homotopy type theory]].

## Existing work

### Homotopy type theory

Work conducted in homotopy type theory:

* [[HoTT book]]

* [[Dan Licata]], [[Michael Shulman]], _Calculating the Fundamental Group of the Circle in Homotopy Type Theory_, ([arXiv:1301.3443](https://arxiv.org/abs/1301.3443))

* [[Favonia|Kuen-Bang Hou (Favonia)]], [[Robert Harper]], _Covering Spaces in Homotopy Type Theory_, ([pdf](https://favonia.org/files/cover-types2016.pdf)).

* [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], _Higher Groups in Homotopy Type Theory_, ([arXiv:1802.04315](https://arxiv.org/abs/1802.04315))

* [[Nicolai Kraus]], [[Thorsten Altenkirch]], _Free Higher Groups in Homotopy Type Theory_, ([arXiv:1805.02069](https://arxiv.org/abs/1805.02069))

* [[Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _Localization in Homotopy Type Theory_, ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

* [[Guillaume Brunerie]], _The James construction and $\pi_4(S^3)$ in homotopy type theory_, ([arXiv:1710.10307](https://arxiv.org/abs/1710.10307))

* [[Dan Licata]], [[Guillaume Brunerie]], _$\pi_n(S^n)$ in Homotopy Type Theory_, Invited Paper, CPP 2013, [PDF](http://dlicata.web.wesleyan.edu/pubs/lb13cpp/lb13cpp.pdf)

* [[Ulrik Buchholtz]], [[Egbert Rijke]], _The Cayley-Dickson Construction in Homotopy Type Theory_, ([arXiv:1610.01134](https://arxiv.org/abs/1610.01134)):

> Construction of the [[quaternionic Hopf fibration]]

* [[Ulrik Buchholtz]], [[Favonia|Kuen-Bang Hou (Favonia)]], _Cellular Cohomology in Homotopy Type Theory_, Logical Methods in Computer Science, Volume 16, Issue 2 (June 1, 2020) ([arXiv:1802.02191](https://arxiv.org/abs/1802.02191), [lmcs:6518](https://lmcs.episciences.org/6518))

* [[Ulrik Buchholtz]], [[Egbert Rijke]], _The real projective spaces in homotopy type theory_, ([arXiv:1704.05770](https://arxiv.org/abs/1704.05770))

* [[Robert Graham]], _Synthetic Homology in Homotopy Type Theory_, ([arXiv:1706.01540](https://arxiv.org/abs/1706.01540))

* [[Guillaume Brunerie]], _On the homotopy groups of spheres in homotopy type theory_, ([arXiv:1606.05916](https://arxiv.org/abs/1606.05916))

* [[Kevin Quirin]], [[Nicolas Tabareau]], _Lawvere-Tierney sheafification in Homotopy Type Theory_, Journal of Formalized Reasoning, ASDD-AlmaDL, 2016, 9 (2), 0.6092/issn.1972-5787/6232. hal-01451710, [PDF](https://hal.inria.fr/hal-01451710/document)

* [[Favonia|Kuen-Bang Hou (Favonia)]], [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], _A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory_, ([arXiv:1605.03227](https://arxiv.org/abs/1605.03227))

* [[Favonia|Kuen-Bang Hou (Favonia)]], [[Michael Shulman]], _The Seifert–van Kampen Theorem in Homotopy Type Theory_, 2016, ([pdf](https://favonia.org/files/vankampen-csl2016.pdf)), ([doi](10.4230/LIPIcs.CSL.2016.22))

* [[Dan Licata]] and [[Eric Finster]], _Eilenberg-MacLane Spaces in Homotopy Type Theory_, LICS 2014, [PDF](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf)

* Floris van Doorn, Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke, and Mike Shulman, _Spectral Sequences in Homotopy Type Theory_. Paper forthcoming. 2018. [github](https://github.com/cmu-phil/Spectral).

* {#Scoccola19} [[Luis Scoccola]], _Nilpotent Types and Fracture Squares in Homotopy Type Theory_ ([arXiv:1903.03245](https://arxiv.org/abs/1903.03245))

* [[Auke B. Booij]], *Analysis in univalent type theory* (2020) $[$[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$

  > (on [[analysis]])

* Kristina Sojakova, _The equivalence of the torus and the product of two circles in homotopy type theory_, ([arXiv:1510.03918](https://arxiv.org/abs/1510.03918))

* [[Ulrik Buchholtz]], _Higher Structures in Homotopy Type Theory_, ([arXiv:1807.02177](https://arxiv.org/abs/1807.02177))

* Nicolai Kraus, Jakob von Raumer, _Path Spaces of Higher Inductive Types in Homotopy Type Theory_, ([arXiv:1901.06022](https://arxiv.org/abs/1901.06022))

* Ulrik Buchholtz, Egbert Rijke, _The long exact sequence of homotopy $n$-groups_, ([arXiv:1912.08696](https://arxiv.org/abs/1912.08696))

* [[Daniel Christensen]], [[Luis Scoccola]], _The Hurewicz theorem in Homotopy Type Theory_, ([arXiv:2007.05833](https://arxiv.org/abs/2007.05833))

* Martin E. Bidlingmaier, Florian Faissole, [[Bas Spitters]], _Synthetic topology in Homotopy Type Theory for probabilistic programming_ ([arXiv:1912.07339](https://arxiv.org/abs/1912.07339))

* Ayberk Tosun, _Formal Topology in Univalent Foundations_, ([pdf](https://odr.chalmers.se/handle/20.500.12380/301098), [slides](https://www.cs.bham.ac.uk/~axt978/talks/lab-lunch-formal-topology.pdf))

* [[Daniel Carranza]], Jonathan Chang, [[Chris Kapulkin]], Ryan Sandford, _2-adjoint equivalences in homotopy type theory_, ([arXiv:2008.12433](https://arxiv.org/abs/2008.12433))

* {#Swan20} [[Andrew Swan]], _On the Nielsen-Schreier Theorem in Homotopy Type Theory_ ([arXiv:2010.01187](https://arxiv.org/abs/2010.01187))

  > (on the [[Nielsen-Schreier theorem]])

* Håkon Robbestad Gylterud, Elisabeth Bonnevier, _Non-wellfounded sets in homotopy type theory_, ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))

* [[Martín Hötzel Escardó]], _Injective types in univalent mathematics_, ([arXiv:1903.01211](https://arxiv.org/abs/1903.01211), published online 5th January 2021 in [Mathematical Structures in Computer Science](https://doi.org/10.1017/S0960129520000225))

* [[Martín Hötzel Escardó]], _The Cantor-Schröder-Bernstein Theorem for ∞-groupoids_, ([arXiv:2002.07079](https://arxiv.org/abs/2002.07079))

* [[Tom de Jong]], [[Martín Hötzel Escardó]], _Domain Theory in Constructive and Predicative Univalent Foundations_, in: _29th EACSL Annual Conference on Computer Science Logic_, [CSL 2021](https://csl2021.fmf.uni-lj.si/), LIPIcs proceedings 183, 2021 ([doi:10.4230/LIPIcs.CSL.2021.28](https://doi.org/10.4230/LIPIcs.CSL.2021.28), [arXiv:2008.01422](https://arxiv.org/abs/2008.01422))

  > (on [[domain theory]])

* [[Martín Hötzel Escardó]] and [Cory Knapp](https://github.com/cmknapp), _Partial Elements and Recursion via Dominances in Univalent Type Theory_, ([LIPIcs:2017:7682](https://drops.dagstuhl.de/opus/volltexte/2017/7682/))

* [[Martín Hötzel Escardó]], _Compact, totally separated and well-ordered types in univalent mathematics_, ([abstract at the author's web page](https://www.cs.bham.ac.uk/~mhe/papers/compact-ordinals-Types-2019-abstract.pdf), also available at the [Types 2019 Book of Abstracts](https://eutypes.cs.ru.nl/eutypes_pmwiki/uploads/Main/books-of-abstracts-TYPES2019.pdf))

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Daniel Grayson]], [[Michael Shulman]], _Construction of the Circle in UniMath_, ([arXiv:1910.01856](https://arxiv.org/abs/1910.01856))

* [[Anders Mörtberg]], Loïc Pujet, _Cubical synthetic homotopy theory_,  CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs January 2020, pp. 158–171, [doi:10.1145/3372885.3373825](https://doi.org/10.1145/3372885.3373825), ([pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf))

### Modal homotopy type theory

Work which uses [[modal homotopy type theory]], with, for example,  [[modalities]] for [[cohesion]], [[differential cohesion]] and [[supergeometry]]:

* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

* [[Felix Wellen]], _[[schreiber:thesis Wellen|Cartan Geometry in Modal Homotopy Type Theory]]_, ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

* [[Felix Wellen]], _Cohesive Covering Theory (Extended Abstract)_, ([pdf](https://hott-uf.github.io/2018/abstracts/HoTTUF18_paper_15.pdf));

* [[David Jaz Myers]], _Good Fibrations through the Modal Prism_, ([arXiv:1908.08034](https://arxiv.org/abs/1908.08034))

See also [[schreiber:Some thoughts on the future of modal homotopy type theory]].

[[!redirects mathematics presented in HoTT]]


mathematics resources

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This page lists some (online) resources for the topic of [[mathematics]].

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{:toc}

## Lists on the $n$Lab

* [[math blogs]] -- list of online communities like blogs and wikis ("Online Resources")

* [[math institutions]] -- links to main institutions and societies and to lists of more institutions

* [[math archives]] -- links to archives of writings and video lectures in areas of relevance to the $n$lab (including physics, philosophy)

* [[math resources by individuals]] -- links to math resource webpages of individuals in areas of importance to the $n$lab

* [[list of mathematics software]]

* [[lists of journals]]

* [[funding]]

## External lists

Some other lists are:

* [Math on the Web](http://www.mathontheweb.org/mathweb/index.html) -- links maintained by American Mathematical Society

* [MathOverflow](http://mathoverflow.net) question: [mathematical resources on the
web](http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web)

* [MathOnline](http://mathonline.andreaferretti.it/) is dedicated to collecting free mathematics resources, like books, lecture notes, videos and more. Using reviews, votes, and a tagging system, all this material is kept organized. (No DNS entry as of 2012-07-14.)

* {#FundingSourceListsOnWikipedia} Wikipedia: [Funding Sources for American Mathematicians](https://en.wikipedia.org/wiki/Draft:Funding_Sources_for_American_Mathematicians)

## Related specific entries

* **mathematics resources**

* [[physics resources]]

* [[philosophy resources]]

[[!redirects math resources]]
[[!redirects maths resources]]
[[!redirects mathematics resources]]
[[!redirects mathematical resources]]


mathematicscontents

* **[[mathematics]]**

  * [[math resources]]

  * [[history of mathematics]]

* [[foundations|Structural Foundations]]

  * [[logic]]
    * [[internal language]]
    * [[classical mathematics]]
    * [[constructive mathematics]]
    * [[predicative mathematics]]
    * [category:foundational axiom](http://ncatlab.org/nlab/list/foundational+axiom)

  * [[set theory]]

    * [[structural set theory]]

  * [[category theory]]
    * [[Categories and Sheaves]]
    * [[topos theory]]

      * [[Sheaves in Geometry and Logic]]
  * [[higher category theory]]

    * [[higher topos theory]]

      * [[Higher Topos Theory|(∞,1)-topos theory]]
        * [[models for ∞-stack (∞,1)-toposes]]
        * [[cohomology]]
    * [[homotopy theory]]
      * [[stable homotopy theory]]
      * [[rational homotopy theory]]
* [[topology and geometry|Topology and Geometry]]
  * [[geometry]] (general list), [[topology]] (general list)
  * [[general topology]]
  * [[differential topology]]
  * [[differential geometry]]
     * [[synthetic differential geometry]]
     * [[symplectic geometry]]
  * [[algebraic geometry]]
  * [[noncommutative algebraic geometry]]
  * [[noncommutative geometry]] (general flavour)
  * [[higher geometry]]
* [[algebra|Algebra]]
  * [[universal algebra]]
  * [[higher algebra]]
  * [[homological algebra]]
  * [[group theory]], [[ring theory]]

    * [[representation theory]]
  * [[algebraic approaches to differential calculus]]
  * [[counterexamples in algebra]]
* [[analysis]]
  * [[nonstandard analysis]]
  * [[functional analysis]]

    * [[operator algebras]]
  * [[Fourier transform]]

* [[Lie theory]]

  * [[infinity-Lie theory - contents|higher Lie theory]]

* [[probability theory]]

* [[discrete mathematics]]

[Edit this sidebar](/nlab/edit/mathematicscontents)


Mather's stability theorem

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## Ingredients

\begin{definition}\label{StableSmoothFunction}
A [[smooth map]] $f \,\colon\, X \to Y$ between [[smooth manifolds]]  is called *stable* if every nearby smooth function, hence every function in some [[open neighbourhood]] of $f$ inside the [[mapping space]] $C^\infty(X,Y) \subset C^0(X,Y)$, is equal to $f$ up to [[conjugation]] with a [[pair]] of [[diffeomorphisms]].
\end{definition}

Similarly, $f$ is *infinitesimally stable* if this statement holds to first order in [[derivatives]], in  a suitable sense.

(due to [Mather 68](#Mather68), see [Ruas 22, Def. 2.2, Def. 3.3](#Ruas22)).

## Statement

\begin{proposition}
A [[proper map|proper]] [[smooth map]] between [[smooth manifolds]] is "stable" (Def. \ref{StableSmoothFunction}) if and only if it is infinitesimally stable.
\end{proposition}

([Mather 70, Thm. 4.1](#Mather70), [Ruas 22, Thm. 3.11](#Ruas22))

## Related statements

* [[inverse function theorem]]

## References

The original articles:

* {#Mather68} [[John N. Mather]], _Stability of $C^\infty$ mappings: I.  The division theorem_,  Annals of Mathematics **87** 1 (1968) 89  $[$[doi:10.2307/1970595](http://dx.doi.org/10.2307/1970595)$]$

* {#Mather69} [[John N. Mather]], _Stability of $C^\infty$ Mappings: II. Infinitesimal Stability Implies Stability_, Annals of Mathematics **89** 2 (1969) 254  $[$[doi:10.2307/1970668](http://dx.doi.org/10.2307/1970668)$]$

* {#Mather70} [[John N. Mather]], *Stability of $C^\infty$ mappings: V. Transversality*, Advances in Mathematics **4** 3 (1970) 301-336 (doi:10.1016/0001-8708(70)90028-9)

Review:

* {#Ruas22} Maria Aparecida Soares Ruas, *Old and new results on density of stable mappings* $[$[arXiv:2201.03888](https://arxiv.org/abs/2201.03888)$]$

A proof in [[synthetic differential topology]] is provided in section 7.3 of

* [[Marta Bunge]], Felipe Gago, Ana Maria San Luis Fernández, _Synthetic Differential Topology_, 2018, ([CUP](http://www.cambridge.org/gb/academic/subjects/mathematics/logic-categories-and-sets/synthetic-differential-topology)) ([excerpt](http://assets.cambridge.org/97811084/47232/excerpt/9781108447232_excerpt.pdf))

following

* Ana Maria San Luis Fernández, _Estabilidad transversal de gérmenes representables infinitesimalmente,_ Ph.D. Thesis, Universidad de Santiago de Compostela (1999) ([abstract page](https://documat.unirioja.es/servlet/tesis?codigo=12609), [on GoogleBooks](https://books.google.ae/books/about/Estabilidad_transversal_de_g%C3%A9rmenes_rep.html?id=itNUAAAAYAAJ))

[[!redirects Mather stability theorem]]


Mather-Yau theorem

## Statement

The isomorphism type of the [[local ring]] of an isolated complex [[hypersurface singularity]] is determined by its Tjurina algebra. In positive characteristic there is an analogue where the local ring is determined by a higher Tjurina algebra instead.

## Literature

The result is proven in

* John N. Mather, Stephen S.-T. Yau, _Criterion for biholomorphic equivalence of isolated hypersurface singularities_, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 10, Phys. Sci., 5946–5947 [https://mathscinet.ams.org/mathscinet-getitem?mr=773820](MR773820) [doi](https://doi.org/10.1073/pnas.78.10.5946)

Some extensions and analogues include

* Gert-Martin Greuel, Thuy Huong Pham, _Mather-Yau theorem in positive characteristic_, J. Algebraic Geom. 26 (2017), 347-355 [doi](https://doi.org/10.1090/jag/669)
* Z. Hua, G. Zhou, _Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras_, Math. Ann. (2022) [doi](https://doi.org/10.1007/s00208-022-02435-3) [arXiv:1803.06128](https://arxiv.org/abs/1803.06128)

category: algebraic geometry


Mathew B. Hastings

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

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

## Selected writings

On the [[quantum adiabatic theorem]] in the context of [[topological phases of matter]]:

* [[Mathew B. Hastings]], [[Xiao-Gang Wen]], *Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance*, Phys. Rev. B **72** (2005) 045141  [[arXiv:cond-mat/0503554](https://arxiv.org/abs/cond-mat/0503554), [doi:10.1103/PhysRevB.72.045141](https://doi.org/10.1103/PhysRevB.72.045141)]

category: people

[[!redirects Mathew Hastings]]


Mathew Bullimore

* [webpage](https://www.durham.ac.uk/staff/mathew-r-bullimore/)

## Selected writings

On [[Coulomb branch]] [[singularities]] in [[D=3 N=4 super Yang-Mills theory]]:

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb Branch of 3d $\mathcal{N}=4$ Theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

  (relation to [[Nahm's equations]] and the [[moduli space of monopoles]])

Discussion of [[topological quantum field theory]] with an eye towards [[supersymmetry|supersymmetric]] field theory, in terms of the [[ordinary homology]] of [[configuration spaces of points]] (see also at _[[correlators as differential forms on configuration spaces of points]]_):

* [[Christopher Beem]], [[David Ben-Zvi]], [[Mathew Bullimore]], [[Tudor Dimofte]], [[Andrew Neitzke]], _Secondary products in supersymmetric field theory_ ([arXiv:1809.00009](https://arxiv.org/abs/1809.00009))

On the [[Witten index]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E.V. Ferrari, Heeyeon Kim, _Twisted Indices of 3d N=4 Gauge Theories and Enumerative Geometry of Quasi-Maps_ ([arXiv:1812.05567](https://arxiv.org/abs/1812.05567))

On the [[Witten index]] for [[D=3 N=2 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E. V. Ferrari, Heeyeon Kim, _The 3d Twisted Index and Wall-Crossing_ ([arXiv:1912.09591](https://arxiv.org/abs/1912.09591))

category: people


Mathieu Anel

* [website](http://mathieu.anel.free.fr/)

Mathieu Anel is one of the two editors of the two volume work: [[New Spaces for Mathematics and Physics]].

## Selected writings

Proving the [[Blakers-Massey theorem]] in any [[(infinity,1)-topos|$(\infty,1)$-topos]] and with the [[(n-connected, n-truncated) factorization system]] allowed to be replaced by more general [[modal homotopy type theory|modalities]]:

* {#AnelBiedermanFinsterJoyal17a} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _A Generalized Blakers-Massey Theorem_, Journal of Topology **13** 4 (2020) 1521-1553 $[$[arXiv:1703.09050](https://arxiv.org/abs/1703.09050), [doi:10.1112/topo.12163](https://doi.org/10.1112/topo.12163)$]$

On [[homotopy type with finite homotopy groups]]

* {#Anel21} [[Mathieu Anel]], _The elementary infinity-topos of truncated coherent spaces_ ([arXiv:2107.02082](https://arxiv.org/abs/2107.02082))

On [[(infinity,1)-topos|$(\infty,1)$-toposes]] understood as *[[logoi]]*:

* {#AnelJoyal19} [[Mathieu Anel]], [[André Joyal]], *Topo-logie*, in *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) 155-257 [[doi:10.1017/9781108854429.007](https://doi.org/10.1017/9781108854429.007), [pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf)]

category: people


Mathieu Dupont

category: people


Mathieu group

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## Idea

The Mathieu groups, denoted $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$ are five of the [[sporadic finite simple groups]] in the [[Happy Family]]. They were first described in the 1860-70s by Émile Mathieu, and the first such groups to be discovered.

They arise as the [[automorphism groups]] of [[Steiner systems]]. The [[order of a group|orders]] of the groups are as follows:

* $M_{11}$ - 7920;
* $M_{12}$ - 95040;
* $M_{22}$ - 443520;
* $M_{23}$ - 10200960;
* $M_{24}$ - 244823040;

The Matthieu group $M_{24}$ is the [[automorphism group]] of the [[binary Golay code]]; this is a [[vector space]] over the [[field]] $\mathbb{F}_2$. The other groups can be obtained as stabilisers of various (sets of) elements of the Golay code, and hence are subgroups of $M_{24}$. The Mathieu groups form the so-called _first generation_ of the _happy family_: the collection of 20 sporadic groups which are subgroups of the [[Monster group]].

## Related concepts

* [[Mathieu moonshine]]

* [[Fischer group]]

## References

* [N-cafe blogpost](http://golem.ph.utexas.edu/category/2013/02/m13.html) on the groupoid $M_{13}$.

* [[John Conway]], Noam D. Elkies and Jeremy L. Martin, "The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$", _Experimental Mathematics_ **15** (2006),  223–236. [Eprint](http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794826).

See also

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

[[!redirects Mathieu groups]]


Mathieu moonshine

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## Idea

> material taken from ([MIF](#Freiburg))

By classical results due to Nikulin, Mukai, Xiao and Kondo in the 1980's and 90's, the finite [[symplectic group|symplectic]] [[automorphism groups]] of [[K3]] surfaces are always subgroups of the [[Mathieu group]] $M_{24}$. This is a [[sporadic finite simple group]] of [[order of a  group|order]] 244823040. However, also by results due to Mukai, each such automorphism group has at most 394 elements and thus is by orders of magnitude smaller than $M_{24}$. On the other hand, according to a recent observation by Eguchi, Ooguri and Tachikawa, the
[[elliptic genus]] of [[K3]] surfaces seems to contain a mysterious footprint of an [[action]] of the entire group $M_{24}$: If one decomposes the [[elliptic genus]] into irreducible [[characters]] of the [[SCFT|N=4 superconformal algebra]], which is natural in view of superconformal field theories ([[SCFT]]s) associated to K3, then the coefficients of the so-called non-[[BPS state|BPS]] characters coincide with the [[dimensions]] of [[representations]] of $M_{24}$.

Mathieu moonshine corresponds to one of the 23 [[Niemeier lattice|Niemeier lattices]] in the generalization to all such lattices known as [[umbral moonshine]].

## Related concepts

* [[Niemeier lattice]]

* [[Moonshine]]

* [[monster group]]

* [[umbral moonshine]]

## References

### General

* [[Matthias Gaberdiel]], Roberto Volpato, _Mathieu Moonshine and Orbifold K3s_ ([arXiv:1206.5143](http://arxiv.org/abs/1206.5143))

* Anne Taormina, [[Katrin Wendland]], _The overarching finite symmetry group of Kummer surfaces in the Mathieu group $M_24$_ ([arXiv:1107.3834](http://arxiv.org/abs/1107.3834))

* [[Matthias Gaberdiel]], [[Stefan Hohenegger]], [[Roberto Volpato]], _Symmetries of K3 sigma models_, Commun.Num.Theor.Phys. **6** (2012) 1-50 [[arXiv:1106.4315](https://arxiv.org/abs/1106.4315)]

* [[Matthias Gaberdiel]], [[Roberto Volpato]], _Mathieu Moonshine and Orbifold K3s_ [[arXiv:1206.5143](https://arxiv.org/abs/1206.5143)]

### Surveys

* [[Terry Gannon]], _Much ado about Mathieu_, ([arXiv:1211.5531](http://arxiv.org/abs/1211.5531))

* [[Robert Volpato]], _Mathieu moonshine and elliptic genus of K3_ (2010) ([pdf](http://th-workshop2010.desy.de/e98837/e98839/e99205/infoboxContent99210/Volpato.pdf))

* {#Freiburg} Mathematisches Institute Freiburg, _Mathieu moonshine_ ([web](http://home.mathematik.uni-freiburg.de/mathphys/mitarbeiter/wendland/publikationen/mathieu.html))


Mathieu Stiénon

Mathieu Philippe Stiénon is a mathematician at Department of Mathematics of Pennsylvania State University.

* page at [PSU](https://pure.psu.edu/en/persons/mathieu-philippe-stienon)

## Selected writings

On [[Atiyah Lie algebroids]]:

* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _From Atiyah classes to homotopy Leibniz algebras_, Commun. Math. Phys. __341__ (2016) 309--349 [doi](https://doi.org/10.1007/s00220-015-2494-6)

On [[Poisson Lie 2-groups]]:

* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _Poisson 2-groups_ ([arXiv](https://arxiv.org/abs/1202.0079))

[[!redirects Mathieu Stienon]]
[[!redirects Mathieu Philippe Stienon]]
[[!redirects Mathieu Philippe Stiénon]]


MathOverflow

[__MathOverflow__](https://mathoverflow.net/) is a website created in September 2009 by [[Anton Geraschenko]] and a group of UC Berkeley PhD students.  It uses the same software as StackOverflow and was named after it.


MathScheme

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## Idea

MathScheme is a project aimed at combining [[computation]] via [[computer algebra]] with [[formal proof]] via [[proof assistants]].

* [MathScheme Web page](http://www.cas.mcmaster.ca/research/mathscheme/)

From their [Mission page](http://www.cas.mcmaster.ca/research/mathscheme/mission.html):

> Sound mathematics is essential in all modern technological developments. As we build increasingly complex systems, mathematical reasoning becomes more difficult and error-prone. This is particularly true for software systems. A mechanized mathematics system (MMS) is required that can support, improve, and — where possible — automate mathematical reasoning.

> There are presently two major types of MMSs, computer algebra systems and computer theorem proving systems. Computer algebra systems provide algorithms for symbolic computation. They are relatively fast and easy to use, but they are not rigorously based, untrustworthy, and narrow in scope. Computer theorem proving systems provide tools for creating formal proofs. They are based on well-defined logical foundations, can support a wide range of mathematics, but are difficult to use and often lack the specific knowledge needed to perform many routine computations.

> While mechanized mathematics has the potential to revolutionize the design, implementation, and analysis of software systems, contemporary MMSs fall well short of this potential. There are no MMSs today that provide highly integrated symbolic computation and formal deduction capabilities. There have been earlier attempts to add the capabilities of one type of MMS to an existing system of the other type or to combine two existing systems of different type, but these attempts have not been very successful. One cannot add fundamental capabilities as an afterthought.

> Our objective is to develop a new approach to mechanized mathematics that combines the strengths of symbolic algebra systems with those of formal deduction systems to yield a system that is both powerful and sound. The first goal of the project is to develop a formal framework that integrates symbolic computation and formal deduction. The framework should allow mathematical knowledge to be represented both declaratively using axioms and procedurally using algorithms, and it should provide a style of mathematical reasoning in which computation and deduction are intertwined.

> The second project goal is to design and implement an MMS based on the formal framework. It should provide services for building formal languages, theories, computations, deductions, and mappings between theories — the services that are necessary for mechanizing the mathematics process.

> Our long-range goal is to build, on top of the MMS, an interactive mathematics laboratory (IML) that provides an integrated set of tools for facilitating and managing mathematical reasoning. The IML is intended to have the capabilities of both contemporary computer algebra systems and computer theorem proving systems and the means to formalize a wide range of mathematical knowledge.

## Related entries

* [[formal proof]]

[[!include proof assistants and formalization projects -- list]]

## References

* [MathScheme Web page](http://www.cas.mcmaster.ca/research/mathscheme/)


Mathys Rennela

* [personal page](http://www.cs.ru.nl/M.Rennela/)

## Selected writings

On [[quantum programming languages]] (*[[EWIRE]]*) for [[quantum circuits]] as [[domain specific embedded programming languages]]:

* {#RennelaStaton17} [[Mathys Rennela]], [[Sam Staton]], *Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory*, Logical Methods in Computer Science **16** 1 (2020) [[arXiv:1711.05159](https://arxiv.org/abs/1711.05159), doi:10.23638/LMCS-16(1:30)2020]

category: people


Matija Bašić

* [webpage](http://web.math.hr/~mbasic/)

[[!redirects Matija Basic]]

category: people


Matija Pretnar

* [webpage](http://matija.pretnar.info/)

## Selected writings

On effect handling via [[module over a monad|modules]] over [[monads in computer science]]:

* [[Gordon D. Plotkin ]], [[Matija Pretnar]], *Handling Algebraic Effects*, Logical Methods in Computer Science, **9** 4 (2013) lmcs:705 [[arXiv:1312.1399](https://arxiv.org/abs/1312.1399), doi:10.2168/LMCS-9(4:23)2013]

category: people


Matilde Marcolli

__Matilde Marcolli__ is a mathematician of Italian origin, currently at Caltech, with a background in mathematical physics. She works on very diverse topics such as gauge theories, [[noncommutative geometry]], arithmetic of quantum statistical mechanics, the combinatorics of [[renormalization]], and [[motives]].

She was a recipient of the Sonja Kovalevskaja prize and worked for several years at the Max Planck Institute for Mathematics in Bonn.

* [web](http://www.its.caltech.edu/~matilde)

* [wikipedia](https://en.wikipedia.org/wiki/Matilde_Marcolli)

* [research works](http://www.its.caltech.edu/~matilde/work.html)

* talks: [slides](http://www.its.caltech.edu/~matilde/slides.html)

## Selected writings

On relation of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]:

* [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036))

Speculation on [[quantum field theory]] over $\mathbb{F}_1$ (via [[wonderful compactifications]] of [[configuration spaces of points]]):

* Dori Bejleri, [[Matilde Marcolli]], *Quantum field theory over $\mathbb{F}_1$*, Journal of Geometry and Physics
**69** (2013) 40-59 [[arXiv:1209.4837](https://arxiv.org/abs/1209.4837), [doi:10.1016/j.geomphys.2013.03.002](https://doi.org/10.1016/j.geomphys.2013.03.002)]

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))

Discussion of [[BTZ black holes]] via [[tensor networks]] in the [[p-adic AdS/CFT correspondence]]:

* 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))

On [[Hopf algebras]] in generative linguistics:

* [[Matilde Marcolli]], [[Noam Chomsky]], Robert Berwick, *Mathematical Structure of Syntactic Merge* ([arXiv:2305.18278](https://arxiv.org/abs/2305.18278))

* [[Matilde Marcolli]], Robert Berwick, [[Noam Chomsky]], *Old and New Minimalism: a Hopf algebra comparison* ([arXiv:2306.10270](https://arxiv.org/abs/2306.10270))

category: people, noncommutative geometry

[[!redirects M. Marcolli]]


matrix

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## Definition

For [[natural numbers]] $n$ and $m$ and a set $X$, an $n\times m$ __matrix__ of elements of $X$ is a function $M:[n]\times[m]\rightarrow X$ from the Cartesian product $[n]\times[m]$ to $X$.

Often one uses the term in a context where one can add and multiply matrices using [[matrix calculus]].  Addition of matrices of the same dimension requires $X$ to have an "addition" operation; multiplication of matrices requires $X$ to also have a "multiplication" operation.  Usually $X$ is at least a [[rig]] and often a [[ring]] or a [[field]].

More generally, for arbitrary sets $A$ and $B$ we can define an $A\times B$-matrix to be a function $A\times B\to X$.  If $X$ has some kind of "infinitary sums" as well as finite "products", then we can also multiply matrices of this sort: e.g. if $X$ is the set of objects of a [[monoidal category]] with arbitrary [[coproducts]].

Note that if the structure on X is such that matrix multiplication is associative and there exist identity matrices, then matrices over X may be taken as the morphisms of a category $Mat_X$ whose composition is matrix multiplication. This is in particular the case if X is a field, and so many basic theorems of linear algebra may be understood as concerning functors from $Vect_X$ into $Mat_X$ and natural transformations between such functors.

## Related concepts

* [[matrix multiplication]], [[matrix calculus]]

* [[square matrix]]

* [[diagonal matrix]]

* [[rank of a matrix]]

* [[inverse matrix]]

* [[matrix equivalence]]

* [[monomial matrix]], [[permutation matrix]]

* [[principal submatrix]]

* [[matrix group]], [[matrix Lie group]]

* [[unitary matrix]]. [[unitary group]]

* [[triangular matrix]]

* [[Smith normal form]]

* [[adjacency matrix]]

* [[linear algebra]], [[general linear group]], [[special linear group]], [[matrix mechanics]], [[matrix theory]], [[matrix Hopf algebra]], [[matrix Lie algebra]], [[matrix Lie group]], [[classical Lie group]], [[universal localization]], [[tensor calculus]], [[moment of inertia]], [[eigenvalue]], [[characteristic polynomial]] (Cayley-Hamilton theorem), [[spectral curve]]

Special cases: [[S-matrix]], [[classical r-matrix]], [[density matrix]], [[hermitian matrix]], [[skew-symmetric matrix]], [[quantum Yang-Baxter matrix]], [[random matrix]], [[skew-symmetric matrix]]

Operations on/with matrices: [[transpose matrix]], [[adjoint matrix]]  [[trace]], [[matrix factorization]], [[Gauss decomposition]], [[Gram-Schmidt process]]

Determinants and [[determinant]] like notions, and special cases: [[quasideterminant]], [[Berezinian]],[[Jacobian]], [[Pfaffian]], [[hafnian]],  [[Wronskian]], [[resultant]], [[discriminant]]

## References

Textbook accounts:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §2 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

[[!redirects matrix]]
[[!redirects matrices]]


matrix algebra

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## Idea

The [[associative algebra]] of [[square matrices]] over some [[ring]] under [[matrix multiplication]] is the coresponding _matrix algebra_.

## Properties

### General

The multiplicative [[identity element]] in a matrix algebra is the [[identity matrix]].

### Relation to groupoid algebras

The algebra of $n \times n$-matrices is equivalently the [[groupoid convolution algebra]] of the [[pair groupoid]] on the set with $n$-elements.

### Norms

The matrix algebra over a [[normed ring]] is naturally itself a normed ring. See at _[normed ring -- Examples -- matrix ring](http://ncatlab.org/nlab/show/normed+ring#MatrixRing)_.

## Related concept

* [[square matrix]]

[[!redirects matrix algebras]]


matrix analysis

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## Idea

[[analysis]] with/of [[matrices]]

## Related concepts

* [[matrix calculus]]

* [[Gram-Schmidt orthogonalization]]

* [[Gershgorin circle theorem]]

## References

Lecture notes:

* {#Meckes19} [[Mark Meckes]], Section 7.1 of: *Lecture notes on matrix analysis*, 2019 ([pdf](https://case.edu/artsci/math/mwmeckes/matrix-analysis.pdf), [[MeckesMatrixAnalysis.pdf:file]])

See also:

* Wikipedia, *[Matrix analysis](https://en.wikipedia.org/wiki/Matrix_analysis)*


matrix calculus

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## Idea

In a [[category]] $C$ with [[biproducts]], [[morphisms]] between [[finite product|finite]] biproducts are naturally encoded in terms of arrays of morphisms between the [[direct sum|direct summands]] of the [[objects]]. The natural operations on  morphisms (addition, composition) correspond to the usual matrix calculus operations on these arrays.

For the special case that $C =$ [[Vect]] this reproduces the standard matrix calculus of linear algebra.

## Rules

Let $f : X \to Y$ be a [[morphism]] in a [[category]] with [[biproducts]] where the objects $X$ and $Y$ are given as [[direct sums]]
$$
  X = \oplus_{j = 1}^m X_j
  \,,
  \;\;
  Y = \oplus_{i = 1}^n Y_i
  \,.
$$

Since a [[biproduct]] is both a [[product]] as well as a [[coproduct]], the morphism $f$ is fixed by all its compositions $f^i_j$ with the product projections $\pi^i : Y \to Y_i$ and the coproduct injections $\iota_j : X_j \to X$:

$$
  f^i_j
  :=
  X_j \stackrel{\iota_j}{\to}
  X \stackrel{f}{\to}
  Y \stackrel{\pi^i}{\to}
  Y_i
  \,.
$$

In **matrix calculus** one therefore writes

$$
  f :=
  \left(
    \array{
       f^1_1 & f^1_2 & \cdots & f^1_m
       \\
       \vdots & \vdots & \ddots & \vdots
       \\
       f^n_1 & f^n_2 & \cdots & f^n_m
    }
  \right)
  \,.
$$

With this notation one has the following rules for computation:

* matrix addition, where $f,g : X \to Y$,

  $$
    (f + g)^i_j = f^i_j + g^i_j
  $$

* matrix multiplication, where $f:X \to Y$, $g:Y \to Z$,

  $$
    (g \circ f)^i_j
    =
    \sum_k
     g^i_k \circ f^k_j
    \,,
  $$

where in each case the sum of morphisms is taken using the canonical [[enriched category|enrichment]] of $C$ in abelian [[monoids]] (as described at [[biproduct]]).

As can be seen in the above formulas, particularly for matrix multiplication, this is a context is which the [[Einstein summation convention]] can be used, with a distinction drawn between upper and lower indices.  Then repeated indices (in formulas with general applicability) will always appear once upper and once lower, summed over.  However, this convention can apply only to the morphisms, not to the objects.

## In dagger categories

If the category $C$ is in addition a [[dagger category]] with an obvious compatibility condition between the dagger operation $(-)^\dagger : C \to C$ and the biproduct structure, then the usual rules of computation for matrices over complex numbers have analogs in $C$.

* conjugation

  $$
    (f^\dagger)_{i j} =  (f_{j i})^\dagger
  $$

Here the distinction between upper and lower indices cannot be maintained, although it is still true that repeated indices will be summed in formulas with general applicability.

## Related concepts

* [[matrix analysis]]

* [[integral transform]]

## References

Discussion in the generality of [[monoidal category|monoidal]] [[category theory]] is in

* [[John Harding]], section 2  of _[[matrixcalculus.pdf:file]]_

Formalization in terms of [[dependent linear type theory]] is in

* _[[schreiber:Type-semantics for quantization]]_

[[!redirects matrix multiplication]]

[[!redirects matrix product]]


matrix equivalence

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## Definition

Two [[matrices]] $A,B \in Mat_{n \times m}(R)$ are called _matrix equivalent_ if there exist [[invertible matrices]] $P \in GL(n,R)$, $Q \in GL(m,R)$ such that $A_2$ [[equality|equals]] the [[matrix product]]

$$
  A_2 \;=\;  P \cdot A_1 \cdot Q
  \,.
$$

## Related concepts

* [[matrix congruence]]

* [[Smith normal form]]

## References

See also

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

[[!redirects matrix equivalences]]

[[!redirects equivalent matrix]]
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matrix factorization

# Matrix factorizations
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There are two different meanings of phrase "matrix factorization" which are closely related

* (typically canonical) [[matrix decomposition]]s of matrices into products of matrices of specific kind, like [[Gauss decomposition]], [[polar decomposition]], LU-decomposition etc.

* decomposition of a ring element, understood as a diagonal matrix, as a product of matrices over a ring, in the sense of Eisenbud and followers.

This entry is dedicated to the latter as it concerns appearance of certain categories in mathematical physics.

## Overview

Matrix factorizations were introduced by [[David Eisenbud]], and they were originally studied in the context of [[commutative algebra]].

Matrix factorizations arise in [[string theory]] as categories of [[D-branes]] for [[Landau-Ginzburg model|Landau-Ginzburg]] [[B-model]]s. This was proposed by Kontsevich and elaborated in the paper of Kapustin-Li.

Matrix factorizations have also been used by Khovanov-Rozansky in knot theory [math/0401268](https://arxiv.org/abs/math/0401268).

__Matrix factorization categories__ are examples of [[Calabi-Yau categories]], hence correspond to [[TCFT]]s by the Costello/Kontsevich/Hopkins-Lurie theorem. The Calabi-Yau structure is elucidated in recent work of Dyckerhoff-Murfet and Polishchuk-Vaintrob.

Dyckerhoff has proved that the [[Hochschild homology]] of the matrix factorizations category of an isolated singularity is the [[Jacobian ring]] of the singularity. See also the work of E. Segal and Caldararu-Tu. In light of the Costello/Kontsevich/Hopkins-Lurie theorem, this result has been anticipated for some time, as the closed state space of a Landau-Ginzburg B-model is the Jacobian ring.

There is also the Calabi-Yau/Landau-Ginzburg correspondence. In some cases, categories of matrix factorizations turn out to be equivalent to categories of coherent sheaves.

For general theory and properties of matrix factorizations, see work of Orlov. For example, matrix factorization categories are related to derived categories of singularities.

## Definition

(Eisenbud 1980) A __matrix factorization__ of an element $x$ in a commutative ring $A$ is an ordered pair of maps of free $A$-modules $(\phi:F\to G,\psi: G\to F)$ such that $\phi\circ\psi = x\cdot 1_G$ and $\psi\circ\phi = x\cdot 1_F$.

Note that if $(\phi,\psi)$ is a matrix factorization of $x$, then $x$ annihilates $Coker\phi$. Often instead of the assumption that $A$-modules $F,G$ are free, one assumes that they are finitely generated projective.

## Related concepts

* [[Landau-Ginzburg model]]

## Literature

* [[David Eisenbud]], _Homological algebra on a complete intersection, with an application to group representations_, Trans. Amer. Math. Soc. __260__:35-64 (1980) [doi](https://doi.org/10.1090/S0002-9947-1980-0570778-7)

* [[Tobias Dyckerhoff]], _Compact generators in the categories of matrix factorizations_,  Duke Math. J. 159(2): 223-274 (2011) [doi](https://doi.org/10.1215/00127094-1415869) [arXiv:0904.4713](https://arxiv.org/abs/0904.4713)

The definition of a [[triangulated category]] of B-branes for the [[Landau-Ginzburg model]] via [[matrix factorization]] was proposed by [[Maxim Kontsevich]] and is written out in

* [[Anton Kapustin]], Yi Li, _D-Branes in Landau-Ginzburg models and algebraic geometry_ (arXiv:[hep-th/0210296](https://arxiv.org/abs/hep-th/0210296))

* {#Orlov} [[Dmitri Orlov]], _Triangulated categories of singularities and D-branes in Landau-Ginzburg models_, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227--248 ([arXiv:math/0302304](https://arxiv.org/abs/math/0302304))

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and triangulated categories of singularities_, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503--531, Progr. Math. __270__, Birkhäuser 2009 arXiv:[math.AG/0503632](https://arxiv.org/abs/math.ag/0503632)

See also

* Junwu Tu, _Matrix factorizations via Koszul duality_, Compositio Mathematica __150__:9 (2014) 1549--1578
[doi](https://doi.org/10.1112/S0010437X14007295) [arXiv:1009.4151](https://arxiv.org/abs/1009.4151)

* J. Burke, M. E. Walker, _Matrix factorizations over projective schemes_, Homology, Homotopy and Applications __14__(2) (2012) 37--61 [arXiv:1110.2918](https://arxiv.org/abs/1110.2918)

* Matthew Ballard, David Favero, [[Ludmil Katzarkov]], _A category of kernels for graded matrix factorizations and its implications for Hodge theory_, Publ.math.IHES 120, 1–111 (2014) [doi](https://doi.org/10.1007/s10240-013-0059-9) [arXiv:1105.3177](https://arxiv.org/abs/1105.3177)

* [[Alexander I. Efimov]], _Cyclic homology of categories of matrix factorizations_,  Intern. Math. Res. Notices __12__ (2018) 3834--3869 [doi](https://doi.org/10.1093/imrn/rnw332) [arXiv:1212.2859](https://arxiv.org/abs/1212.2859)

A matrix factorization of a potential W which is a section of a line bundle on an algebraic stack is studied in

* [[Alexander Polishchuk]], [[Arkady Vaintrob]], _Matrix factorizations and singularity categories for stacks_, Annales de l'Institut Fourier __61__:7 (2011) 2609-2642 [doi](https://doi.org/10.5802/aif.2788) [numdam](http://www.numdam.org/articles/10.5802/aif.2788)

* A. Polishchuk, [[Arkady Vaintrob]], _Matrix factorizations and cohomological field theories_, J. Reine Angew. Math. 714 (2016) 1--122

* [[Alexander Polishchuk]], _Homogeneity of cohomology classes associated with Koszul matrix factorizations_, Compositio Math. __152__ (2016) 2071--2112
[doi](https://doi.org/10.1112/S0010437X16007557) [arXiv:1409.7115](https://arxiv.org/abs/1409.7115)

* [[Alexander Efimov]], [[Leonid Positselski]], _Coherent analogues of matrix factorizations and relative singularity categories_, Algebra & Number Theory __9__ (2015) 1159--1292 [doi](https://doi.org/10.2140/ant.2015.9.1159)

On a connection to the representation theory of loop groups and families of Dirac operators (and the [[string 2-group]])

* {#FreedTeleman14} [[Daniel S. Freed]], [[Constantin Teleman]], _Dirac families for loop groups as matrix factorizations_, Comptes Rendus Mathematique __353__:5 (2015) 415-419 [doi](https://doi.org/10.1016/j.crma.2015.02.011) [arXiv:1409.6051](https://arxiv.org/abs/1409.6051)

On a [[bicategory]] of [[Landau-Ginzburg models]] with [[matrix factorizations]] as [[1-morphisms]]:

* [[Nils Carqueville]], [[Daniel Murfet]], *Adjunctions and defects in Landau-Ginzburg models*, Adv. Math. **289** (2016) 480-566 ([doi:10.1016/j.aim.2015.03.033](https://doi.org/10.1016/j.aim.2015.03.033), [arXiv:1208.1481](https://arxiv.org/abs/1208.1481))

On a category of [[matrix factorizations]] in the context of [[Landau-Ginzburg models]], described in terms of [[linear logic]] and the [[geometry of interactions]]:

* [[Daniel Murfet]], _The cut operation on matrix factorisations_, J. Pure Appl. Alg. **222** 7 (2018) 1911-1955 ([arXiv:1402.4541](https://arxiv.org/abs/1402.4541), [doi:10.1016/j.jpaa.2017.08.014](https://doi.org/10.1016/j.jpaa.2017.08.014))

Constructing $A_\infty$-category of matrix factorizations

* [[Daniel Murfet]], _Constructing $A_\inf$-categories of matrix factorisations_, [arXiv:1903.07211](https://arxiv.org/pdf/1903.07211)

Generalization of matrix factorizations to higher matrix factorizations:

* [[David Eisenbud]], [[Irena Peeva]], _Matrix factorizations for complete intersections and minimal free resolutions_, [arXiv:1306.2615](https://arxiv.org/abs/1306.2615)

Generalization to factorization categories

* Matthew Ballard, Dragos Deliu, David Favero, M. Umut Isik, [[Ludmil Katzarkov]], _Resolutions in factorization categories_, [arXiv:1212.3264](https://arxiv.org/abs/1212.3264)

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