Max Kelly

__Gregory Maxwell Kelly__ (1930-2007) was a [[category theory|category theorist]], the founder of the Australian category theory school.

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

## Selected writings

Precursor discussion towards the notion of [[enriched categories]]:

* [[Gregory Maxwell Kelly]]. _Tensor products in categories_. Journal of Algebra **2** 1 (1965) 15-37 [doi:10.1016/0021-8693(65)90022-0]

Introducing the notion of [[closed categories]]:

* {#EilenbergKelly66} [[Samuel Eilenberg]], [[G. Max Kelly]], *Closed Categories*, pp. 421-562 in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Introducing the notions of [[adjunctions]] in general [[2-categories]] and of [[enriched adjoint functors]]:

* [[Max Kelly]], *Adjunction for enriched categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) [[doi:10.1007/BFb0059145](https://doi.org/10.1007/BFb0059145)]

On [[enriched functor categories]]:

* [[Brian Day]], [[Max Kelly]], *Enriched functor categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106** (1969) 178-191 [[doi:10.1007/BFb0059146](https://doi.org/10.1007/BFb0059146), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0059146.pdf)]

On [[limits]] and [[colimits]] and [[exponential objects]] in the [[category]] [[Top]] of [[topological spaces]]:

* {#DayKelly} [[Brian Day]], [[G. Max Kelly]], *On topological quotients preserved by pullback or products*, Mathematical Proceedings of the Cambridge Philosophical Society **67** 3 (1970) 553 - 558  ([doi:10.1017/S0305004100045850](https://doi.org/10.1017/S0305004100045850))

On [[continuous functors]] (and introducing the notion of [[orthogonal factorization systems]]):

* [[Peter Freyd]], [[Max Kelly]], *Categories of continuous functors*, J. Pure. Appl. Algebra **2** (1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

Introducing the notion of [[weighted limits]] in [[enriched category theory]]:

* [[Francis Borceux]], [[Gregory Maxwell Kelly]], *A notion of limit for enriched categories*, Bulletin of the Australian Mathematical Society **12** 1 (1975) 49-72 [[doi:10.1017/S0004972700023637](https://doi.org/10.1017/S0004972700023637)]

The [[coherence theorem]] for [[compact closed categories]] (cf. *[[coherence theorem for monoidal categories]]*):

* {#KellyLaplaza80} [[Max Kelly]], [[M. L. Laplaza]], *Coherence for compact closed categories*, Journal of Pure and Applied Algebra, **19** 193-213 (1980) [doi:10.1016/0022-4049(80)90101-2, [pdf](https://core.ac.uk/download/pdf/82696829.pdf)]

Proof that the [[funny tensor product]] of [[categories]] is the only other [[symmetric closed monoidal category|symmetric closed monoidal structure]] on [[Cat]] besides the [[cartesian monoidal category|cartesian monoidal structure]]:

* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), doi:10.1016/0022-4049(80)90082-1]

On [[enriched category theory]]:

* [[Max Kelly]], _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)

  Republished as: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 ([tac:10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

On [[sketches]] and [[locally presentable categories]] in the generality of [[enriched category theory]]:

* {#Kelly82} [[G. Max Kelly]], *Structures defined by finite limits in the enriched context, I*, Cahiers de topologie et géométrie différentielle **23** 1 (1982)  3-42 [[numdam:CTGDC_1982__23_1_3_0](http://archive.numdam.org/item/CTGDC_1982__23_1_3_0)]

On [[essential localizations]]:

* [[Max Kelly]], [[F. W. Lawvere]] *On the complete lattice of essential localizations*, Bull. Soc. Math. Belgique **41** (1989) 289–319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

On [[reflective factorization systems]]:

* {#CHK} C. Cassidy, M. Hébert, [[Max Kelly]], _Reflective subcategories, localizations, and factorization systems_,  J. Austral. Math Soc. (Series A) 38 (1985), 287--329 ([doi:10.1017/S1446788700023624](https://doi.org/10.1017/S1446788700023624))

* {#CJKP} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures 5 (1997), 1--58 ([doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444))

On [[monoid objects in monoidal categories]]:

* [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553))

On [[Cartesian bicategories]]:

* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, Theory and Applications of Categories, Vol. 19, 2008, No. 6, pp 93-124. ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921), [tac:19-06](http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html))

On the [[Australian Category Theory Seminar]]:

* {#Kelly} [[G. Maxwell Kelly]], _The beginnings of category theory in Australia_, in: *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) 1-6 [[doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

On [[monadicity]] over [[graphs]] of [[categories]] with (co)limits:

 * [[G. M. Kelly]], [[I. J. Le Creurer]], _On the monadicity over graphs of categories with limits_, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **38** 3 (1997) 179-191 [[numdam](http://www.numdam.org/item/CTGDC_1997__38_3_179_0/)]

 * [[G. M. Kelly]], [[Stephen Lack]], _On the monadicity of categories with chosen colimits_, Theory and Applications of Categories **7** 7 (2000) 148-170 [[TAC](http://www.tac.mta.ca/tac/volumes/7/n7/7-07abs.html)]

 * [[Jiří Adámek]], [[G. M. Kelly]], _$M$-Completeness is seldom monadic over graphs_, Theory and Applications of Categories **7** 8 (2000) 171-205 [[TAC](http://www.tac.mta.ca/tac/volumes/7/n8/7-08abs.html)]

## Further publications

1. Topics in homology theory (PhD Thesis, University of Cambridge, 1957; supervisors S. Wylie and M. G. Barratt).
1. Single-space axioms for homology theory, Proc. Cambridge Phil. Soc. 55 (1959) 10–22.
1. The exactness of Čech homology over a vector space, Proc. Cambridge Phil. Soc. 57 (1961) 428–429.
1. On manifolds containing a submanifold whose complement is contractible, Proc. Cambridge Phil. Soc. 57 (1961) 507–515.
1. Observations on the Künneth theorem, Proc. Cambridge Phil. Soc. 59 (1963) 575–587.
1. On the radical of a category, J. Austral. Math. Soc. 4 (1964) 299–307.
1. Complete functors in homology: I. Chain maps and endomorphisms, Proc. Cambridge Phil. Soc. 60 (1964) 721–735.
1. Complete functors in Homology: II. The exact homology sequence, Proc. Cambridge Phil. Soc. 60 (1964) 737–749.
1. On Mac Lane’s conditions for coherence of natural associativities, commutativities, etc. , J. Algebra 1 (1964) 397–402.
1. A lemma in homological algebra, Proc. Cambridge Phil. Soc. 61 (1965) 49–52.
1. Chain maps inducing zero homology maps, Proc. Cambridge Phil. Soc. 61 (1965), 847–854.
1. (with S. Eilenberg) A generalization of the functorial calculus, J. Algebra 3 (1966), 366–375.

1. Monomorphisms, epimorphisms, and pullbacks, J. Austral. Math. Soc. 9 (1969) 124–142.
1. Adjunction for enriched categories, Lecture Notes in Math. 106 (Springer-Verlag, 1969) 166–177.
1. (with B. J. Day) Enriched functor categories, Lecture Notes in Math. 106 (Springer-Verlag, 1969) 178–191.
1. (with B. J. Day) On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Phil. Soc. 67 (1970) 553–558.
1. (with S. E. Dickson) Interlacing methods and large indecomposables, Bull. Austral. Math. Soc. 3 (1970) 337–348.
1. (with S. Mac Lane) Coherence in closed categories, J. Pure Appl. Algebra 1 (1971) 97–140;Erratum Ibid. 1 (1971) 219.
1. An Introduction to Algebra and Vector Geometry (Reed Education, Sydney, 1972).
1. (with P. J. Freyd) Categories of continuous functors I, J. Pure Appl. Algebra 2 (1972) 169–191; Erratum Ibid. 4 (1974) 121.
1. (with S. Mac Lane) Closed coherence for a natural transformation, Lecture Notes in Math. 281 (Springer-Verlag, 1972) 1–28.
1. Many-variable functorial calculus I, Lecture Notes in Math. 281 (Springer-Verlag, 1972) 66–105.
1. An abstract approach to coherence, Lecture Notes in Math. 281 (Springer-Verlag, 1972) 106–147.
1. A cut-elimination theorem, Lecture Notes in Math. 281 (Springer-Verlag, 1972) 196–213.
1. Cohérence pour les catégories à distributivité, Cahiers de topologie et géométrie différentielle 14 (1973) 36–37.
1. (with R. Street) Review of the elements of 2-categories, Lecture Notes in Math. 420 (Springer-Verlag, 1974) 75–103.
1. On clubs and doctrines, Lecture Notes in Math. 420 (Springer-Verlag, 1974) 181–256.
1. Doctrinal adjunction, Lecture Notes in Math. 420 (Springer-Verlag, 1974) 257–280.
1. Coherence theorems for lax algebras and for distributive laws, Lecture Notes in Math. 420 (Springer-Verlag, 1974) 281–375.
1. (with F. Borceux) A notion of limit for enriched categories, Bulletin Austral. Math. Soc. 12 (1975) 49–72.
1. Quelques observations sur les demonstrations par récurrence transfinie en algébre catégorique, Cahiers de topologie et géométrie différentielle 16 (1975) 259–263.
1. (with A. Pultr) On algebraic recognition of direct-product decompositions, J. Pure Appl. Algebra 12 (1978) 207–224.

1. Saunders Mac Lane and category theory, in Saunders Mac Lane, Selected Papers (Springer-Verlag, New York, Heidelberg, Berlin, 1979) 527–543.

1. Examples of non-monadic structures on categories, J. Pure Appl. Algebra 18 (1980) 59–66.
1. (with M. L. Laplaza) Coherence for compact closed categories, J. Pure Appl. Algebra 19 (1980) 193–213.
1. A unified treatment of transfinite construction for free algebras, free monoids, colimits, associated sheaves, and so on, Bulletin Austral. Math. Soc. 22 (1980) 1–83.
1. (with V. Koubek) The large limits that all good categories admit, J. Pure Appl. Algebra 22 (1981) 253–263.
1. Basic Concepts of Enriched Category Theory (London Math. Soc. Lecture Notes Series 64, Cambridge Univ. Press, 1982).

1. On the essentially-algebraic theory generated by a sketch, Bulletin Austral. Math. Soc. 26 (1982) 45–56.
1. Two addenda to the author’s transfinite constructions article, Bulletin Austral. Math. Soc. 26 (1982) 221–237.
1. (with S. Kasangian and F. Rossi) Cofibrations and the realization of nondeterministic automata, Cahiers de topologie et géométrie différentielle 24 (1983) 23–46.
1. (with E. J. Dubuc) A presentation of topoias algebraic relative to categories or graphs, J. Algebra 81 (1983) 420–433.
1. A note on the generalized reflexion of Guitart and Lair, Cahiers de topologie et géométrie différentielle 24 (1983) 155–159.
1. (with F. Rossi) Topological categories with many symmetric monoidal closed structures, Bulletin Austral. Math. Soc. 31 (1985) 41–59.
1. (with C. Cassidy and M. Hébert) Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (Series A) (1985) 287–329; Corrigenda Ibid. 41 (1986) 286.
1. (with G. B. Im) On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986) 1–18.
1. (with G. B. Im) Some remarks on conservative functors with left adjoints, J. Korean Math. Soc. 23 (1986) 19–33. 250 Historical Records of Australian Science, Volume 21 Number 2
1. A survey of totality for ordinary and enriched categories, Cahiers de topologie et géométrie différentielle catégoriques 27 (1986) 109–132.
1. (with G. B. Im) A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986) 75–88.
1. (with F. Borceux) On locales of localizations, J. Pure Appl. Algebra 46 (1987) 1–34.
1. (with G. B. Im) Adjoint-triangle theorems for conservative functors, Bulletin Austral. Math. Soc. 36 (1987) 133–136.
1. On the ordered set of reflective subcategories, Bulletin Austral. Math. Soc. 36 (1987) 137–152.
1. (with M. H. Albert) The closure of a class of colimits, J. Pure Appl. Algebra 51 (1988) 1–17.
1. (with R. Paré) A note on the Albert-Kelly paper “The closure of a class of colimits”, J. Pure Appl. Algebra 51 (1988) 19–25.
1. Elementary observations on 2-categorical limits, Bulletin Austral. Math. Soc. 39 (1989) 301–317.
1. (with R. Blackwell and A. J. Power) Two dimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1–41.

1. (with G. J. Bird, A. J. Power and R. H. Street) Flexible limits for 2-categories, J. Pure Appl. Algebra 61 (1989) 1–27.
1. (with A. Carboni and R. J. Wood) A 2-categorical approach to change of base and geometric morphisms I, Cahiers de topologieet géométrie différentielle catégoriques 32 (1991) 47–95.
1. A note on relations relative to a factorization system, Lecture Notes in Math. 1448 (Springer-Verlag, 1991) 249–261.
1. On clubs and data-type constructors, in Applications of Categories to Computer Science (Proc. LMS Symposium, Durham 1991) (Cambridge Univ. Press 1992) 163–190.
1. (with S. Lack) Finite-product-preserving functors, Kan extensions, and strongly-finitary 2-monads, Applied Categorical Structures 1 (1993) 85–94.
1. (with S. Lack and R. F. C. Walters) Coinverters and categories of fractions for categories with structure, Applied Categorical Structures1 (1993) 95–102.
1. (with A. J. Power) Adjunctions whose counits are coequalizers and presentations of finitary enriched monads, J. Pure Appl. Algebra 89 (1993) 163–179.
1. (with A. Carboni and M. C. Pedicchio) Some remarks on Maltsev and Goursat categories, Applied Categorical Structures 1 (1993) 385–421.
1. (with G. Janelidze) Galois theory and a general notion of central extension, J. Pure Appl. Algebra 97 (1994) 135–161.
1. (with A. Carboni, G. Janelidze and R. Paré) On localization and stabilization for factorization systems, Applied Categorical Structures 5 (1997) 1–58.
1. (with G. Janelidze) The reflectiveness of covering morphisms in algebra and geometry, Theory and Applications of Categories 3 (1997) 132–159.
1. (with I. Le Creurer) On the monadicity over graphs of categories with limits, Cahiersde topologie et géométrie différentielle catégoriques38 (1997) 179–191.
1. (with S. Lack) On property-like structures, Theory and Applications of Categories 3 (1997) 213–250.
1. (with A. Carboni, D. Verity and R. J. Wood) A 2-categorical approach to change of base and geometric morphisms II, Theory and Applications of Categories 4 (1998) 82–136.
1. (with S. Kasangian and V. Vighi) A bicategorical approach to information flow and security, in Categorical Studies in Italy, Rendiconti del Circolo Matematico di Palermo 64 (2000) 99–122.
1. (with S. Lack) On the monadicity of categories with chosen colimits, Theory and Applications of Categories 7 (2000) 148–170.
1. (with J. Adámek) M-completeness is seldom monadic over graphs, Theory and Applications of Categories 7 (2000) 171–205.
1. (with G. Janelidze) Central extensions in Malt’sev varieties, Theory and Applications of Categories 7 (2000) 219–226.
1. (with E. Faro) On the canonical algebraic structure of a category, J. Pure Appl. Algebra 154 (2000) 159–176.
1. (with G. Janelidze) Central extensions in universal algebra: a unification of three notions, Algebra Universalis 44 (2000) 123–128.
1. (with S. Lack) V-Cat is locally presentable or locally bounded when V is so, Theory and Applications of Categories 8 (2001) 555–575.
1. (with G. Janelidze) A note on actions of a monoidal category, Theory and Applications of Categories 9 (2001) 61–91.
1. (with A. Labella, R. Street and V. Schmitt) Categories enriched on both sides, J. Pure Appl. Algebra 168 (2002) 53–98.
1. (with S. Lack) Monoidal functors generated by adjunctions, with applications to transport of structure along an equivalence, Fields Institute Communications 43 (2004) 319–340. Gregory Maxwell Kelly 1930–2007 251 ([[Kelly-Lack_MonoidalFunctors.pdf:file]])
1. On the operads of J. P. May, Reprints in Theory and Applications of Categories 13 (2005) 1–13.
1. (with V. Schmitt) Notes on enriched categories with colimits of some class, Theory and Applications of Categories 14 (2005) 399–423.
1. (with F. Borceux and G. Janelidze) On the representability of actions in a semi-abelian category, Theory and Applications of Categories14 (2005) 244–286.
1. (with F. Borceux and G. Janelidze) Internal object actions, Commentationes Mathematicae Universitatis Carolinae 46 (2005) 235–255.
1. Basic concepts of enriched category theory, Reprint of the 1982 original [Cambridge Univ. Press, Cambridge], Reprints in Theory and Applications of Categories 10 (2005).
1. The beginnings of category theory in Australia, in “Categories in algebra, geometry and mathematical physics”, Contemporary Math. 431 (Amer. Math. Soc. , Providence, RI, 2007) 1–6.
1. (with A. Carboni, R. F. C. Walters and R. J. Wood) Cartesian bicategories II, Theory and Applications of Categories 19 (2008) 93–124.

category: people

[[!redirects G. M. Kelly]]
[[!redirects Gregory Max Kelly]]
[[!redirects Gregory Maxwell Kelly]]
[[!redirects G. Maxwell Kelly]]

[[!redirects G.M. Kelly]]
[[!redirects G. Max Kelly]]


Max Lieblich

* [webpage](http://www.math.washington.edu/~lieblich/)

category: people


Max Newman

Max Newman was one of the fathers of [[computer science]], working at Manchester on the first electronic programmable computers.

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

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

## Selected writings

Introducing key lemma of abstract [[rewriting|rewriting theory]], now often called *[[Newman's lemma]]*:

* [[Maxwell Herman Alexander Newman]], _On theories with a combinatorial definition of "equivalence"_, Annals of Mathematics **43** 2 (1942) 223–243 [[doi:10.2307/2269299](https://doi.org/10.2307/2269299)]

category:people

[[!redirects Max H. A. Newman]]
[[!redirects Maxwell H. A. Newman]]

[[!redirects Maxwell Herman Alexander Newman]]


Max Niedermaier

* [webpage](http://www.physicsandastronomy.pitt.edu/people/max-niedermaier)

## related $n$Lab entries

* [[asymptotic safety]]

category: people


Max Planck

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

## Selected writings

Laying the seed of [[quantum physics]] (and introducing what came to be called [[Planck's constant]], [p. 164](https://www.gutenberg.org/files/40030/40030-pdf.pdf#page=178)):

* [[Max Planck]] (transl. M. Martius) *The Theory of Heat Radiation* (1914) [[pdf](https://www.gutenberg.org/files/40030/40030-pdf.pdf)]

## Related entries

* [[quantum mechanics]]

* [[Planck's constant]]

* [[Planck length]]

* [[Planck mass]]

category: people


Max Riegler

* [webpage](https://sites.google.com/view/maxriegler)

## Selected writings

Discussion of [[holographic entanglement entropy]] in asymptotically flat [[3d quantum gravity]] via [[Wilson line observables]] in [[Chern-Simons theory]]:

* Arjun Bagchi, Rudranil Basu, Daniel Grumiller, [[Max Riegler]], _Entanglement entropy in Galilean conformal field theories and flat holography_, Phys. Rev. Lett. 114, 111602 (2015) ([arXiv 1410.4089](https://arxiv.org/abs/1410.4089))

* Rudranil Basu, [[Max Riegler]], _Wilson Lines and Holographic Entanglement Entropy in Galilean Conformal Field Theories_, Phys. Rev. D 93, 045003 (2016) ([arXiv:1511.08662](https://arxiv.org/abs/1511.08662))

* [[Wout Merbis]], [[Max Riegler]], _Geometric actions and flat space holography_ ([arXiv:1912.08207](https://arxiv.org/abs/1912.08207))

category: people


Max S. New

[[!redirects Max New]]

* [webpage](http://maxsnew.github.io/)

I am a computer scientist, currently working on the semantics of [[gradual typing]].

category: people


Max Tegmark

* [website at institute](http://web.mit.edu/physics/people/faculty/tegmark_max.html)

* [personal website](http://space.mit.edu/home/tegmark/home.html)

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

## Selected writings

On the [[interpretation of quantum mechanics]]:

* [[Max Tegmark]], *The Interpretation of Quantum Mechanics: Many Worlds or Many Words?*, Fortsch. Phys. **46** (1998) 855-862 [[arXiv:quant-ph/9709032](https://arxiv.org/abs/quant-ph/9709032)]

category: people


Max Zeuner

* [website](https://www.su.se/english/profiles/maze1512-1.450461)

## Selected writings

On [[affine schemes]] in  [[univalent foundations]]:

* [[Anders Mörtberg]], [[Max Zeuner]], *A Univalent Formalization of Affine Schemes* ([arXiv:2212.02902](https://arxiv.org/abs/2212.02902))

## Talks

* [[Max Zeuner]], *A univalent formalization of affine schemes*, 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Zeuner-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=nLP7GjL1Buc))

category: people


Max Zimet

* [webpage](https://bhi.fas.harvard.edu/people/maxwell-zimet)

## Selected writings

Systematic construction of [[Ricci tensor|Ricci flat]] [[Riemannian metrics]] on [[K3]] [[orbifolds]]:

* [[Shamit Kachru]], [[Arnav Tripathy]], [[Max Zimet]], _K3 metrics_ ([arXiv:2006.02435](https://arxiv.org/abs/2006.02435))

* [[Arnav Tripathy]], [[Max Zimet]], _A plethora of K3 metrics_ ([arXiv:2010.12581](https://arxiv.org/abs/2010.12581))

category: people


Max Zorn

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

## Related $n$Lab entries

* [[Zorn's lemma]]

category: people


max-entropy

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

# Contents
* table of contents
{: toc}

## Idea

A notion of [[entropy]].

In the context of [[probability theory]], the **max-entropy** or **Hartley entropy** of a [[probability distribution]] on a [[finite set]] is the [[logarithm]] of the [[number]] of outcomes with non-[[zero]] [[probability]].

In [[quantum probability theory]] this means that the max entropy is the logarithm of the [[rank]] of the [[density matrix]].

## Properties

### Relation to Shannon entropy

The ordinary [Shannon entropy](entropy#entropy_of_a_partition_of_a_discrete_probability_space) of a probability distribution is never greater than the max-entropy.

### Relation to Renyi entropy

The Hartley entropy of a finite probability distribution $(p_i)_{i = 1}^n$ is the special case of [[Renyi entropy]]

$$
  S_\alpha(p)
  \;=\;
  \frac{1}{1 - \alpha}
  ln
  \left(
  \underoverset
    {i = i}
    {n}
    {\sum}
    (p_i)^\alpha
  \right)
$$

for $\alpha = 0$.

[[!include Renyi entropies -- table ]]

## Related entries

* [[min-entropy]]

## References

Review:

* Scholarpedia, *[Quantum entropies -- Hartley entropy](http://www.scholarpedia.org/article/Quantum_entropies#Max_entropy_.28Hartley_entropy.29)*

* J. Aczél, B. Forte and C. T. Ng, Advances in Applied Probability
Advances in Applied Probability, Vol. 6, No. 1 (Mar., 1974), pp. 131-146 ([jstor:1426210](https://www.jstor.org/stable/1426210))

[[!redirects max-entropies]]

[[!redirects Hartley entropy]]
[[!redirects Hartley entropies]]

[[!redirects maximum entropy]]


max-plus algebra

# The max-plus algebra and min-plus algebra
* table of contents
{: toc}

## Definitions

The __max-plus algebra__, or __tropical semiring__, is the [[rig]] based on the set of real numbers extended by $-\infty$ (the [[upper reals]]), with addition $x \oplus y = max(x,y)$ and multiplication $x \otimes y = x + y$.
Similarly, the __min-plus algebra__, or __rig of costs__, is the [[rig]] based on the set of real numbers extended by $\infty$ (the [[lower reals]]), with addition $x \oplus y = min(x,y)$ and multiplication $x \otimes y = x + y$.  The two rigs are isomorphic, with an [[isomorphism]] (either way) given by $x \mapsto -x$; the difference is just a matter of the desired perspective.

## Applications

The max-plus algebra is an [[idempotent semiring]] and [[dioid]] that is used in the modelling of timed systems.  Typically in a simple example, the completion time of a production system will be given by a system of equations that have 'max' occurring in them. (The next process in a system cannot start until all its component parts have been themselves completed.) The use of the max-plus notation completely linearises many systems.

The min-plus algebra may be used to draw an analogy between the use of [[action form|action]] in [[classical mechanics|classical]] vs [[quantum mechanics|quantum]] systems.  While the latter involves a [[sum of histories|sum of a multiplicative action over classical trajectories]], the former involves a [[least action principle|minimisation of an additive action over classical trajectories]].  There is a similar analogy between [[statics]] and [[statistical mechanics]].

## References

* [[Stéphane Gaubert]], [Introduction aux Systèmes Dynamiques à Événements Discrets](http://amadeus.inria.fr/gaubert/PAPERS/POLY12-02-1999.pdf)

* [[Stéphane Gaubert]], _Methods and Applications of $(max,+)$ Linear Algebra_ [Report 3088](https://hal.inria.fr/inria-00073603/document), January 1997, INRIA.

* [[John Baez]] & [[David Corfield]], [Changing the rig](http://math.ucr.edu/home/baez/corfield/2006/05/changing-rig.html)

* [[Jean-Pierre Quadrat]], _[Max-Plus Algebra and Applications to System Theory and Optimal Control](http://www.cmap.polytechnique.fr/~gaubert/PAPERS/ICM94-preprint.pdf)_, Proccedings of the International Congress of Mathematicians , Zurich 1994,  Birkhauser , 1995.

## Resources

There are several research groups, world wide, with research in this area and with good websites, including simulation, and calculational, tools. One way into the network of these sites is [here](http://www.maxplus.org/).

[[!redirects max-plus algebra]]
[[!redirects min-plus algebra]]
[[!redirects tropical semiring]]
[[!redirects rig of costs]]


Maxence Mayrand

Postdoc at the University of Toronto.

* [website](https://www.math.toronto.edu/mayrand/)


Maxim Grigoriev

* [personal page](https://itmp.msu.ru/en/institute/scientific-members/grigoriev-m-a)

## Selected writings

On (generalizations of) [[BV-formalism]]:

* [[Maxim Grigoriev]], *Local gauge theories as presymplectic gauge PDEs*, talk at *Higher Structures and Field Theory*, ESI, Vienna (Aug 2022) [[webpage](https://www.esi.ac.at/events/t985), video:[YT](https://youtu.be/LNb4iFLqrb4)]

On [[L-infinity algebras|$L_\infty$-algebras]] from [[BV-formalism]] in [[local field theory|local]] [[gauge field|gauge]] [[field theory]]:

* [[Maxim Grigoriev]], [[Dmitry Rudinsky]], *Notes on the $L_\infty$-approach to local gauge field theories*, Journal of Geometry and Physics **190** (2023) 104863 [[arXiv2303.08990](https://arxiv.org/abs/2303.08990), [doi:10.1016/j.geomphys.2023.104863](https://doi.org/10.1016/j.geomphys.2023.104863)]

category: people


Maxim Kontsevich

__Maxim Kontsevich__ (also _Максим Концевич_, born 1964 August 25) is a Russian-French mathematician, a Professor at [IHES](http://www.ihes.fr), a recipient of the [1998 Fields Medal](http://159.226.47.99:8080/general/prize/medal/1998.htm), the 2008 Crafoord Prize, the 2012 Shaw prize and the new prize for fundamental physics 2012.

* wikipedia [English](http://en.wikipedia.org/wiki/Maxim_Kontsevich), [German](http://de.wikipedia.org/wiki/Maxim_Lwowitsch_Konzewitsch)

* [official CV page](http://193.51.104.7/~maxim/CVAnglais.html), [autobiography](http://www.shawprize.org/en/shaw.php?tmp=3&twoid=92&threeid=210&fourid=352&fiveid=173)

#Contents#
* table of contents
{:toc}

## Work

The [Fields Medal 1998](http://159.226.47.99:8080/general/prize/medal/1998.htm) was awarded for solutions of "four problems in geometry" concerning the subjects:

1. [[intersection theory]] on [[compactification|compactified]] [[moduli spaces]] $\mathcal{M}_{g,n}$ of punctured [[Riemann surfaces]] and the [[Witten conjecture]] (obtained by relating two 2d [[string]]/[[quantum gravity]] models),

1. [[knot invariants]],

1. [[deformation quantization]]

   Kontsevich's formula for the [[formal deformation quantization]] of any [[Poisson manifold]] given by the [[n-point function|3-point function]] of [[open strings]] in the [[Poisson sigma-model]]

1. [[quantum cohomology]]/[[mirror symmetry]];

   Kontsevich's work for formulating [[mirror symmetry]] in [[string theory]] by means of [[A-infinity categories]] as an equivalence of formalizations of [[topological string]] [[quantum field theory]] now known as _[[TCFT]]_ formulations of the [[A-model]] and the [[B-model]]:

Cf. T. R. Ramadas, _The Work of The Fields
Medallists: 1998; 3. Maxim Kontsevich_, [pdf](http://www.ias.ac.in/resonance/July1999/pdf/July1999ResearchNews.pdf)

Some of the articles of MK can be found at [the arXiv](http://arxiv.org/) and most of others at

* MK's [official publications page](http://193.51.104.7/~maxim/publicationsanglais.html) (there are also several unlisted preprints at [the MPI server](http://www.mpim-bonn.mpg.de/Research/MPIM+Preprint+Series/) from collaboration with [[Alexander Rosenberg]]).

Much of Kontsevich's work has never been fully written and has been known via his public lectures and communications to other mathematicians; most notably, [[motivic integration]] was introduced in his lecture in 1996 at Orsay and never published by him, though the subject influenced and is present implicitly again in his newer (written) works. He has solved the Witten conjecture on the connection between the [[KdV equation]] and [[quantum gravity]] theory (in terms of [[moduli space]]s of [[Riemann surface]]s; cf. also [[Airy function]]).

Kontsevich has made contributions to various parts of [[mathematics]] and [[mathematical physics]] which inspired much of his research. One should emphasise his work on the mathematical formulation of [[conformal field theory]] (esp. 1988--1992 and again from 2002 in connection with the [[stochastic Loewner equation]] with R. Friedrich and Suhov), [[Gromov-Witten invariant]]s (since collaboration with [[Yuri Manin]] in Bonn, early 1990s), [[knot theory]] ([[Vassiliev invariant]]s, [[quantum groups]]), $A_\infty$-[[A-infinity-category|categories]], [[Landau-Ginzburg model]]s in [[algebraic geometry]], the [[AKSZ theory|AKSZ model]] in [[quantum field theory]], introduction of [[formal noncommutative symplectic geometry]] along with ideas about a generalization of formal Lie calculus and [[Koszul duality]] for dg-operads, discovery of [[graph complex]] and [[graph homology]] and its role in various problems of geometry and topology including moduli spaces and a geometric explanation of the origin of [[Rozansky-Witten invariant]]s, fundamental works in [[noncommutative algebraic geometry]], introducing [[homological mirror symmetry]], foundations of [[deformation theory]], work on generalizations of [[determinant]] and [[trace]] for [[linear operator]]s (with Vishik), [[Donaldson-Thomas invariant]]s (with Soibelman), [[tropical geometry]], [[nonabelian Hodge theory]], [[noncommutative motive]]s, various constructions with [[supersymmetry]] (esp. in [[geometry]]), [[derived noncommutative algebraic geometry]] and so on.

Much of Kontsevich's research is based on insights into the relation between [[classical physics]] and [[quantum physics]] and [[quantization|quantizing]] various constructions even in pure [[mathematics]]. His most famous work in this area includes the _[[Kontsevich formality theorem]]_ solving the problem of [[deformation quantization]] of [[Poisson manifolds]] using [[homological algebra]] and [[operad]]s; this created a number of new directions of research in mathematics. [[quantization|Quantization]] ideas are also present in works related to the [[geometry]] of [[Weyl algebra]], including the progress on the [[Jacobian conjecture]]. In addition to finished results, Kontsevich introduced a number of stimulating conjectures which strongly influence modern mathematics.

## Selected writings

On [[graph complexes]] and their [[quasi-isomorphism]] to differential forms on ([[Fulton-MacPherson compactifications]] of) [[configuration spaces of points]], via assignment of [[Chern-Simons propagators]]/[[Chern-Simons theory]]-[[Feynman amplitudes]], regarded [[correlator as differential form on configuration space of points|as differential form on configuration space of points]]:

* {#Kontsevich94} [[Maxim Kontsevich]], pages 11-12 of _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

* {#Kontsevich99b} [[Maxim Kontsevich]], around Def. 15 and Lemma 3 in _Operads and Motives in Deformation Quantization_, Lett. Math. Phys. 48 35-72, 1999 ([arXiv:math/9904055](https://arxiv.org/abs/math/9904055))

Specifically on [[Vassiliev knot invariants]]:

* [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))

On [[wall crossing]] phenomena:

* Lecture: [[wall crossing in Aarhus 2010]]

* Introductory Lecture: [Calabi-Yau Motives](https://www.youtube.com/watch?v=0M-jXPi_t1I), 2015 Breakthrough Prize in Mathematics Symposium

On [[Wick rotation]] in [[quantum field theory]] in terms of complex metrics:

* [[Maxim Kontsevich]], [[Graeme Segal]], *Wick rotation and the positivity of energy in quantum field theory*, The Quarterly Journal of Mathematics **72** 1-2 (2021) 673–699 [[arXiv:2105.10161](https://arxiv.org/abs/2105.10161), [doi:10.1093/qmath/haab027](https://doi.org/10.1093/qmath/haab027)]

category: people

[[!redirects Maxim Kontsevich]]
[[!redirects Maksim Kontsevich]]
[[!redirects M. Kontsevich]]
[[!redirects Maksim Koncevič]]
[[!redirects Максим Концевич]]
[[!redirects Kontsevich]]
[[!redirects Koncevič]]
[[!redirects Концевич]]


Maxim Pospelov

* [Institute page](https://cse.umn.edu/physics/maxim-pospelov)

* [GoogleScholar page](https://scholar.google.ca/citations?user=WYm3vy0AAAAJ&hl=en)

## Selected writings

On searches for [[dark matter]]:

* Torsten Bringmann, [[Maxim Pospelov]], *Novel direct detection constraints on light dark matter*, Phys. Rev. Lett. **122** (2019) 171801 [[arXiv:1810.10543](https://arxiv.org/abs/1810.10543), [doi:10.1103/PhysRevLett.122.171801](https://doi.org/10.1103/PhysRevLett.122.171801)]

* [[Maxim Pospelov]], _Dark matter identification efforts_, talk at *[[Strings 2022]]* [[indico:4940821](https://indico.cern.ch/event/1085701/contributions/4940821), [video](https://ustream.univie.ac.at/media/core.html?id=118664ce-20a5-4b62-a3b3-735ce5deea20)]

category: people


Maxim V. Polyakov

* [GoogleScholar page](https://scholar.google.com/citations?user=_n-f_HUAAAAJ&hl=ru)

# Selected writings

On the [[D-term]] in [[quantum hadrodynamics]]:

* [[Maxim V. Polyakov]], C. Weiss, *Skewed and double distributions in pion and nucleon* $[$[arXiv:hep-ph/9902451](https://arxiv.org/abs/hep-ph/9902451)$]$

* [[Maxim V. Polyakov]], [[Peter Schweitzer]], *D-term, strong forces in the nucleon, and their applications* $[$[arXiv:1801.05858](https://arxiv.org/abs/1801.05858), [spire:1648787](https://inspirehep.net/literature/1648787)$]$

* {#PolyakovSchweitzer18} [[Maxim V. Polyakov]], [[Peter Schweitzer]], *Forces inside hadrons: pressure, surface tension, mechanical radius, and all that*, Int. J. Mod. Phys. A **33** (2018) 1830025 $[$[doi:10.1142/S0217751X18300259](https://doi.org/10.1142/S0217751X18300259), [arXiv:1805.06596](https://arxiv.org/abs/1805.06596)$]$

category: people

[[!redirects Maxim Polyakov]]


Maxim Zabzine

* [website](http://www.teorfys.uu.se/people/maxim/)

category: people


maximal central extension

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

#Contents#
* table of contents
{:toc}

## Idea

Given some [[algebra|algebraic]] object like an [[associative algebra]] or [[group]] or [[Lie algebra]] which with a notion of _[[central extension]]_, one may ask for the largest possible such, the _maximal central extension_. This makes sense if one considers [[subobjects]] of a given algebraic object.

## Related concepts

* [[universal central extension]]

## References

* {#MiyakeOpolka91} Katsuya Miyake, Hans Opolka, _On nilpotent extensions of algebraic number fields I_, Nagayo Math. J., Vol. 125 (1991), 1-14 ([Euclid:1118783087](https://projecteuclid.org/euclid.nmj/1118783087))

>  The lower central series of the absolute Galois group of a field is obtained by iterating the process of forming the maximal central extension of  the maximal nilpotent extension of a given class, starting with the maximal   abelian extension. The  purpose of this paper is to give a  cohomological   description of this central series in case of an algebraic number field.

[[!redirects maximal central extensions]]


maximal compact subgroup

+-- {: .rightHandSide}
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###Context###
#### Group Theory
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=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $G$ a [[topological group]] a _compact subgroup_ is a topological [[subgroup]] $K \subset G$ which is a [[compact group]].

+-- {: .num_defn}
###### Definition

A [[compact topological space|compact]] [[subgroup]] $K \hookrightarrow G$ is called **maximal compact** if it is not properly contained in another compact subgroup of $G$.

=--

If $G$ is a Lie group, it is unique up to conjugation. In a $p$-adic group, there may be finitely many conjugacy classes of maximal compact subgroups.

## Properties

+-- {: .num_defn #AlmostConnected}
###### Definition

A [[locally compact topological group]] $G$ is called **[[almost connected topological group|almost connected]]** if the [[quotient]] [[topological space]] $G/G_0$ (of $G$ by the [[connected component]] of the neutral element) is  [[compact topological space|compact]].

=--

See for instance ([Hofmann-Morris, def. 4.24](#HofmannMorris)).

+-- {: .num_defn }
###### Example

Every [[compact topological space|compact]] and every [[connected topological space|connected]] [[topological group]] is almost connected.

Also every [[quotient]] of an almost connected group is almost connected.

=--

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

Let $G$ be a [[locally compact topological space|locally compact]] [almost connected](#AlmostConnected) [[topological group]].

Then

* $G$ has a maximal compact subgroup $K$;

* the [[coset]] space $G/K$ is [[homeomorphism|homeomorphic]] to a [[Euclidean space]].

=--

This is due to ([Malcev](#Malcev)) and ([Iwasawa](#Iwasawa)). See for instance ([Stroppel, theorem 32.5](#Stroppel)).

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

Let $G$ be a [[locally compact topological space|locally compact]] [almost connected](#AlmostConnected) [[Lie group]].

Then a [[compact topological space|compact]] subgroup $K \hookrightarrow G$ is maximal compact precisely if the [[coset]] space $G/K$ is [[contractible space|contractible]]

(in which case, due to theorem \ref{MalcevIwasawa}, it is necessarily [[homeomorphism|homeomorphic]] to a [[Euclidean space]]).

=--

This is ([Antonyan, theorem 1.2](#Antonyan)).

+-- {: .num_remark}
###### Remark

In particular, in the above situation the [[subgroup]] inclusion

$$
  K \hookrightarrow G
$$

is a [[homotopy equivalence]] of [[topological spaces]].

=--

## Examples

### For Lie groups
 {#ExamplesForLieGroups}

The following table lists some [[Lie groups]] and their maximal compact Lie subgroups (e.g. [Conrad](#Conrad)). See also _[[compact Lie group]]_.

|            Lie group                         |     maximal compact subgroup    |
|----------------------------------------------|---------------------------------|
| [[real numbers|real]] [[general linear group]] $GL(n, \mathbb{R})$  | [[orthogonal group]] $O(n)$    |
| its connected component $GL(n,\mathbb{R})_0$ | [[special orthogonal group]] $SO(n)$ |
| [[complex numbers|complex]] [[general linear group]] $GL(n, \mathbb{C})$  | [[unitary group]] $U(n)$        |
| [[complex numbers|complex]] [[special linear group]] $SL(n, \mathbb{C})$  | [[special unitary group]] $SU(n)$        |
| [[symplectic group]] $Sp(2n,\mathbb{R})$                | [[unitary group]] $U(n)$        |
| [[complex symplectic group]] $Sp(2n,\mathbb{C})$ | [[compact symplectic group]] $Sp(n)$ |
| [[Narain group]] $O(n,n)$ | [[direct product group]] of two [[orthogonal groups]] $O(n) \times O(n)$ |
| [[unitary group]] $U(p,q)$ | $U(p) \times U(q)$ |
| special [[Lorentz group|Lorentz]]/[[anti de Sitter group|AdS]] etc. group $SO(p,q)$ | $S\big(O(p) \times O(q)\big)$ |
| Lorentz / AdS [[pin group]] $Pin(q,p)$ | $Pin(q) \times Pin(q) / \{(1,1), (-1,-1)\}$ |

The following table lists specifically the maximal compact subgroups of the "$E$-series" of Lie groups culminating in the  [[exceptional Lie groups]] $E_n$.

| $n$ | [[real form]] $E_{n(n)}$ | maximal compact subgroup $H_n$ | $dim(E_{n(n)})$ | $dim(E_{n(n)}/H_n ) $ |
|---|---|---|---|---|
| 2 | $SL(2, \mathbb{R}) \times \mathbb{R}$ | $SO(2)$ | 4 | 3 |
| 3 | $SL(3,\mathbb{R}) \times SL(2,\mathbb{R})$ | $SO(3) \times SO(2)$ | 11 | 7 |
| 4 | $SL(5, \mathbb{R})$ | $SO(5)$ | 24 | 14 |
| 5 | $Spin(5,5)$  | $(Sp(2) \times Sp(2))/\mathbb{Z}_2$ | 45 | 25|
| 6 | [[E6|E6(6)]] | $Sp(4)/\mathbb{Z}_2$ | 78 | 42 |
| 7 | [[E7|E7(7)]] | $SU(8)/\mathbb{Z}_2$ | 133 | 70 |
| 8 | [[E8|E8(8)]] | $Spin(16)/\mathbb{Z}_2$ | 248 | 128 |

### Counterexamples

A maximal compact subgroup may not exist at all without the almost connectedness assumption. An example is the [[Prüfer group]] $\mathbb{Z}[1/p]/\mathbb{Z}$ endowed with the discrete ($0$-dimensional) smooth structure. This is a union of an increasing sequence of finite cyclic groups, each obviously compact.

## Related concepts

* [[closed subgroup]]

* [[locally compact topological group]]

* [[compact Lie group]]

* [[maximal subgroup]]

* [[maximal torus]]

* [[reduction and lift of structure groups]]

## References

Textbooks accounts:

* M. Stroppel, _Locally compact groups_, European Math. Soc., (2006)
  {#Stroppel}

* Karl Hofmann, Sidney Morris, _The Lie theory of connected pro-Lie groups_, Tracts in Mathematics 2, European Mathematical Society,  (2000)
 {#HofmannMorris}

See also

* Wikipedia, _[Maximal compact subgroup](http://en.wikipedia.org/wiki/Maximal_compact_subgroup)_

Original articles:

* {#Malcev} A. Malcev, _On the theory of the Lie groups in the large_, Mat.Sbornik N.S. vol. 16 (1945) pp. 163-189

* {#Iwasawa} K. Iwasawa, _On some types of topological groups_, Ann. of Math. vol.50 (1949) pp. 507-558.

* M. Peyrovian, _Maximal compact normal subgroups_, Proceedings of the American Mathematical Society, Vol. 99, No. 2, (1987) ([jstor:pss/2046647](http://www.jstor.org/pss/2046647))

* [[Karl Heinrich Hofmann]], [[Christian Terp]], *Compact Subgroups of Lie Groups and Locally Compact Groups*, Proceedings of the American Mathematical Society Vol. 120, No. 2 (Feb., 1994), pp. 623-634 ([jstor:2159906](https://www.jstor.org/stable/2159906), [doi:10.2307/2159906](https://doi.org/10.2307/2159906))

* {#Antonyan} Sergey A. Antonyan, _Characterizing maximal compact subgroups_ ([arXiv:1104.1820v1](http://arxiv.org/abs/1104.1820v1))

The maximal compact subgroups inside the (indefinite) rotation groups

* {#Conrad} [[Brian Conrad]], _Examples of maximal compact subgroups_ ([pdf](http://virtualmath1.stanford.edu/~conrad/210CPage/handouts/maxcompact.pdf))

[[!redirects maximal compact subgroups]]

[[!redirects compact subgroup]]
[[!redirects compact subgroups]]


maximal consistent formulae

# $\Lambda$-maximal consistent formulae
* table of contents
{: toc}

## Definition

Given a [[normal modal logic]], $\Lambda$,  a set, $\Gamma$, of formulae is said to be __$\Lambda$-consistent__ if $\neg(\Gamma\vdash_\Lambda \bot)$, i.e., $\bot$ is not [[modal logic|deducible]] from $\Gamma$.

A set, $\Gamma$, of formulae is said to be __$\Lambda$-maximal__ if it is consistent and, for any $\phi \in \mathcal{L}_\omega(n)$ either $\phi\in \Gamma$ or $\neg\phi\in \Gamma$.

+-- {.un_remark #important}
###### Important

If $\Gamma$ is a $\Lambda$-maximal set of formulae, then within the [[Lindenbaum-Tarski algebra]], $\mathfrak{A}^\Lambda_\omega$, the set $x_\Lambda = \{{\|\phi\|} \mid \phi \in \Lambda\}$ is an [[ultrafilter]].
=--

## Canonical frame

Let $S^\Lambda_\omega = \{\Gamma \mid \Gamma \in \Lambda maximal\}$,  then $\Gamma\leftrightarrow x_\Gamma$ is a bijection between $S^\Lambda_\omega$ and the set of [[ultrafilters]] of $\mathfrak{A}^\Lambda_\omega$.

This set forms the set of states / worlds for the **canonical frame** of $\Lambda$.  The relations are given by

$$R_i \Gamma\Delta \;iff\; \Diamond_i\Delta \subseteq \Gamma.$$

[[!redirects maximal consistent]]
[[!redirects maximal consistent formula]]
[[!redirects maximal consistent formulas]]
[[!redirects maximal consistent formulae]]


maximal element

# Maximal elements
* table of contents
{: toc}

## Idea

An element of a [[poset]] (or [[proset]]) is __maximal__ if no other (inequivalent) element is greater.  A [[top element|maximum]] must be maximal, and a maximal element of a [[toset]] must be a maximum.  However, it's easy to find posets with maximal elements that aren\'t maxima, or even with a unique maximal element that isn\'t a maximum.  The existence of a maximal element is often given by [[Zorn's lemma]].

## Definition

Let $P$ be a [[preordered set]] and $x$ an element of $P$.  Then $x$ is __maximal__ in $P$ if, whenever $x \leq y$ in $P$, we have $y \leq x$.  Dually, $x$ is __minimal__ in $P$ if, whenever $y \leq x$ in $P$, we have $x \leq y$.

## Properties

### Elementary properties

If $P$ has a [[top element]], then this is the unique (up to equivalence) maximal element of $P$.

Suppose that $P$ is [[totally ordered set|totally ordered]].  Then a maximal element of $P$ is the same as a [[top element]] of $P$.

Suppose that $P$ is [[finite set|finite]] and has a unique maximal element $x$.  Then $x$ is a top element of $P$.

### Deep properties

According to [[Zorn's Lemma]], if every [[totally ordered set|totally ordered]] [[subset]] of $P$ has an [[upper bound]] in $P$, then $P$ has a maximal element.

## Examples

Let $P$ be $\{a,b,c\}$ with $a \leq b$, $a \leq c$, and no other nontrivial ordering.  Then $b$ and $c$ are both maximal in $P$ (but of course not tops).

Let $P$ be the [[disjoint union]] of $\mathbb{N}$ (the poset of [[natural numbers]]) and a [[singleton]] \{a\}.  Then $a$ is the unique maximal element of $P$ but still not a top.

[[!redirects maximal element]]
[[!redirects maximal elements]]
[[!redirects minimal element]]
[[!redirects minimal elements]]


maximal geodesic

Let $M$ be a [[differentiable manifold]], let $\Del$ be an [[affine connection]] on $M$, and let $p$ be a [[point]] in $M$.  Given a [[tangent vector]] $x$ at $p$, there is are several [[geodesics]] $\gamma$ on $M$ tangent to $x$ at $p$.  Note that $\gamma$ is formally a [[differentiable map]] to $M$ from some [[interval]] ${]a,b[}$, in particular a [[partial function|partial]] differentiable map to $M$ from the [[real line]] $\mathbb{R}$.  (Here, $a$ is an [[upper real number]] so may be $-\infty$; and $b$ is a [[lower real number]] so may be $\infty$.)

If $\gamma\colon {]a,b[} \to M$ and $\gamma'\colon {]a',b'[} \to M$ are geodesics (on $M$ tangent to $x$ at $p$), then they agree on their common domain ${]max(a,a'),min(b,b')[}$.  Accordingly, If ${]a,b[} \subseteq {]a',b'[}$, then there is at most one extension of $\gamma$ to ${]a',b'[}$ that remains a geodesic.  If $M$ is [[complete manifold|complete]] (and perhaps in any case), then $\gamma$ may be extended (uniquely) to all of $\mathbb{R} = {]-\infty,\infty[}$.  Regardless, there is a unique *[[maximal partial function|maximal]]* geodesic (on $M$ tangent to $x$ at $p$).  This is *the* __maximal geodesic on $M$ tangent to $x$ at $p$__.


maximal ideal

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+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _maximal ideal_ (in say a [[commutative ring]] $R$) is an [[ideal]] $M$ which is maximal among [[proper ideals]]. (This is a second-order definition, as it quantifies over subsets of $R$.)

Equivalently, an ideal $M \subseteq R$ is maximal if the [[quotient]] ring $R/M$ is a [[field]]. This suggests a [[first-order logic|first-order]] definition: an ideal $M$ is maximal if $\forall x \in R. \neg (x \in M) \Rightarrow \exists y \in R. x y = 1$.

## Properties

### General

+-- {: .num_prop #EveryProperIdealisContainedInAMaximalOne}
###### Proposition
**(every proper ideal is contained in a maximal one)**

Assuming the [[axiom of choice]] then:

Let $R$ be a [[commutative ring]] and let $I \subset R$ be a proper ideal. Then $R$ contains a maximal ideal $\mathfrak{m}$ containing $I$, i.e. $I \subset \mathfrak{m}$.

=--

(See also at _[[prime ideal theorem]]_.)

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

Write $PropIdl(R)_{\subset}$ for the set of proper ideals of $R$, [[partially ordered set|partially ordered]] by inclusion. We claim that every chain in $PropIdl(R)_{\subset}$ has an upper bound ([def.](#Zorn's+lemma#ChainAndUpperBound)). This then implies the statement by [[Zorn's lemma]] (equivalent to the [[axiom of choice]]).

To show the claim, assume that $\mathcal{C} \subset PropIdl(R)_{\subset}$ is a chain. We have to produce an $I \in PropIdl(R)$ such that for all $c \in C$ then $c \subset I$.

We claim that such $I$ is provided by the union:

$$
  I \coloneqq  \underset{J \in \mathcal{C}}{\cup} J
  \,.
$$

It is clear that if this is indeed a proper ideal, then it is an upper bound of the chain.

To see first of all that this $I$ is an ideal, consider $x_1, x_2 \in I$. There are thus $J_1, J_2 \in \mathcal{C}$ with $x_1 \in J_1$ and $x_2 \in J_2$. Since a chain is [[totally ordered set|total order]] by definition, either $J_1 \subset J_2$ or $J_2 \subset J_1$. We may assume the former without restriction, otherwise rename $1 \leftrightarrow 2$. Therefore now $x_1, x_2 \in J_2$ and so we may add them there and find that $x_1 + x_2 \in J_2 \subset I$. Similarly if $r \in R$ then $r x_i \in J_2 \subset I$.

Finally to see that this idea $I$ is indeed proper. But since all the $J_i$ are proper, neither of them contains $1 \in R$, and hence $I$ does not contain $1 \in R$.

=--

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

In [[classical mathematics]] then:

Every maximal ideal is a [[prime ideal]].

=--

### Relation to points in the spectrum

Assuming [[axiom of choice|AC]] and [[excluded middle|EM]], then

Maximal ideals in the [[spectrum of a commutative ring]] $Spec(R)$ correspond precisely to the [[closed points]] in the [[Zariski topology]] on $Spec(R)$ ([this prop.](Zariski+topology#MaximalIdealsAreClosedPoints)).

Closed points are at the heart of the definition of _[[schemes]]_. A scheme $X$ is a sheaf with respect to the [[Zariski topology]] that admits a covering by open embeddings of affine schemes, where "covering" means that every *closed* point $p: Spec(F) \to X$ ($F$ a *field*) factors through one of the embeddings.

## Related concepts

* [[minimal ideal]]

* [[residue field]]

## References

* Mathematics StackExchange, _[Meaning of closed points of a scheme](http://math.stackexchange.com/questions/942/meaning-of-closed-points-of-a-scheme)_

[[!redirects maximal ideals]]


maximal ideal theorem

A __maximal ideal theorem__ is a theorem stating that every [[proper ideal]] is contained in some [[maximal ideal]].  A maximal ideal theorem is typically equivalent to the [[axiom of choice]] (AC), or a slightly weaker version such as the [[ultrafilter principle]] (UF).

We say 'a' maximal ideal theorem (MIT) instead of 'the' maximal ideal theorem, since we have not said what the ideals are in.  There are several examples:

*  The MIT for [[rings]] is equivalent to AC.

*  The MIT for [[Heyting algebras]] is equivalent to AC.

*  The MIT for [[Boolean algebras]] is equivalent to UF.

*  The MIT for [[rigs]], which subsumes all of the above, is equivalent to AC.

One typically proves a maximal ideal theorem with [[Zorn's Lemma]], unless one is specifically trying to use something weaker.

Compare the [[prime ideal theorem]].


maximal partial function

A __maximal partial function__ is a [[partial function]], say from $X$ to $Y$, which is [[maximal element|maximal]] in the poset of partial functions from $X$ to $Y$.  Explicitly, $f\colon X \nrightarrow Y$ is maximal if, given any $g\colon X \nrightarrow Y$, if $\dom f \subseteq \dom g$ and $f(a) = g(a)$ whenever $a \in \dom f$, then $\dom f = \dom g$ (and so $f = g$).

All [[total functions]] are maximal, as are all partial functions whose [[codomain]] $Y$ is the [[empty set]].  Assuming the [[law of excluded middle]], these are the only examples.  Outside of the context of [[constructive mathematics]], therefore, (and usually in constructive math too) we are concerned with maximal partial functions subject to some restrictions, such as the __maximal [[partial sections]]__ of some given function from $Y$ to $X$, the maximal [[continuous function|continuous]] partial functions given some [[topological structures]] on $X$ and $Y$, the __maximal [[local sections]]__ of some given continuous function from $Y$ to $X$ given some topological structures on $X$ and $Y$, etc.  Then we are not working in the poset of *all* partial functions from $X$ to $Y$ but in a given [[subposet]].

More abstractly, we may also consider the __maximal [[partial morphisms]]__ in a general [[category]].  This includes, for example, the maximal continuous partial functions between two topological spaces.

[[!redirects maximal partial function]]
[[!redirects maximal partial functions]]
[[!redirects maximal partial mapping]]
[[!redirects maximal partial mappings]]
[[!redirects maximal partial map]]
[[!redirects maximal partial maps]]

[[!redirects maximal partial section]]
[[!redirects maximal partial sections]]
[[!redirects maximal local section]]
[[!redirects maximal local sections]]

[[!redirects maximal partial morphism]]
[[!redirects maximal partial morphisms]]


maximal sieve

A [[sieve]] over an object $c$ of category $C$ is the __maximal sieve__ over $c$ if it contains all morphisms with [[target]] $c$. In other words, it agrees with the class of all objects of the [[slice category]] $C/c$.  One of the axioms of [[Grothendieck topology|Grothendieck topologies]] says that any maximal sieve (that is the maximal sieve for any object in $C$) is a [[covering sieve]].

A maximal sieve is any sieve generated by an [[identity morphism]] in $C$ (recall that a sieve generated by a family of morphisms $\{g_i:c_i\to c\}_{i\in I}$ is the class of all morphisms of the form $g_i\circ h$ where $h:e\to c_i$ is a morphism in $C$).  If a sieve over $c$ is considered as a subpresheaf of the representable presheaf $Hom(-,c)$, then a sieve is maximal iff it is $Hom(-c)$.


maximal spectrum

#Contents#
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{:toc}

## Definition

Given a [[ring]], or a $k$-[[associative algebra|algebra]] (unital or not) $A$, its __maximal spectrum__ $Spec_m A$ is the set of its [[maximal ideals]].

## Properties

If $k$ is a [[field]], and $R$ is a finitely generated [[noetherian ring|noetherian]] commutative unital $k$-[[associative algebra|algebra]] without [[nilpotent elements]], then $Spec_m A$ equipped with the [[Zariski topology]] is a [[noetherian topological space]]; the [[algebraic variety|varieties]] in the classical sense (cf. chapter 1 of Hartshorne) are exactly the spectra of such $k$-algebras. A more appropriate spectrum for general commutative unital rings is the [[prime spectrum]].

## Related concepts

* In [[functional analysis]], there is a slight variant of this notion, defined using (automatically continuous) characters, the [[Gel'fand spectrum]] of a commutative [[Banach algebra]], where however the [[topology]] is much richer, indeed [[compactum|compact Hausdorff]] ([[locally compact space|locally compact Hausdorff]], in the non-unital case).

* In [[analytic geometry]] one also uses [[analytic spectra]].

[[!redirects maximal spectrum]]
[[!redirects maximal spectra]]


maximal subgroup

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## Idea

A _maximal subgroup_ of a given [[group]] $G$ is a [[subgroup]] which is not all of $G$ and not contained in any other [[subgroup]] of $G$.

Hence a maximal subgroup is a [[maximal element]] of the [[lattice of subgroups]] after removing the trivial subgroup $G \subset G$ itself. More concisely it is a [[atom|coatom]] of $G$'s subgroup lattice.

## Related concepts

* [[maximal compact subgroup]]

* [[maximal torus]]

## References

See also

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

[[!redirects maximal subgroups]]


maximal torus

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## Definition

Let $G$ be a [[compact topological group|compact]] [[Lie group]].

A **[[torus]]** in $G$ is an [[abelian group|abelian]] [[subgroup]] $T \hookrightarrow G$ which is [[connected topological space|connected]] (and [[compact topological group|compact]]). Note that any compact connected abelian Lie group must be a torus, hence the name.

A **maximal torus** is a subgroup which is maximal with these properties.

The [[Lie algebra]] of a maximal torus of $G$ is a [[Cartan subalgebra]] of the Lie algebra of $G$.

## Properties
 {#Properties}

Suppose throughout that the [[compact Lie group]] $G$ is [[connected topological space|connected]].

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

Given a choice of maximal torus $T \hookrightarrow G$, then each element $g \in G$ is [[adjoint action|conjugate]] to an element of its maximal torus, i.e. there exists $h_g \in G$ such that $h_g g h_g^{-1} \in T \hookrightarrow G$.

=--

(e.g. [Johansen, theorem 2.7.3](#Johansen))

+-- {: .num_remark}
###### Remark

For the [[unitary group]] $U(n)$, by example \ref{InUnitaryGroup},  prop. \ref{CartanTheorem} is the [[spectral theorem]] that unitary matrices may be diagonalized.

=--

+-- {: .num_cor}
###### Corollary

Any two choices of maximal tori are conjugate.

=--

(e.g. [Johansen, corollary 2.7.5](#Johansen))

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

Any two elements of a maximal torus $T\hookrightarrow G$ are conjugate in $G$ precisely if they are conjugate via the [[Weyl group]].

=--

(e.g. [Johansen, prop 2.7.13](#Johansen))

+-- {: .num_cor}
###### Corollary

The [[conjugacy classes]] of $G$ are in bijection to $T/W(G,T)$.

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+-- {: .num_prop}
###### Proposition
Under the assumptions on $G$, the [[exponential map]] $\exp: \mathfrak{g} \to G$ is [[surjection|surjective]].
=--

+-- {: .proof}
###### Proof
The [[exponential map]] $\mathbb{R} \to S^1$ is surjective; similarly, if $\mathfrak{h}$ is a [[Cartan subalgebra]] of $\mathfrak{g}$, then the exponential map $\mathfrak{g} \to G$ maps $\mathfrak{h}$ homomorphically onto a maximal torus. Also the exponential map $\mathfrak{g} \to G$ preserves conjugation by any element of $G$, i.e., $\exp(g x g^{-1}) = g\exp(x)g^{-1}$. Then surjectivity follows from Proposition \ref{CartanTheorem}.
=--

## Examples

+-- {: .num_example #InUnitaryGroup}
###### Example

The maximal torus of a [[unitary group]] $U(n)$ is the subgroup $U(1)^n\hookrightarrow U(n)$ of diagonal matrices with unitary entries.

=--

+-- {: .num_example}
###### Example

The maximal torus of the [[special unitary group]] $SU(n+1)$ is again $U(1)^n$ (one dimension lower) included as the subgroup of diagonal matrices whose first $n$ diagonal entries are arbitrary elements of $U(1)$ and whose last diagonal entry is the inverse of their product.

=--

+-- {: .num_example}
###### Example

The maximal torus of the [[special orthogonal group]] $SO(2n)$ and $SO(2n+1)$ is $U(1)^n \simeq SO(2)^n$ included block-diagonal (with, in the odd-dimensional case, the remaining block entry being the unit).

=--

## Related concepts

* [[maximal subgroup]]

* [[maximal compact subgroup]]

* [[Cartan subalgebra]]

* [[weight (in representation theory)]], [[root (in representation theory)]]

* [[Weyl group]]

* [[orbit method]]

## References

e.g.

* {#Johansen} Troels Roussauc Johansen, section 2.6 of _Character Theory for Finite Groups and Compact Lie Groups_ [pdf](http://www.math.upb.de/~johansen/character-theory.pdf)

[[!redirects maximal tori]]


maximally mixed quantum state

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## Idea

In [[quantum information theory]], by the *maximally mixed state* of a [[quantum systems]] represented by a [[finite-dimensional Hilbert space]] $\mathscr{H}$ one means the [[mixed state]] whose [[density matrix]] is a multiple of the [[identity matrix]] on $\mathscr{H}$.

Hence if $\big(\left\vert w \right\rangle\big)_{w \colon W}$ is any [[orthonormal linear basis]] for $\mathscr{H}$, then the maximally mixed state is (in [[bra-ket notation]]) given by

$$
  \frac{1}{dim(\mathscr{H})}
  \,
  \underset{w}{\sum}
  \left\vert w \right\rangle
  \left\langle w \right\vert
  \;\;\colon\;\;
  \mathscr{H} \otimes \mathscr{H}^\ast
  \,,
$$

representing the uniform [[probability distribution]] on $W$.

## Properties

### Relation to quantum channels

A [[quantum channel]] which preserves the maximally mixed state is called a *[[unital quantum channel]]*.

A [[quantum channel]] with an [environmental representation](quantum+channel#QuantumChannelsAndDecoherence) given by coupling to a maximally mixed state of the environment is called a *[[unistochastic quantum channel]]* or (ambiguously) *noisy quantum operation*. Such operation on single [[qbits]] constitute the "[[DQC1]]-model" of [[quantum computation]].

## Related concepts

* [[open quantum systems]]

* [[quantum noise]]

## References

See references at *[[unistochastic quantum channel]]*.

[[!redirects maximally mixed quantum states]]

[[!redirects maximally mixed state]]
[[!redirects maximally mixed states]]


Maximilian Kreuzer

* [webpage](http://hep.itp.tuwien.ac.at/~kreuzer/)

## related $n$Lab entries

* [[discrete torsion]]

category: people


Maximilian Rupprecht

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

## Selected writings

On the [[Nicolai map]]:

* [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Universal form of the Nicolai map*, Phys. Rev. D **104** 021701 (2021) [[arXiv:2104.00012](https://arxiv.org/abs/2104.00012), [doi:10.1103/PhysRevD.104.L021701](https://doi.org/10.1103/PhysRevD.104.L021701)]

* [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Is the Nicolai map unique?*,  J. High Energ. Phys. **2022** 139 (2022) [[arXiv:2207.09471](https://arxiv.org/abs/2207.09471), doi:10.1007/JHEP09(2022)139]

On the Nicolai map for the [[CP^N sigma-model|$\mathbb{C}P^n$ sigma-model]]:

* [[Lorenzo Casarin]], [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Nicolai maps with four-fermion interactions* [[arXiv:2310.19946](https://arxiv.org/abs/2310.19946)]

category: people


Maximilian Schlosshauer

* [personal page](http://faculty.up.edu/schlosshauer/)

* [InSpire page](https://inspirehep.net/authors/2053956)

## Selected writings

On [[quantum decoherence]]:

* [[Maximilian Schlosshauer]], *Decoherence, the measurement problem, and interpretations of quantum mechanics*,  	Rev. Mod. Phys. **76** (2004) 1267-1305 [[arXiv:quant-ph/0312059](https://arxiv.org/abs/quant-ph/0312059), [doi:10.1103/RevModPhys.76.1267](https://doi.org/10.1103/RevModPhys.76.1267)]

  > (with regards to the [[measurement problem]] and [[interpretations of quantum mechanics]])

* [[Maximilian Schlosshauer]], *Decoherence and the Quantum-To-Classical Transition*, The Frontiers Collection, Springer (2007) [[doi:10.1007/978-3-540-35775-9](https://doi.org/10.1007/978-3-540-35775-9)]

* [[Maximilian Schlosshauer]], *Quantum Decoherence*, Phys. Rep. **831** (2019) 1-57 [[arXiv;1911.06282](https://arxiv.org/abs/1911.06282), [doi:10.1016/j.physrep.2019.10.001](https://doi.org/10.1016/j.physrep.2019.10.001)]

On [[quantum Bayesianism]]/[[QBism]]

* {#FuchsSchlosshauerStacey14} [[Christopher A. Fuchs]], [[Maximilian Schlosshauer]] (foreword), [[Blake C. Stacey]] (ed.) *My Struggles with the Block Universe* [[arXiv:1405.2390](https://arxiv.org/abs/1405.2390)]

## Related entries

* [[density matrix]]

category: people


Maximilien Péroux

* [website](http://homepages.math.uic.edu/~mholmb2/)

category: people


Maxwell's equations

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## Idea

In the context of [[electromagnetism]], _Maxwell's equations_ are the [[equations of motion]] for the electromagnetic [[field strength]] [[electric current]] and [[magnetic current]].

## Three dimensional formulation

$E$ is here the (vector of) strength of electric field and $B$
the strength of magnetic field; $Q$ is the charge and $j_{el}$ the density of the electrical current; $\epsilon_0$, $c$, $\mu_0$ are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: $\mu_0 \epsilon_0  = 1/c^2$).

### Integral formulation in vacuum

__[[Gauss' law]] for electric fields__
$$
\textstyle{\int}_{\partial V} E\cdot d A =  \frac{Q}{\epsilon_0}
$$
where $\partial V$ is a closed surface which is a boundary of a 3d domain $V$ (physicists say "volume") and $Q = \textstyle{\int}_V \rho d V$ the charge in the domain $V$; $\cdot$ denotes the scalar (dot) product. Surface element $d A$ is $\vec{n} d |A|$, i.e. it is the scalar surface measure times the unit vector of normal outwards.

__No magnetic monopoles__ (Gauss' law for magnetic fields)
$$
\textstyle{\int}_\Sigma B\cdot d A =  0
$$
where $\Sigma$ is any closed surface.

__Faraday's law of induction__
$$
\textstyle{\oint}_{\partial \Sigma} E\cdot d s = - \frac{d}{d t} \textstyle{\int}_\Sigma B\cdot d A
$$

The line element $d s$ is the differential (or 1-d measure on the boundary) of the length times the unit vector in counter-circle direction (or parametrize the curve with $s$ being a vector in 3d space, express magnetic field in the same parameter and calculate the
integral as a function of parameter: $\cdot$ is a scalar ("dot") product).

__Ampère-Maxwell law__ (or generalized Ampère's law; Maxwell added the second term involving derivative of the flux of electric field
to the Ampère's law which described the magnetic field due electric current).
$$
\textstyle{\oint}_{\partial \Sigma} B\cdot d s = \mu_0 I + \mu_0 \epsilon_0 \frac{d}{d t} \textstyle{\int}_\Sigma E\cdot d A
$$
where $\Sigma$ is a surface and $\partial \Sigma$ its boundary; $I$ is the total current through $\Sigma$ (integral of the component of $j_{el}$ normal to the surface).

### Differential equations
 {#DifferentialEquations}

Here we put units with $c = 1$. By $\rho$ we denote the density of the charge.

In [[pre-metric electromagnetism|pregeometric form]], Maxwell's equations are [[differential equations]] for *four* fields ([[vector fields]]) called

* *electric field* $\vec E$

* *magnetic flux density* $\vec B$

* *magnetic field* $\vec H$

* *displacement field* $\vec D$ (or similar)

and state:

* **([[magnetic Gauss law]])**

  $div B = 0$

  ("there is no source for magnetic [[flux]], hence no [[magnetic monopoles]]")

* **Faraday's law**:

  $\frac{d}{d t} B + rot E = 0$

* **([[electric Gauss law]])**:

  $div D = \rho$

  ("the source of electric [[flux]] is electric current")

* **generalized Ampère's law**

  $- \frac{d}{d t} D + rot H = j_{el}$

In order to complete this [[pregeometric electromagnetism|pregeometric]] form to the actual [[equations of motion]] of the [[electromagnetic field]], these four fields are to be subjected to a constraint called the *[[constitutive equation]]* which expresses $(\vec D, \vec H)$ as a [[function]] of $(\vec E, \vec B)$.

In [[vacuum]] and in the absence of [[background field|background]] [[gravity]], this [[constitutive relation]]:

* equates the displacement field $\vec D$ with the electric field $\vec E$ times a constant $\epsilon_0$

  called the "*permitivity of the vacuum*",

* equates the magnetic flux $\vec B$ with the magnetic field $\vec H$ times a constant $\mu_0$

  called the "*permeability of the vacuum*".

But for [[electromagnetic fields]] inside [[dielectric media]] other [[constitutive relations]] appear. For small field strengths these are typically [[linear functions]] $C$ (e.g. [de Lange & Raab 2006 (19)](#deLangeRaab06))

$$
  (\vec D, \vec H)
    \;\overset{!}{=}\;
  C\big((\vec E, \vec B)\big)
  \,,
$$

but in general the constitutive [[relation]] can be a non-linear or even be a "[[multi-valued function]]" (namely when there are hysteresis effects in the [[dielectric medium]]).

Similarly (interestingly), in the presence of [[background field|background]] [[gravity]] (such as for [[electromagnetic fields]] in and around a star) there is a linear such relation depending on the [[pseudo-Riemannian metric]]. This is most transparently expressed in terms of the [[Hodge star operator]] acting on the electromagnetic fields re-packaged as a [[Faraday tensor]] [[differential 2-form]] (see [below](#InTermsOfFaradayTensor)).

## In terms of Faraday tensor $F$
 {#InTermsOfFaradayTensor}

This is adapted from _[electromagnetic field -- Maxwell's equations](electromagnetic+field#MaxwellEquations)_, for more see the references [below](#ReferencesForMaxwellEquationsViaDifferentialForms).

In modern language, the insight of ([Maxwell, 1865](#Maxwell1865)) is that locally, when physical [[spacetime]] is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional [[Minkowski space]] $U \subset (\mathbb{R}^4, g = diag(-1,1,1,1))$, the electric field $\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right]$ and magnetic field $\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]$ combine into a differential [[differential form|2-form]]

$$
  \begin{aligned}
    F & \coloneqq E \wedge d t + B
      \\
      & \coloneqq
        E_1 d x^1 \wedge d t +
        E_2 d x^2 \wedge d t +
        E_3 d x^3 \wedge d t
     \\
& \phantom{\coloneqq} +
        B_1 d x^2 \wedge d x^3 +
        B_2 d x^3 \wedge d x^1 +
        B_3 d x^1 \wedge d x^2
  \end{aligned}
$$

in $\Omega^2(U)$ and the [[electric charge]] density and current density combine to a differential 3-form

$$
\begin{aligned}
    j_{el} & \coloneqq j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\
& \coloneqq    j_1 d x^2 \wedge d x^3 \wedge d t +
        j_2 d x^3 \wedge d x^1 \wedge d t +
        j_3 d x^1 \wedge d x^2 \wedge d t
        -
        \rho \; d x^1 \wedge d x^2 \wedge d x^3
\end{aligned}
$$

in $\Omega^3(U)$ such that the following two equations of differential forms are satisfied

$$
  \begin{aligned}
    d F = 0
    \\
    d \star F  = j_{el}
  \end{aligned}
  \,,
$$

where $d$ is the de Rham differential operator and $\star$ the [[Hodge star]] operator. If we decompose $\star F$ into its components as before as

$$
\begin{aligned}
\star F
&= -D + H\wedge dt \\
&= -D_1 \; d x^2 \wedge d x^3
   -D_2 \; d x^3 \wedge d x^1
   -D_3 \; d x^1 \wedge d x^2
     \\
& \phantom{=} +
     H_1 \; d x^1 \wedge d t
     +
     H_2 \; d x^2 \wedge d t
     +
     H_3 \; d x^3 \wedge d t
  \end{aligned}
$$

then in terms of these components the Maxwell equations  read as follows:

$d F = 0$ gives the [[magnetic Gauss law]] and Faraday's law

$d \star F = j_{el}$ gives [[Gauss's law]] and Ampère-Maxwell law

## In terms of the Faraday bivector $F$

In the [[geometric algebra]] formalism of electromagnetism, one works in the 4-dimensional [[Clifford algebra]] $\mathrm{Cl}^{1, 3}(\mathbb{R})$ representing [[spacetime]], where the orthonormal basis vectors $\{\gamma_i\}$ have signature $(+, -, -, -)$ with $\gamma_0$ representing the time dimension and the other three basis vectors representing the spacial dimensions. The pseudoscalar of $\mathrm{Cl}^{1, 3}(\mathbb{R})$ is represented by the product of all the basis vectors
$$I = \prod_{i} \gamma_i = \gamma_0 \gamma_1 \gamma_2 \gamma_3$$
The basis [[bivectors]] of $\mathrm{Cl}^{1, 3}(\mathbb{R})$ come in two sets, the *timelike bivectors* $\sigma_1 = \gamma_1 \gamma_0$, $\sigma_2 = \gamma_2 \gamma_0$, $\sigma_3 = \gamma_3 \gamma_0$, and the *spacelike bivectors* $I \sigma_1 = -\gamma_2 \gamma_3$, $I \sigma_2 = \gamma_1 \gamma_3$, $I \sigma_3 = \gamma_1 \gamma_2$. The bivector subalgebra of $\mathrm{Cl}^{1, 3}(\mathbb{R})$ is equivalent to the three-dimensional [[Clifford algebra]] $\mathrm{Cl}^{3, 0}(\mathbb{R})$ corresponding to the relative space in the rest frame defined by $\gamma_0$, where the basis timelike bivectors correspond to the basis relative vectors and the basis spacelike bivectors correspond to the basis relative bivectors of the rest frame.

Let $x$ be a vector in $\mathrm{Cl}^{1, 3}(\mathbb{R})$. Then we define the coordinates of $x$ relative to the basis $\{\gamma_i\}$ to be $x^i = \gamma_i \cdot x$. $x^0$ is also denoted as $t$ since it represents the time coordinate. The spacetime vector derivative is defined as the operator

$$\nabla = \gamma_0 \frac{\partial}{\partial t} + \gamma_1 \frac{\partial}{\partial x^1} + \gamma_2 \frac{\partial}{\partial x^2} + \gamma_3 \frac{\partial}{\partial x^3}$$

The relative vector derivative is defined as the operator

$$\mathbf{\nabla} = \sigma_1 \frac{\partial}{\partial x^1} + \sigma_2 \frac{\partial}{\partial x^2} + \sigma_3 \frac{\partial}{\partial x^3}$$

Assuming the use of [[natural units]] where $c = \epsilon_0 = 1$ and ignoring the polarization and magnatization fields for the time being, Maxwell's equations are written as:

* electric Gauss's law:
$$\mathbf{\nabla} \cdot E = \rho$$

* Faraday's law:
$$\mathbf{\nabla} \wedge E = -\frac{\partial}{\partial t}\left(I B\right)$$

* magnetic Gauss's law:
$$\mathbf{\nabla} \cdot B = 0$$

* Ampère-Maxwell law:
$$\mathbf{\nabla} \wedge B = I\left(\mathbf{J} + \frac{\partial}{\partial t}\left(E\right)\right)$$

where $E$ and $B$ are the relative electric and magnetic relative fields, $\rho$ is the density of the charge, and $\mathbf{J}$ is the relative current of the charge.

The *Faraday bivector* is the bivector $F = E + I B$. By multiplying the last two equations by the pseudoscalar, one gets
$$\mathbf{\nabla} \cdot (I B) = 0$$
$$\mathbf{\nabla} \wedge (I B) = -\mathbf{J} - \frac{\partial}{\partial t}\left(E\right)$$

and by adding the four equations together, one gets the equation
$$\mathbf{\nabla} \cdot (E + I B) + \mathbf{\nabla} \wedge (E + I B) = \rho -\mathbf{J} -\frac{\partial}{\partial t}\left(E + I B\right)$$

which then becomes

$$\mathbf{\nabla} \cdot F + \mathbf{\nabla} \wedge F + \frac{\partial}{\partial t}\left(F\right) = \rho - \mathbf{J}$$

Given a timelike bivector $v$ and a general bivector $M$ in $\mathrm{Cl}^{1, 3}(\mathbb{R})$, $v M = v \cdot M + v \wedge M$. Thus, the above equation could be simplified even further to

$$\mathbf{\nabla} F + \frac{\partial}{\partial t}\left(F\right) = \rho - \mathbf{J}$$

or

$$\left(\mathbf{\nabla} + \frac{\partial}{\partial t}\right)\left(F\right) = \rho - \mathbf{J}$$

Now, the spacetime current is given by $J = \gamma_0 (\rho - \mathbf{J})$, which is a vector in spacetime. The spacetime vector derivative is related to the relative vector derivative by the following equation:

$$\nabla = \gamma_0 (\mathbf{\nabla} + \frac{\partial}{\partial t})$$

Thus, by left multiplying each side by $\gamma_0$, one gets

$$\gamma_0 \left(\mathbf{\nabla} + \frac{\partial}{\partial t}\right)\left(F\right) = \gamma_0 (\rho - \mathbf{J})$$

or

$$\nabla F = J$$

where $\nabla$ is the spacetime vector derivative, $F$ is the Faraday bivector, and $J$ is the spacetime current.

## Related concepts

* [[Hodge-Maxwell theorem]]

* [[Kirchhoff's laws]]

* [[Yang-Mills equations]]

## References

For more see the references at *[[electromagnetism]]*.

### General

Maxwell's equations originate in:

* {#Maxwell1865} [[James Clerk Maxwell]], *A Dynamical Theory of the Electromagnetic Field*, Philosophical Transactions of the Royal Society of London **155** (1865) 459-512 [[doi:10.1098/rstl.1865.0008](https://doi.org/10.1098/rstl.1865.0008), [jstor:108892](https://www.jstor.org/stable/108892), [Wikipedia entry](http://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field)]

* [[James Clerk Maxwell]], *A Treatise on Electricity and Magnetism*, Clarendon Press Series, Macmillan & Co. (1873),  Cambridge University Press (2010) [[ark:/13960/t9s17v886](https://archive.org/details/electricandmagne01maxwrich), [doi:10.1017/CBO9780511709333](https://doi.org/10.1017/CBO9780511709333), [pdf](https://www.aproged.pt/biblioteca/MaxwellI.pdf), [Wikipedia entry](https://en.wikipedia.org/wiki/A_Treatise_on_Electricity_and_Magnetism)]

Some history and reflection:

* A. C. T. Wu, [[Chen Ning Yang]], *Evolution of the concept of vector potential in the description of the fundamental interactions*,  International Journal of Modern Physics A **21** 16 (2006) 3235-3277 [[doi:10.1142/S0217751X06033143](https://doi.org/10.1142/S0217751X06033143)]

* {#Dyson07} [[Freeman Dyson]], _Why is Maxwell’s Theory so hard to understand?_, Proceedings of [The Second European Conference on Antennas and Propagation, EuCAP 2007](https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4446147) ([doi: 10.1049/ic.2007.1146](https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4446147))

* [[Chen Ning Yang]], *The conceptual origins of Maxwell’s equations and gauge theory*, Phyics Today **67** 11 (2014) [[doi:10.1063/PT.3.2585](https://doi.org/10.1063/PT.3.2585), [pdf](http://home.ustc.edu.cn/~lxsphys/2021-3-18/The%20conceptual%20origins%20of%20Maxwell%27s%20equations%20and%20gauge%20theory.pdf)]

For Maxwell's equations in the generality of [[dielectric media]], see the references there, such as:

* G. Russakoff, *A Derivation of the Macroscopic Maxwell Equations*, American Journal of Physics **38** (1970) 1188–1195 [[doi:10.1119/1.1976000](https://doi.org/10.1119/1.1976000)]

* {#deLangeRaab06} O. L. de Lange, R. E. Raab *Surprises in the multipole description of macroscopic electrodynamics*, American Journal of Physics **74** (2006) 301–312 [[doi:10.1119/1.2151213](https://doi.org/10.1119/1.2151213)]

On the Maxwell [[Green's function]] ([[propagator]]) and numerical solutions:

* Boris Lo, Victor Minden, Phillip Colella, *A real-space Green's function method for the numerical solution of Maxwell's equations*,  **11** 2 (2016) 143–170 [[doi:10.2140/camcos.2016.11.143](https://doi.org/10.2140/camcos.2016.11.143), [pdf](https://msp.org/camcos/2016/11-2/camcos-v11-n2-p01-s.pdf)]

[[!include electromagnetism in terms of differential forms -- references]]

### Via "geometric algebra: (Clifford algebra)

Formulation of Maxwell's equations via "[[geometric algebra]]":

* {#DoranLasenby03} [[Chris Doran]], [[Anthony Lasenby]], _Geometric algebra for physicists_, Cambridge University Press (2003) ([pdf](http://catdir.loc.gov/catdir/samples/cam033/2002035182.pdf))

* {#Arthur11} John W. Arthur, *Understanding Geometric Algebra for Electromagnetic Theory*, John Wiley & Sons Inc. (2011). (ISBN:978-0470941638, [doi:10.1002/9781118078549](https://doi.org/10.1002/9781118078549))

[[!redirects Maxwell equation]]
[[!redirects Maxwell's equation]]

[[!redirects Maxwell equations]]
[[!redirects Maxwell's equations]]

[[!redirects permittivity of the vacuum]]


May > history

< [[May]]
[[!redirects May -- history]]


May recognition theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The original _May recognition theorem_ ([May 72](#May72)) characterizes the [[homotopy types]] of $n$-fold [[iterated loop spaces]] as "grouplike" [[En-algebras]] in the [[classical model structure on topological spaces|classical homotopy theory of spaces]] ([[∞Grpd]]), hence as [[∞-groups]] with $(n-1)$-fold abelian structure ([[braided ∞-groups]] for $n = 2$, [[sylleptic ∞-group|sylleptic ∞-groups]] for $n=3$, etc.)

The statement generalizes from [[∞Grpd]] to any [[(∞,1)-topos]] ([Lurie 09b, 1.3](#Lurie09b), [Lurie 17, 5.2.6](#Lurie17)).

## Statement

+-- {: .num_prop #RecognitionForGroupsInAnInfinityTopos}
###### Proposition
**(Recognition of [[group objects in an (∞,1)-category|group objects]] in an [[(∞,1)-topos]])**

Let $\mathbf{H}$ be an [[(∞,1)-topos]]  Then the operation $\Omega$ of forming [[loop space objects]] constitutes an [[equivalence of (∞,1)-categories]]

$$
  Groups(\mathbf{H})
  \underoverset
    {\underset{\;\;\mathbf{B}\;\;}{\longrightarrow}}
    {\overset{\Omega}{\longleftarrow}}
    {\simeq}
  \mathbf{H}^{\ast/}_{\geq 1}
$$

between

1. [[∞-groups]] (i.e. [[groupoid object in an (infinity,1)-category|group objects]]) in $\mathbf{H}$

1. [[pointed object|pointed]]$\;$[[n-connected object of an (infinity,1)-topos|connected objects]] in $\mathbf{H}$.

The inverse equivalence is the [[delooping]] $\mathbf{B}$, see at _[[looping and delooping]]_.

=--

This is [Lurie 09a, Theorem 7.2.2.11](#Lurie09a).

More generally:

+-- {: .num_prop #RecognitionForAbelianGroupsInAnInfinityTopos}
###### Proposition
**(Recognition of abelian [[group objects in an (∞,1)-category|group objects]] in an [[(∞,1)-topos]])**

Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $n \in \mathbb{N}$, $n \geq 1$. Then the operation $\Omega^n$ of forming $n$-fold [[iterated loop space|iterated]] [[loop space objects]] constitutes an [[equivalence of (∞,1)-categories]]

$$
  Groups_{E_n}(\mathbf{H})
  \underoverset
    {\underset{\;\;\mathbf{B}^n\;\;}{\longrightarrow}}
    {\overset{\Omega^n}{\longleftarrow}}
    {\simeq}
  \mathbf{H}^{\ast/}_{\geq n}
$$

1. [[∞-groups]] (i.e. [[groupoid object in an (infinity,1)-category|group objects]]) in $\mathbf{H}$ with [[En-algebra]]-[[mathematical structure|structure]],

1. [[pointed object|pointed]] [[n-connected object of an (infinity,1)-topos| $(n-1)$-connected objects]] in $\mathbf{H}$.

=--

This is [Lurie 09b, Theorem 1.3.6](#Lurie09b), [Lurie 17, Theorem 6.2.6.15](#Lurie17).

## Related concepts

* [[looping and delooping]]

* [[Quillen equivalence between simplicial groups and reduced simplicial sets]]

* [[recognition of relative loop spaces]]

## References

* {#May72} [[Peter May]], _The geometry of iterated loop spaces_, Lecture Notes in Mathematics, Springer 1972 ([doi:10.1007/BFb0067491](https://link.springer.com/book/10.1007/BFb0067491), [pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))

* {#Lurie09a} [[Jacob Lurie]], _[[Higher Topos Theory]]_ (2009)

* {#Lurie09b} [[Jacob Lurie]], _Derived Algebraic Geometry VI: $E_k$ Algebras_ ([arXiv:0911.0018](https://arxiv.org/abs/0911.0018))

* {#Lurie17} [[Jacob Lurie]], _[[Higher Algebra]]_ (2017)