James construction

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### Context
#### Topology
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[[!include topology - contents]]
=--
#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[free construction|free]] [[topological monoid]] $F X$ on a [[topological space]] $X$ is canonically [[filtered topological spaces|filtered]] by the length of words.  Given instead a [[pointed topological space]] $(X,x)$, there is also a reduced version by taking $F X$ and identifying $x$ with the identity of $F X$. This latter [[filtered topological space]] is known as the _[[James construction]]_ $J(X,x)$ ([James 55](#James55)).

## Details

 The [[James construction]] $J(X,x)$ may be constructed [[homotopy theory|homotopy-theoretically]] ([Brunerie 13](#Brunerie13), [Brunerie 17](#Brunerie17)). Recall that, for $K$ a finite [[simplicial complex]], for $(X,A)$ a pair of spaces, its _polyhedral product_ $(X,A)^K$ is defined as the union $\bigcup_{\sigma\in S(K)}(X,A)^\sigma$ as a subspace of the Cartesian product $X^{V(K)}$. Here, for $\sigma\in S(K)$ a simplex of $K$, the subspace $(X,A)^\sigma$ consist of those $x\in X^{V(K)}$ such that, for each vertex $v$ in the complement of $\sigma$, the coordinate projection $\proj_v x$ lies in $A$. Equivalently, the polyhedral product $(X,A)^K$ can be considered as a homotopy colimit of these $(X,A)^\sigma$ over the poset $S(K)$ of simplexes $\sigma$ of $K$, where the maps are the respective inclusions.

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

For $X$ a space equipped with a basepoint $x$, define a filtered space $fil_\bullet$ as follows. Set $\fil_0$ as $\{x\}$.
For $k\ge 1$, require that the following square is homotopy pushout:
$$
  \array{
 &&&&
     (X,x)^{\partial \Delta[k-1]}
     &&&&
      \\
      &
      && inc \swarrow
       &
       & \searrow
      &&
     \\
&& X^k &&&& fil_{k-1}
\\
      &
      && {}_{p_k}\searrow
       &
       & \swarrow_{j_k}
      &&
     \\
&&&&
     fil_k
     &&&&
  }
$$
where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over $S(\partial \Delta[k-1])$ given by the maps
$$(X,x)^\sigma \xrightarrow{\sim} X^{\dim(\sigma)+1} \xrightarrow{p_{\dim(\sigma)+1}} fil_{\dim(\sigma)+1}$$
for each simplex $\sigma$ of the boundary simplicial complex $\partial \Delta[k-1]$ of the standard $(k-1)$-simplex. The homotopy colimit $fil_\infty$ of the sequence of maps $fil_0 \stackrel{j_1}{\to} fil_1 \stackrel{j_2}{\to} \ldots$ is called the **James construction** on $(X, x)$.
=--

+-- {: .num_prop #TypeOfJames}
###### Proposition
 For $(X,x)$ a pointed space, if $(X,x)$ is path-connected, then $fil_\infty \simeq \Omega\Sigma X$.
=--

## Properties

### Relation to configuration spaces

The [[James construction]] of $X$ is [[homotopy equivalence|homotopy equivalent]] to the [[configuration space (mathematics)|configuration space]] $C(\mathbb{R}^1, X)$ of points in the [[real line]] with "[[charges]]" taking values in $X$.

(e.g. [Bödigheimer 87, Example 9](#Boedigheimer87))

## See also

* [[James construction type]]

## References

The construction is due to

* {#James55} [[Ioan James]], _Reduced product spaces_, Annals of Mathematics, Second Series, 62: 170–197 (1955)

Review:

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lectures 2,3 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* [[Dylan Wilson]], _James construction_, 2017 ([pdf](https://web.archive.org/web/20170829072533/http://www.math.uchicago.edu/~dwilson/pretalbot2017/james-construction.pdf))

* Wikipedia, _[James reduced product](https://en.wikipedia.org/wiki/James_reduced_product)_

Discussion via [[configuration space (mathematics)|configuration spaces]] includes

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf))

Discussion via [[homotopy type theory]] includes the following

* {#Brunerie13} [[Guillaume Brunerie]] _The James Construction and $\pi_4(S^3)$_, talk at the Institute of Advanced Studies on March 27, 2013 ([recording](https://video.ias.edu/univalent/1213/0327-GuillaumeBrunerie))

* {#Brunerie17} [[Guillaume Brunerie]], _The James construction and $\pi_4(S^3)$ in homotopy type theory_ ([arXiv:1710.10307](https://arxiv.org/abs/1710.10307))

[[!redirects James constructions]]


James construction type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _James construction type_ is an [[axiom|axiomatization]] of the [[James construction]] in the context of [[homotopy type theory]].

## Definition

As a [[higher inductive type]], the James construction type $J A$ of a [[pointed type]] $(A, a)$ is given by the following constructors

* $\epsilon_{J A} : J A$
* $\alpha_{J A} : A \to J A \to J A$
* $\delta_{J A} : \prod_{x : J A} x = \alpha_{J A}(a)(x)$

In [[Coq]] pseudocode this becomes

    Inductive JamesConstruction (A : PointedType) : Type
      | epsilon : JamesConstruction A
      | alpha : A -> JamesConstruction A -> JamesConstruction A
      | delta : forall (x : JamesConstruction A) x = alpha point x

We can see that $J A$ is simply the [[free monoid]] on $A$. The higher inductive type is recursive which can make it difficult to study. This however can be remedied by defining a sequence of types $(J_n A)_{n: \mathbb{N}}$ together with maps $(i_n : J_n A \to J_{n+1} A)_{n:\mathbb{N}}$ such that the type $J_\infty A$ defined as the [[sequential colimit]] of $(J_n A)_{n:\mathbb{N}}$ is equivalent to $J A$.

## Properties ##

There is an equivalence of types $J A \simeq \Omega \Sigma A$ if $A$ is [[0-connected]].

## Related concepts

* [[higher inductive type]]

* [[pointed type]]

* [[free monoid]], [[list]]

* [[James construction]]

## References

* {#Brunerie16} [[Guillaume Brunerie]], _On the homotopy groups of spheres in homotopy type theory_ ([arxiv:1606.05916](https://arxiv.org/abs/1606.05916))

* {#LumsdaineShulman17} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], _Semantics of higher inductive types_ ([arXiv:1705.07088](https://arxiv.org/abs/1705.07088))

* {#Brunerie17} [[Guillaume Brunerie]], _The James construction and $\pi_4(S^3)$ in homotopy type theory_ ([arXiv:1710.10307](https://arxiv.org/abs/1710.10307))

[[!redirects James construction types]]


James Cranch

* [website](http://www.srcf.ucam.org/~jdc41/)

category: people


James Cruickshank

* [webpage](http://maths.nuigalway.ie/~jc/)

## Selected writings

On [[twisted cohomology]], [[equivariant cohomology]] and ([[stable cohomotopy|stable]]) [[twisted cohomotopy|twisted]]/[[equivariant cohomotopy|equivariant]] [[Cohomotopy]]:

* {#Cruickshank99} [[James Cruickshank]], _Twisted  Cobordism and its Relationship to Equivariant Homotopy Theory_, 1999 ([pdf](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf), [[Cruickshank99.pdf:file]])

  > (the [[equivariant Pontrjagin theorem]] for [[free action|free]] [[G-manifolds]])

* {#Cruickshank03} [[James Cruickshank]], _Twisted homotopy theory and the geometric equivariant 1-stem_, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

> (Lemma 5.2 is the [[twisted Pontrjagin theorem]])

## Related $n$Lab entries

* [[twisted cohomology]], [[equivariant cohomology]]

* [[cohomotopy]]/[[framed manifold]] [[cobordism cohomology theory]]

* [[stable cohomotopy]]

* [[twisted cohomotopy]], [[equivariant cohomotopy]]

category: people


James D. Steward

* Wikipedia entry

## Selected writings

On [[positive-definite functions]] and their understanding/generalization to [[states on star-algebras]]:

* {#Steward76} [[James D. Steward]], *Positive definite functions and generalizations, an historical survey*, The Rocky Mountain Journal of Mathematics **6** 3 (1976) 409-434 [[jstor:44236118](https://www.jstor.org/stable/44236118)]

category: people

[[!redirects James Steward]]


James Davis

* [webpage](http://www.indiana.edu/~jfdavis/)

## Selected writings

On [[equivariant K-theory]] as a [[Spectra]]-valued [[presheaf]] on the [[orbit category]] (a [[naive G-spectrum]], see also at _[[proper equivariant homotopy theory]]_):

* [[James Davis]], [[Wolfgang Lück]], _Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory_, K-Theory 15:201–252, 1998 ([pdf](https://jfdmath.sitehost.iu.edu/teaching/m721F19/assembly.pdf))

On [[G-spaces]], [[equivariant homotopy theory]] and [[equivariant cohomology]]:

* [[Alejandro Adem]], [[James Davis]], *Topics in Transformation Groups*, Chapter 1 in: *Handbook of Geometric Topology*, 2001, Pages 1-54 ([doi:10.1016/B978-044482432-5/50002-0](https://doi.org/10.1016/B978-044482432-5/50002-0), [pdf](https://personal.math.ubc.ca/~adem/topics.pdf))

## Related $n$Lab pages

* [[Atiyah-Hirzebruch spectral sequence]] (in [[equivariant cohomology theory]])

* [[algebraic topology]] (for [book](https://jfdmath.sitehost.iu.edu/teaching/m623/book.pdf)  with [[Paul Kirk]]).

category: people

[[!redirects Jim Davis]]

[[!redirects James F. Davis]]


James Dolan

James Dolan is a category theorist currently unaffiliated with a university. In the past he has been associated with University of California Riverside and Macquarie University.  He has a [website](http://math.ucr.edu/~jdolan/) with some notes about math on it.  He also has a [[James Dolan:HomePage|personal web]] here on the $n$Lab.

## Related entries

* [[categorification]]
* [[generalized tangle hypothesis]]
* [[cobordism hypothesis]]
* [[stabilization hypothesis]]
* [[opetope]]
* [[groupoid cardinality]]
* [[2-rig]]
* [[core-natural transformation]]
* [[microcosm principle]]
* [[periodic table]]
* [[stuff type]]
* [[stuff, structure, property]]
* [[walking structure]]
* ...many others

category: people

[[!redirects Jim Dolan]]
[[!redirects james dolan]]


James Dugundji

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

## Selected writings

On generalizing the [[Tietze extension theorem]] to [[codomains]] which are [[locally convex topological vector spaces]]:

* {#Dugundji51} [[James Dugundji]], *An extension of Tietze's theorem*, Pacific Journal of Mathematics **1** 3 (1951) [[doi:10.2140/pjm.1951.1.353](http://dx.doi.org/10.2140/pjm.1951.1.353), [pdf](https://msp.org/pjm/1951/1-3/pjm-v1-n3-p04-p.pdf)]

On [[CW complexes]] and [[absolute neighbourhood retracts]]:

* {#Dugundji52} [[James Dugundji]], _Note on CW polytopes_, Portugaliae mathematica (1952) **11** 1 (1952) 7-10-b ([dml:114693](https://eudml.org/doc/114693))

On [[general topology]] (including [[compactly generated topological spaces]]):

* [[James Dugundji]], *Topology*, Allyn and Bacon 1966 ([pdf](https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf))

category: people


James Eells

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

## Selected writings

On [[Hurewicz fibrations]] and [[Hurewicz connections]]

* [[James Eells]], Jr., _Fibring spaces of maps_, in Richard Anderson (ed.) _Symposium on infinite-dimensional topology_

On [[twistor fibrations]]:

* [[James Eells]], [[Simon Salamon]], _Twistorial construction of harmonic maps of surfaces into four-manifolds_,  Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 4, p. 589-640 ([numdam:ASNSP_1985_4_12_4_589_0](http://www.numdam.org/item/ASNSP_1985_4_12_4_589_0))

category: people


James Francese

Mathematician working on special topics in geometry and foundations


James Freitag

James Freitag is a mathematician at University of Illinois at Chicago. In words from his

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

his work is "a mixture of model theory, algebra, and number theory".

## Selected writing

* [[James Freitag]], _Algebraic relations between solutions of Painlevé equations_, [arXiv:1710.03304](https://arXiv.org/abs/1710.03304)
* Guy Casale, [[James Freitag]], Joel Nagloo, _Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups_, Annals of Mathematics __192__ (2020), 721–765 [doi](https://doi.org/10.4007/annals.2020.192.3.2)
* James Freitag, Rémi Jaoui, Rahim Moosa, _When any three solutions are independent_, Invent. math. __230__ (2022) 1249--1265 [arXiv:2110.08123](https://arxiv.org/abs/2110.08123) [doi](https://doi.org/10.1007/s00222-022-01143-8)

category: people


James Glimm

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

## related $n$Lab entries

* [[constructive quantum field theory]]

category: people


James Gray

* [webpage](http://www1.phys.vt.edu/~grayphys/)

## Selected writings

On [[heterotic string theory|heterotic]] [[string phenomenology]] with [[heterotic line bundle models]]:

* {#AndersonGrayLukasPalti11} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_, Phys. Rev. D 84, 106005 (2011) ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* {#AndersonGrayLukasPalti12} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic Line Bundle Standard Models_ JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757))

Scan of [[SU(5)]] [[GUT]] [[perturbative string theory vacuum|vacua]] among [[heterotic line bundle models]]:

* [[Lara Anderson]], [[Andrei Constantin]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic $SU(5)$ GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

* [[James Gray]], Juntao Wang, _Jumping Spectra and Vanishing Couplings in Heterotic Line Bundle Standard Models_, JHEP11 (2019) 073 ([arXiv:1906.09373](https://arxiv.org/abs/1906.09373))

On [[moduli stabilization]]:

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Burt Ovrut]], _Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories_ ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704))

category: people


James Halverson

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

## Selected writings

* {#CveticHalversonLinLiuTian19} [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, _A Quadrillion Standard Models from F-theory_ ([arXiv:1903.00009](https://arxiv.org/abs/1903.00009))

  (realizations of the [[MSSM]] in [[F-theory]])

On [[machine learning]] the detection of [[unknots]]:

* [[Sergei Gukov]], [[James Halverson]], [[Fabian Ruehle]], [[Piotr Sułkowski]], *Learning to Unknot*, Mach. Learn.: Sci. Technol. **2** (2022) 025035 [[arXiv:2010.16263](https://arxiv.org/abs/2010.16263), [doi:10.1088/2632-2153/abe91f](https://iopscience.iop.org/article/10.1088/2632-2153/abe91f)]

## Related entries

* [[M-theory on G2-manifolds]]

* [[F-theory]]

* [[string phenomenology]]

category: people


James Hartle

Physicist at the University of California, Santa Barbara.

* [website](http://web.physics.ucsb.edu/~hartle/)

##Selected writings

An elaboration of [[Murray Gell-Mann|Murray Gell-Mann's]] _[[Nature conformable to herself]]_ ([Gell-Mann 96](https://ncatlab.org/nlab/show/Murray+Gell-Mann#Gell-Mann96))

* [[James Hartle]], _How Nature is Conformable to Herself: A View from Quantum Cosmology_, ([arXiv:1909.08724](https://arxiv.org/abs/1909.08724))

##Related pages

* [[no boundary proposal]]

category: people


James Ladyman

* [webpage](http://www.bristol.ac.uk/school-of-arts/people/james-a-ladyman/)

## Selected writings

On the understanding of [[transport]] in [[homotopy type theory]] as expressing [[Leibniz]]'s principle of "[[indiscernibility of identicals]]" (aka "substitution of equals" or "substitutivity"), as well as the converse implication of  [[path induction]] from transport (together with the uniqueness principle for id-types):

* {#LadymanPresnell15} [[James Ladyman]], [[Stuart Presnell]], *Identity in Homotopy Type Theory, Part I: The Justification of Path Induction*, Philosophia Mathematica **23** 3 (2015) 386–406 [[doi:10.1093/philmat/nkv014](https://doi.org/10.1093/philmat/nkv014), [pdf](http://philsci-archive.pitt.edu/11079/1/Identity_in_HTT_public.pdf)]

## Related entries

* [[philosophy of science]]

* [[homotopy type theory]]

category: people


James Lepowsky

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

## Selected writings

On the [[representation categories]] of (rational) [[vertex operator algebras]] as ([[modular tensor categories|modular]]) [[fusion categories]]:

* {#HuangLepowskiZhang14} [[Yi-Zhi Huang]], [[James Lepowsky]], Lin Zhang, *Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules*,  In: Bai, Fuchs, Huang, Kong, Runkel, Schweigert (eds.), *Conformal Field Theories and Tensor Categories* Mathematical Lectures from Peking University. Springer (2014) $[$[arXiv:1012.4193](https://arxiv.org/abs/1012.4193), [doi:10.1007/978-3-642-39383-9_5](https://doi.org/10.1007/978-3-642-39383-9_5)$]$

## Related nLab entries

* [[vertex operator algebra]]

* [[Moonshine]]

category: people


James Liu

* [webpage](https://lsa.umich.edu/physics/people/faculty/jimliu.html)

## Selected writings

On [[U-duality]]:

* {#LiuMinasian97} [[James Liu]], [[Ruben Minasian]], _U-branes and $T^3$ fibrations_, Nucl.Phys. B510 (1998) 538-554 ([arXiv:hep-th/9707125](http://arxiv.org/abs/hep-th/9707125))

On [[string-string duality]] and the [[I8]]-anomaly term:

* {#DuffLiuMinasian95} [[Mike Duff]], [[James Liu]], [[Ruben Minasian]], _Eleven Dimensional Origin of String/String Duality: A One Loop Test_, Nucl.Phys. B452 (1995) 261-282 ([arXiv:hep-th/9506126](https://arxiv.org/abs/hep-th/9506126))

On [[higher curvature corrections]] in [[type II supergravity]] and in [[D=11 supergravity]]:

* [[James Liu]], [[Ruben Minasian]], _Higher-derivative couplings in string theory: dualities and the B-field_, Nucl. Phys. B 2013 ([arXiv:1304.3137](https://arxiv.org/abs/1304.3137))

* [[James Liu]], [[Ruben Minasian]], Raffaele Savelli, Andreas Schachner, *Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles* $[$[arXiv:2205.11530](https://arxiv.org/abs/2205.11530)$]$

## Related $n$Lab entries

* [[U-duality]]

* [[string-string duality]]

* [[I8]]

* [[8-dimensional supergravity]]

category: people


James MacManus

* [InSpire page](https://inspirehep.net/authors/2684559)

## Selected writings

On the relation between [[functorial quantum field theory]] (axiomatizing the [[Schrödinger picture]] of [[quantum field theory]]) and [[algebraic quantum field theory]] (axiomatizing the [[Heisenberg picture]]):

* {#BunkMacManusSchenkel23} [[Severin Bunk]], [[James MacManus]], [[Alexander Schenkel]], *Lorentzian bordisms in algebraic quantum field theory* [[arXiv:2308.01026](https://arxiv.org/abs/2308.01026)]

category: people


James McClure

* [webpage](https://www.math.purdue.edu/~mcclurej/)

## Selected writings

On [[H-infinity ring spectra]]:

* {#McLure86} [[James McClure]], _$H_\infty$-ring spectra via space-level homotopy theory_ ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/BMMS8.pdf)), chapter VII in R. Bruner, [[Peter May]], [[James McClure]], [[Mark Steinberger]], _$H_\infty$-Ring Spectra and their Applications_, Lecture Notes in Mathematics 1176, Springer 1986 ([doi:10.1007/BFb0075405](https://link.springer.com/book/10.1007/BFb0075405), [pdf](http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf))

On [[equivariant stable homotopy theory]]:

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], [[Mark Steinberger]] (with contributions by [[James McClure|J.E. McClure]]), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics  **1213** (1986) [[pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778)]

On [[generalized cup products]]:

* [[James E. McClure]], [[Jeffrey H. Smith]],
  _Multivariable cochain operations and little $n$-cubes_
([arXiv:math/0106024](https://arxiv.org/abs/math/0106024))

## Related $n$Lab entries

* [[ring spectrum]]

* [[H-infinity ring spectrum]]

* [[KU]]

category: people

[[!redirects James E. McClure]]


James Milne

__James Milne__ is an [[algebraic geometry|algebraic geometer]].

See [www.jmilne.org](http://www.jmilne.org)

## Selected writings:
 {#SelectedWritings}

On [[étale cohomology]]:

* _[[Étale Cohomology]]_, Princeton Mathematical Series __33__, 1980. xiii+323 pp. ([jstor:j.ctt1bpmbk1](https://www.jstor.org/stable/j.ctt1bpmbk1))

* _[[Lectures on Étale Cohomology]]_  ([pdf](http://www.jmilne.org/math/CourseNotes/LEC.pdf))

On [[Tannakian categories]]:

* [[Pierre Deligne]], [[James Milne]]: *Tannakian categories*,  in: *Hodge Cycles, Motives, and Shimura Varieties*, Lecture Notes in Mathematics **900**, Springer (1982) [[doi:10.1007/978-3-540-38955-2_4](https://doi.org/10.1007/978-3-540-38955-2_4), [webpage](http://www.jmilne.org/math/xnotes/tc.html)]

On [[Shimura varieties]]:

* [[James Milne]], _Introduction to Shimura Varieties_, 2004, ([pdf](https://www.jmilne.org/math/articles/2005aX.pdf))

On [[algebraic groups]]:

* {#Milne17} [[James Milne]], *Algebraic Groups -- The theory of group schemes of finite type over a field*, Cambridge University Press 2017  ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))

On [[algebraic number theory]]:

* [[James Milne]], *Algebraic number theory*, 2020 ([pdf](https://www.jmilne.org/math/CourseNotes/ANT.pdf))

On [[group theory]]:

* [[James Milne]], *Group theory* (2021) [[web](https://www.jmilne.org/math/CourseNotes/gt.html), [pdf](https://www.jmilne.org/math/CourseNotes/GT.pdf)]

## Related entries

* [[étale cohomology]]

* [[ℓ-adic cohomology]]

* [[Weil conjecture]]

category:people

[[!redirects J. Milne]]

[[!redirects Milne]]


James Montaldi

* [webpage](https://personalpages.manchester.ac.uk/staff/j.montaldi/)

## related $n$Lab entries

* [[permutation representation]]

* [[character table of 2T]]

category: people


James Munkres

James Raymond Munkres was a professor at MIT.

He got his Ph.D. degree in 1956 from the University of Michigan, advised by [[Edwin E. Moise]].

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

## Selected writings

On [[general topology|general]] [[topology]]:

* {#Munkres75} [[James Munkres]], _Topology_, Prentice Hall (1975, 2000) [ISBN:0-13-181629-2, [pdf](http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf)]

On [[differential topology]]:

* [[James R. Munkres]], _Elementary Differential Topology_, Annals of Mathematics Studies 54 (1966), Princeton University Press ([doi:10.1515/9781400882656](http://dx.doi.org/10.1515/9781400882656)).

On [[algebraic topology]]:

* [[James Munkres]], *Elements of Algebraic Topology*, Addison-Wesley (1984) [[pdf](https://people.dm.unipi.it/benedett/MUNKRES-ETA.pdf)]

## Related entries

* [[topology]]

* [[Poincaré duality space]]

category: people

[[!redirects James R. Munkres]]
[[!redirects James Raymond Munkres]]


James Norris

* [webpage](http://www.statslab.cam.ac.uk/Dept/People/norris.html)

## Related entries

* [[path integral]]

category: people


James R. Norris

James Ritchie Norris is a mathematician at the University of Cambridge.

He got his PhD in 1985 from the University of Oxford, advised by David Edwards.

## Selected writings

On the [[Feynman–Kac formula]] on [[smooth manifolds]]:

* [[James R. Norris]], _A complete differential formalism for stochastic calculus in manifolds_, Séminaire de Probabilités XXVI, Lecture Notes in Mathematics (1992), 189–209.  [doi](http://dx.doi.org/10.1007/bfb0084322).

[[!redirects James Ritchie Norris]]
[[!redirects James Norris]]
[[!redirects J. R. Norris]]
[[!redirects J. Norris]]

category: people


James Robin Wootton

* [institute page](https://researcher.watson.ibm.com/researcher/view.php?person=zurich-JWO)

## Selected writings

On [[topological quantum computation]] with [[abelian anyons]]:

* [[James Robin Wootton]], *Dissecting Topological Quantum Computation*, 2010 ([pdf](https://etheses.whiterose.ac.uk/1163/1/main.pdf), [[WoottonDissectingTQC.pdf:file]])

  > "non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models."

[[!redirects James Wootton]]


James Simons

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

## Selected writings

On [[Chern-Weil theory]] and introducing [[Chern-Simons forms]]:

* [[Shiing-Shen Chern]], [[James Simons]], Section 2 of: _Characteristic Forms and Geometric Invariants_, Annals of Mathematics Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69 ([jstor:1971013](https://www.jstor.org/stable/1971013))

On [[Cheeger-Simons differential characters]] and [[secondary invariants]]:

* {#CheegerSimons85} [[Jeff Cheeger]],  [[James Simons]], _Differential characters and geometric invariants_, pages 50–80 in: _Geometry and Topology_, Lecture Notes in Mathematics, vol 1167. Springer (1985)  ([pdf](http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf), [doi:10.1007/BFb0075212](https://doi.org/10.1007/BFb0075212))

category: people


James Sparks

* [webpage](https://www.maths.ox.ac.uk/people/james.sparks)

## Selected writings

* {#Sparks10} [[James Sparks]], _Sasaki-Einstein Manifolds_, Surv. Diff. Geom. 16 (2011) 265-324 ([arXiv:1004.2461](https://arxiv.org/abs/1004.2461))

* {#GukovSparks02} [[Sergei Gukov]], [[James Sparks]], _M-Theory on $Spin(7)$ Manifolds_, Nucl.Phys. B625 (2002) 3-69 ([arXiv:hep-th/0109025](https://arxiv.org/abs/hep-th/0109025))

On [[M-theory on 8-manifolds]] and [[conifold transition]]:

* [[Sergei Gukov]], [[James Sparks]], [[David Tong]], *Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds*, Class. Quant. Grav. **20** (2003) 665-706 [[arXiv:hep-th/0207244](https://arxiv.org/abs/hep-th/0207244)]

On relating the [[Freed-Witten anomaly]] to [[shifted C-field flux quantization]]:

* [[James Sparks]], _Global Worldsheet Anomalies from M-Theory_, JHEP 0408:037, 2004 ([arXiv:hep-th/0310147](https://arxiv.org/abs/hep-th/0310147))

## Related $n$Lab entries

* [[Sasaki-Einstein manifold]]

* [[M-theory on 8-manifolds]]

* [[Spin(7)-manifolds]]

* [[conifold transition]]

category: people


James Sutherland Frame

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=15590)

## Selected writings

Introducing the [[hook length formula]] in the [[representation theory of the symmetric group]]:

* {#FrameRobinsonThrall54} [[James Sutherland Frame]], [[Gilbert de Beauregard Robinson]], [[Robert McDowell Thrall]], *The Hook Graphs of the Symmetric Group*, Canadian Journal of Mathematics, Volume 6, 1954, pp. 316 - 324 ([doi:10.4153/CJM-1954-030-1](https://doi.org/10.4153/CJM-1954-030-1))

category: people


James W. Alexander

[[!redirects J. W. Alexander]]
James Waddell Alexander was an American [[topology|topologist]] and [[geometry|geometer]].

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

## Selected writings

On $n$-[[manifolds]] as [[branched covers]] of the [[n-sphere]]:

* {#Alexander20} [[James W. Alexander]], _Note on Riemann spaces_, Bull. Amer. Math. Soc. 26 (1920) 370–373

Introducing [[Alexander's trick]]:

* [[James W. Alexander]], *On the Deformation of an $n$ Cell*, Proc Natl Acad Sci USA. **9** 12 (1923) 406–407 [[doi:10.1073%2Fpnas.9.12.406](https://doi.org/10.1073%2Fpnas.9.12.406) ]

Introducing the [[Alexander polynomial]]:

* {#Alexander28} [[James W. Alexander]], *Topological invariants of knots and links*, Trans. Amer. Math. Soc. **30** (1928) 275-306 $[$[doi:10.1090/S0002-9947-1928-1501429-1](https://doi.org/10.1090/S0002-9947-1928-1501429-1)$]$

Introducing [[singular homology]]/[[singular cohomology]] via [[chain homology]]/[[cochain cohomology]]:

* [[J. W. Alexander]], _On the chains of a complex and their duals_, Proc. Nat. Acad. Sei. USA **21** (1935), 509–511 ([doi:10.1073/pnas.21.8.50](https://doi.org/10.1073/pnas.21.8.509))

* [[J. W. Alexander]], _On the ring of a compact metric space_, Proc. Nat. Acad. Sci. USA **21** (1935), 511–512 ([doi:10.1073/pnas.21.8.511](https://doi.org/10.1073/pnas.21.8.511))

* [[J. W. Alexander]], _On the connectivity ring of an abstract space_, Ann. of Math. **37** (1936), 698–708 ([doi:10.2307/1968484](https://doi.org/10.2307/1968484), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/alexcon.pdf))

## Related n-Lab entries

* [[Alexander polynomial]]

* [[Alexander-Čech duality]]

* [[Alexander-Whitney map]]

* [[branched cover]]

category:people


James Wallbridge

* [website](http://www.math.harvard.edu/people/WallbridgeJames.html)

## Publications

* _Higher Tannaka duality_ PhD thesis, Adelaide/Toulouse (2011) [Adelaide University repository](http://digital.library.adelaide.edu.au/dspace/bitstream/2440/69436/1/02whole.pdf) or [arXiv:1204.5787](http://arxiv.org/abs/1204.5787)

  (on [[Tannaka reconstruction]] in the context of [[(∞,1)-category theory]])

category: people


James Webb Space Telescope

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

\tableofcontents

## Idea

The **James Webb Space Telescope** is a telescope which observes in the infrared part of the electromagnetic spectrum.

So far, it has observed distant [[galaxies]] at high redshift, as well as cepheids in SN type IA supernovae, both of which are important in testing the [[standard model of cosmology]], the former for the validity of galaxy formation using [[dark matter]], and the latter for resolving the [[Hubble tension]].

## See also

* [[standard model of cosmology]]

## References

* [[Wenlong Yuan]], [[Adam G. Riess]], [[Stefano Casertano]], [[Lucas M. Macri]], *A First Look at Cepheids in a SN Ia Host with JWST* ([arXiv:2209.09101](https://arxiv.org/abs/2209.09101))

* Michael Boylan-Kolchin, *Stress Testing ΛCDM with High-redshift Galaxy Candidates*, ([arXiv:2208.01611](https://arxiv.org/abs/2208.01611))

* B. E. Robertson et al. *Discovery and properties of the earliest galaxies with confirmed distances*. ([arXiv:2212.0448](https://arxiv.org/abs/2212.04480))

* Nikita Lovyagin, Alexander Raikov, Vladimir Yershov, Yuri Lovyagin, *Cosmological Model Tests with JWST*. ([arXiv:2212.06575](https://arxiv.org/abs/2212.06575))

* Deng Wang, Yizhou Liu, *JWST high redshift galaxy observations have a strong tension with Planck CMB measurements*. ([arXiv:2301.00347](https://arxiv.org/abs/2301.00347))

## External links

* [website](https://jwst.nasa.gov/)

* Wikipedia, *[James Webb Space Telescope](https://en.wikipedia.org/wiki/James_Webb_Space_Telescope)*

[[!redirects JWST]]


James Wells

* [webpage](http://www-personal.umich.edu/~jwells/)

## Selected writings

On [[naturalness]]:

* {#Wells18a} [[James Wells]], _Naturalness, Extra-Empirical Theory Assessments, and the Implications of Skepticism_ ([arXiv:1806.07289](https://arxiv.org/abs/1806.07289))

* {#Wells18} [[James Wells]], _Finetuned Cancellations and Improbable Theories_ ([arXiv:1809.03374](https://arxiv.org/abs/1809.03374))

* [[James Wells]], *Evaluation and Utility of Wilsonian Naturalness* ([arXiv:2107.06082](https://arxiv.org/abs/2107.06082))

On [[split supersymmetry]]:

* [[James Wells]], _Implications of Supersymmetry Breaking with a Little Hierarchy between Gauginos and Scalars_,  Proceedings of [SUSY 2003](http://www.physics.arizona.edu/susy2003/) ([arXiv:hep-ph/0306127](https://arxiv.org/abs/hep-ph/0306127))

On [[gauge coupling unification]]:

* {#EllisWells15} Sebastian A. R. Ellis, [[James Wells]], _Visualizing gauge unification with high-scale thresholds_, Phys. Rev. D 91, 075016 (2015) ([arXiv:1502.01362](https://arxiv.org/abs/1502.01362))

* {#EllisWells17} Sebastian A. R. Ellis, [[James Wells]], _High-scale Supersymmetry, the Higgs Mass and Gauge Unification_, Phys. Rev. D 96, 055024 (2017) ([arXiv:1706.00013](https://arxiv.org/abs/1706.00013))

On [[theory (physics)]]:

* [[James Wells]], _Lexicon of Theory Qualities_ ([pdf](http://www-personal.umich.edu/~jwells/prms/prm8.pdf))

On the [[standard model of particle physics]]:

* [[James Wells]], _The Once and Present Standard Model of Elementary Particle Physics_ ([arxiv:1911.04604](https://arxiv.org/abs/1911.04604))

## Related $n$Lab entries

* [[gauge coupling unification]]

* [[theory (physics)]]

* [[naturalness]]

* [[split supersymmetry]]

category: people


Jamie Vicary

* [website](http://www.comlab.ox.ac.uk/people/Jamie.Vicary/)

## Selected writings

On [[quantum measurement]] formulated in [[quantum information theory via dagger-compact categories ]] in terms of [[Frobenius algebras]]:

* [[Bob Coecke]], [[Duško Pavlović]], [[Jamie Vicary]],  *A new description of orthogonal bases*, Mathematical Structures in Computer Science **23** 3 (2012) 555- 567 [[arXiv:0810.0812](https://arxiv.org/abs/0810.0812), [doi:10.1017/S0960129512000047](https://doi.org/10.1017/S0960129512000047)]

On [[quantum computation]] via [[quantum information theory in terms of dagger-compact categories]]:

* [[Jamie Vicary]], *The Topology of Quantum Algorithms*, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 $[$[arXiv:1209.3917](https://arxiv.org/abs/1209.3917), [doi:10.1109/LICS.2013.14](https://doi.org/10.1109/LICS.2013.14)$]$

On the online proof assistant *[[Globular]]* for higher dimensional [[rewriting]] via [[semistrict]] [[globular set|globular]] [[higher categories]] ([[associative n-categories]]):

* [[Krzysztof Bar]], [[Aleks Kissinger]], [[Jamie Vicary]], *Globular: an online proof assistant for higher-dimensional rewriting*,  Logical Methods in Computer Science **14** 1 (2018) [[doi:10.23638/LMCS-14(1:8)2018](https://doi.org/10.23638/LMCS-14(1:8)2018), [arXiv:1612.01093](https://arxiv.org/abs/1612.01093)]

On [[associative n-categories]] and their [[formal proof|formalization]] in [[proof assistants]] (cf.: [[Globular]], [[homotopy.io]]):

* {#ReutterVicary19} [[David Reutter]], [[Jamie Vicary]], _High-level methods for homotopy construction in associative $n$-categories_, LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune **62** (2019) 1–13 [[arXiv:1902.03831](https://arxiv.org/abs/1902.03831), [doi:10.1109/LICS52264.2021.9470575](https://doi.org/10.1109/LICS52264.2021.9470575)]

* [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]

On [[monoidal category]]-theory with an eye towards [[quantum information theory 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)]

  based on:

  {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], *Lectures on categorical quantum mechanics* (2012) [[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]]

On [[weak omega-category|weak $\omega$-categories]] via [[computads]] construed as [[inductive types]]:

* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]

## Related entries

* [[quantum information theory via dagger-compact categories]]

* [[classical structure]]

category: people

[[!redirects Jamie Vicary]]
[[!redirects JamieVicary]]


Jan de Boer

* [webpage](http://staff.science.uva.nl/~jdeboer/)

## Selected writings

On [[D=3 N=2 super Yang-Mills theory]]:

* [[Jan de Boer]], [[Kentaro Hori]], [[Yaron Oz]], _Dynamics of N=2 Supersymmetric Gauge Theories in Three Dimensions_, Nucl. Phys. B500:163-191, 1997 ([arXiv:hep-th/9703100](https://arxiv.org/abs/hep-th/9703100))

On [[mirror symmetry]] between [[Higgs branches]]/[[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] (with emphasis of [[Hilbert schemes of points]]):

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))

On [[topological M-theory]] and [[D=7 Chern-Simons theory]]:

* [[Jan de Boer]], [[Paul de Medeiros]], Sheer El-Showk, Annamaria Sinkovics, _Open $G_2$ Strings_ ([arXiv:hep-th/0611080](http://arxiv.org/abs/hep-th/0611080))

On [[string theory]] [[perturbative string theory vacuum|vacua]] [[KK-compactification|compactified]] on [[orbifolds]] and [[orientifolds]]:

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv. Theor. Math. Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

On [[flat space holography]]:

* [[Jan de Boer]], Sergey N. Solodukhin, *A holographic reduction of Minkowski space-time*, Nucl. Phys. B665 (2003) 545-593 ([arXiv:hep-th/0303006](https://arxiv.org/abs/hep-th/0303006))

On [[exotic branes]] (in [[string theory]]) as [[defect branes]] with [[U-duality]]-group [[monodromy]]:

* [[Jan de Boer]], [[Masaki Shigemori]], _Exotic Branes in String Theory_, Physics Reports **532** (2013) 65-118 ([arXiv:1209.6056](https://arxiv.org/abs/1209.6056), [doi:j.physrep.2013.07.003](https://doi.org/10.1016/j.physrep.2013.07.003))

Discussion of [[BTZ black hole|BTZ]] [[black hole entropy]] and more generally of [[holographic entanglement entropy]] in [[3d quantum gravity]]/[[AdS3/CFT2]] via [[Wilson line observables]] in [[Chern-Simons theory]]:

* [[Jan de Boer]], Juan I. Jottar, _Entanglement Entropy and Higher Spin Holography in $AdS_3$_, JHEP 1404:089, 2014 ([arXiv:1306.4347](https://arxiv.org/abs/1306.4347))

On [[topological crystalline insulators]] and their classification by [[equivariant K-theory]]:

On the [[K-theory classification of topological phases of matter]] for the case of [[topological crystalline insulators]] classified by [[equivariant K-theory]]:

* [[Jorrit Kruthoff]], [[Jan de Boer]], Jasper van Wezel, [[Charles L. Kane]], [[Robert-Jan Slager]], *Topological Classification of Crystalline Insulators through Band Structure Combinatorics*, Phys. Rev. X **7** (2017) 041069 $[$[doi:10.1103/PhysRevX.7.041069](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.041069)$]$

* [[Luuk Stehouwer]], [[Jan de Boer]], [[Jorrit Kruthoff]], [[Hessel Posthuma]], *Classification of crystalline topological insulators through K-theory*, Adv. Theor. Math. Phys, **25** 3 (2021) 723–775 [[arXiv:1811.02592](https://arxiv.org/abs/1811.02592), [doi:10.4310/ATMP.2021.v25.n3.a3](https://doi.org/10.4310/ATMP.2021.v25.n3.a3)]

category: people


Jan Gutowski

Jan Gutowski is working on [[black hole]]/[[black brane]] solutions in [[supergravity]] at Surrey University.

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

* SFTC grant _[Black holes in supergravity](http://www.gow.stfc.ac.uk/gow/sm/prsn.asp?cx=05&sc=0&so=oa&pl=list)_

## Selected writings

On [[D=5 supergravity]]:

* {#GGHPR02} [[Jerome Gauntlett]], [[Jan Gutowski]], [[Christopher Hull]], Stathis Pakis, [[Harvey Reall]], _All supersymmetric solutions of minimal supergravity in five dimensions_, Class. Quant. Grav. 20 (2003) 4587-4634 [[arXiv:hep-th/0209114](http://arxiv.org/abs/hep-th/0209114), [doi:10.1088/0264-9381/20/21/005](https://doi.org/10.1088/0264-9381/20/21/005)]

## Related entries

* [[11-dimensional supergravity]], [[5-dimensional supergravity]], [[type II supergravity]]

* [[BPS state]]

* [[calibration]]

category: people


Jan Hamhalter

* [webpage](http://math.feld.cvut.cz/hamhalte/)

## related entries

* [[poset of commutative subalgebras]]

category: people

[[!redirects J. Hamhalter]]


Jan Jaworowski

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

## Selected writings

On [[cohomotopy]]:

* [[Jan Jaworowski]], _Generalized cohomotopy groups as limit groups_, Fundamenta Mathematicae 50 (1962), 393-402 ([doi:10.4064/fm-50-4-333-340](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/4), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm50/fm50133.pdf))

On [[equivariant Tietze extension theorems]]:

* {#Jaworowski73} [[Jan Jaworowski]], _Equivariant extensions of maps_, Pacific J. Math. Volume 45, Number 1 (1973), 229-244 ([euclid:1102947720](https://projecteuclid.org/euclid.pjm/1102947720))

* {#Jaworowski73b} [[Jan Jaworowski]], _Extending equivariant maps for compact Lie group actions_, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 698-701 ([euclid:1183534741](https://projecteuclid.org/euclid.bams/1183534741))

* {#Jaworowski76} [[Jan Jaworowski]], _Extensions of $G$-maps and euclidean $G$-retracts_, Math Z (1976) 146: 143 ([doi:10.1007/BF01187702](https://doi.org/10.1007/BF01187702))

* {#Jaworowski81} [[Jan Jaworowski]], _An equivariant extension theorem and $G$-retracts with a finite structure_, Manuscripta Math (1981) 35: 323 ([doi:10.1007/BF01263266](https://doi.org/10.1007/BF01263266))

category: people


Jan Kneissler

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=146327)

## Selected writings

Computing [[Vassiliev invariants]]

* [[Jan Kneissler]], _On spaces of connected graphs I: Properties of Ladders_, Proc. Internat. Conf. "Knots in Hellas '98", Series on Knots and Everything, vol. 24 (2000), 252-273 ([arXiv:math/0301018](https://arxiv.org/abs/math/0301018))

* [[Jan Kneissler]], _On spaces of connected graphs II: Relations in the algebra Lambda_, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 667-674 ([arXiv:math/0301019](https://arxiv.org/abs/math/0301019))

* [[Jan Kneissler]], _On spaces of connected graphs III: The Ladder Filtration_, Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 675-686 ([arXiv:math/0301020](https://arxiv.org/abs/math/0301020))

via [[stringy weight systems]]:

* [[Jan Kneissler]], Section 1.3 of: _The number of primitive Vassiliev invariants up to degree 12_ ([arXiv:q-alg/9706022](https://arxiv.org/abs/q-alg/9706022))

category: people


Jan Louis

* [webpage](http://www.desy.de/~jlouis/)

category: people


Jan Manschot

* [personal page](https://www.maths.tcd.ie/~manschot/)

* [InSpire page](https://inspirehep.net/authors/1039454)

## Selected writings

On [[Donaldson-Witten theory]]:

* [[Jan Manschot]], *Four-Manifold Invariants and Donaldson-Witten Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14709](https://arxiv.org/abs/2312.14709)]

category: people


Jan Myrheim

* [institute page](https://www.ntnu.edu/employees/jan.myrheim)

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

## Selected writings

Early discussion of what came to be known as [[anyon]] [[braid group statistics]], and preconceiving the role of the [[configuration space of points]]:

* {#LeinaasMyrheim77} [[Jon Magne Leinaas]], [[Jan Myrheim]], _On the theory of identical particles_, _К теории тождествениых частиц_, Nuovo Cim B 37, 1–23 (1977) ([doi:10.1007/BF02727953](https://doi.org/10.1007/BF02727953))

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

* {#Myrheim99} [[Jan Myrheim]], *Anyons*, p. 265-414 in: *Topological aspects of low dimensional systems*, Les Houches LXIX, Springer (1999) $[$[doi:10.1007/3-540-46637-1](https://doi.org/10.1007/3-540-46637-1), [[Myrheim-Anyons.pdf:file]]$]$

category: people


Jan Nekovar

* [webpage](https://www.imj-prg.fr/~jan.nekovar/)

## related $n$Lab entries

* [[Beilinson conjecture]]

* [[absolute Hodge cohomology]]

category: people


Jan Pieter van der Schaar

* [institute page](https://www.uva.nl/profiel/s/c/j.p.vanderschaar/j.p.vanderschaar.html)

* [inspire page](https://inspirehep.net/authors/990060)

## Selected writings

On [[brane intersection]] laws of [[M-branes]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** (1997) 2757 [[doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

On [[M9-branes]]:

* [[Eric Bergshoeff]], [[Jan Pieter van der Schaar]], *On M-9-branes*, Class. Quant. Grav. **16** (1999) 23-39 [[arXiv:hep-th/9806069](http://arxiv.org/abs/hep-th/9806069), [doi:10.1088/0264-9381/16/1/002](https://doi.org/10.1088/0264-9381/16/1/002)]

category: people


Jan Plefka

* [webpage](http://fakultaeten.hu-berlin.de/mnf1/mitarbeiter/1680831)

## Selected writings

On the [[BMN matrix model]]:

* [[Nakwoo Kim]], [[Jan Plefka]], _On the Spectrum of PP-Wave Matrix Theory_, Nucl. Phys. B643:31-48, 2002 ([arXiv:hep-th/0207034](https://arxiv.org/abs/hep-th/0207034))

On the [[BMN matrix model]] as a [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] on the [[3-sphere]]:

* [[Nakwoo Kim]], [[Thomas Klose]], [[Jan Plefka]], _Plane-wave Matrix Theory from N=4 Super Yang-Mills on $\mathbb{R} \times S^3$_, Nucl. Phys. B671:359-382, 2003 ([arXiv:hep-th/0306054](https://arxiv.org/abs/hep-th/0306054))

On [[higher curvature corrections]] in [[D=11 supergravity]]/[[M-theory]] via analysis of [[superparticle]] [[scattering amplitudes]]:

* [[Kasper Peeters]], [[Jan Plefka]], [[Steffen Stern]], _Higher-derivative gauge field terms in the M-theory action_, JHEP 0508 (2005) 095 ([arXiv:hep-th/0507178](https://arxiv.org/abs/hep-th/0507178))

category: people


Jan Raphael Høegh-Krohn

[[!redirects Raphael Høegh-Krohn]]

* [Wikipedia entry](https://en.wikipedia.org/wiki/Raphael_H%C3%B8egh-Krohn)

* [Mathematics genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=33449)

## selected writings

* {#HoeghKrohn74} [[Raphael Høegh-Krohn]], _Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time_, Communications in Mathematical Physics 38.3 (1974): 195-224 ([pdf](https://www.duo.uio.no/bitstream/handle/10852/44072/1973-22.pdf))

## related $n$Lab entries

* [[Euclidean field theory]]

* [[thermal quantum field theory]]

category: people

[[!redirects Raphael Hoegh-Krohn]]
[[!redirects Jan Raphael Hoegh-Krohn]]


Jan Rutten

* [webpage](http://homepages.cwi.nl/~janr/)

* [papers](http://homepages.cwi.nl/~janr/papers/)

## Selected publications

* {#Rutten05} [[Jan Rutten]], *A coinductive calculus of streams*, Mathematical Structures in Computer Science, Mathematical Structures in Computer Science, Volume 15, Issue 1, February 2005, pp. 93-147 ([doi:10.1017/S0960129504004517](https://doi.org/10.1017/S0960129504004517))

[[!redirects JJMM Rutten]]
[[!redirects J.J.M.M. Rutten]]

category:people


Jan Schouten

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

## Selected writings

Introducing what came to be known as the [[Schouten bracket]]:

* [[Jan Schouten]], _Über Differentialkonkomitanten zweier kontravarianten Grössen_, Indagationes Mathematicae **2** (1940) 449–452

* [[Jan Schouten]], _On the differential operators of the first order in tensor calculus_, In: Convegno Int. Geom. Diff. Italia. (1953) 1–7

category: people

[[!redirects Jan Arnoldus Schouten]]
[[!redirects Jan A. Schouten]]
[[!redirects J. A. Schouten]]


Jan Slovák

* [webpage](http://www.math.muni.cz/~slovak/index.eng.html)

## Selected writings

On [[differential geometry]] with focus on [[natural bundles]]:

* {#KolarSlovakMichor93} [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], _[[Natural operations in differential geometry]]_, Springer (1993) [[book webpage](http://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3)
[pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)]

## Related entries

* [[parabolic geometry]]

* [[Cartan connection]]

category: people

[[!redirects Jan Slovak]]


Jan Spalinski

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=3095)

## Selected writings

On [[homotopy theory]] and [[model categories]]

* [[William Dwyer]], [[Jan Spalinski]], *[[Homotopy theories and model categories]]* ([[DwyerSpalinski_HomotopyTheories.pdf:file]])

  in: [[Ioan Mackenzie James|I. M. James]], *[[Handbook of Algebraic Topology]]*, North Holland 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

category: people


Jan Steinebrunner

* [personal page](https://www.jan-steinebrunner.com/)

## Selected writings

On [[hermitian functorial field theory]] and the [[spin-statistics theorem]]:

* [[Luuk Stehouwer]], [[Jan Steinebrunner]], _Reflection Structures and Spin Statistics in Low Dimensions_ [[arXiv:2301.06664](https://arxiv.org/abs/2301.06664)]

On [[dagger categories]]:

* [[Luuk Stehouwer]], [[Jan Steinebrunner]], *Dagger categories via anti-involutions and positivity* [[arXiv:2304.02928](https://arxiv.org/abs/2304.02928)]

category: people


Jan Stienstra

* [webpage](http://www.staff.science.uu.nl/~stien101/)

category: people


Jan Swart

* [webpage](http://staff.utia.cas.cz/swart/)

## Selected writings

On [[quantum probability theory]]:

* [[Jan Swart]], _Introduction to Quantum Probability_, 2017 ([pdf](http://staff.utia.cas.cz/swart/dict.pdf), [[SwartQuantumProbability.pdf:file]])

category: people


Jan Swoboda

* [institute page](https://www.mathi.uni-heidelberg.de/~diffgeo/Jan.html)

## Selected writings

On [[Higgs bundles]]:

* [[Jan Swoboda]], *Moduli Spaces of Higgs Bundles -- Old and New*, Dtsch. Math. Ver. **123** (2021) 65–130 [[doi:10.1365/s13291-021-00229-1](https://doi.org/10.1365/s13291-021-00229-1)]

category: people


Jan Troost

* [webpage](http://www.phys.ens.fr/~troost/)

category: people


Jan van Leeuwen

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

* [GoogleScholar page](https://scholar.google.com/citations?user=7s_pSa4AAAAJ)

* [list](https://webspace.science.uu.nl/~leeuw112/list-vLW.html) of joint writings with [[Jiří Wiedermann]]

## Selected writings

On [[computation]]:

* {#vanLeeuwenWiedermann15} [[Jan van Leeuwen]], [[Jiří Wiedermann]]: *What is Computation -- An Epistemic Approach*, p. 1-13 in Proceedings of *[SOFSEM 2015](http://www.sofsem.cz/sofsem15/): Theory and Practice of Computer Science*, Lecture Notes in Computer Science **8939**, Springer (2015) [[pdf:10.1007/978-3-662-46078-8](https://doi.org/10.1007/978-3-662-46078-8), talk slides: [pdf](http://www.sofsem.cz/sofsem15/files/presentations/invited/vanLeeuwenWiedermann.pdf)]

* {#vanLeeuwen15} [[Jan van Leeuwen]], *The Philosophy of Computation*, p. xix-xxx in: Proceedings of *IIT.SRC 2015 -- Student Research Conference* Bratislava (2015) [full proceedings: [pdf](http://www2.fiit.stuba.sk/iitsrc/iit-src2015-proceedings.pdf), [web](https://www.fiit.stuba.sk/iit-src2015/proceedings.html?page_id=4340); article: [pdf](https://webspace.science.uu.nl/~leeuw112/Bratislava-2015.pdf); slides: [pdf](https://www.fiit.stuba.sk/buxus/docs/IIT.SRC/2015/iit-src-2015_leeuwen_philosophy-of-computation.pdf)]

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    "caption": "(from [van Leeuwen 2015](#vanLeeuwen15))"
\end{imagefromfile}

* {#vanLeeuwenWiedermann17} [[Jan van Leeuwen]], [[Jiří Wiedermann]]: *Knowledge, Representation and the Dynamics of Computation*, pp. 69-89 in: *Representation and Reality in Humans, Other Living Organisms and Intelligent Machines*, Studies in Applied Philosophy, Epistemology and Rational Ethics **28**, Springer (2017) [[doi:10.1007/978-3-319-43784-2_5](https://doi.org/10.1007/978-3-319-43784-2_5), [[LeeuwenWiedermann-Computation.pdf:file]]]

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    },
    "caption": "(from [van Leeuwen and Wiedermann 2017](#vanLeeuwenWiedermann17))"
\end{imagefromfile}

* {#vanLeeuwenWiedermann19} [[Jan van Leeuwen]], [[Jiří Wiedermann]], *Understanding Computation -- A General Theory of Computational Processes*, Technical Report UU-CS-2019-012, Utrecht University (2019) [[dspace:1874/390075](https://dspace.library.uu.nl/handle/1874/390075), [pdf](http://www.cs.uu.nl/research/techreps/repo/CS-2019/2019-012.pdf), [pdf](https://dspace.library.uu.nl/bitstream/handle/1874/390075/2019_012.pdf?sequence=1&isAllowed=y)]

category: people


Jan Vysoky

Researcher at CAS Prague.

* personal [page](http://physics.fjfi.cvut.cz/~vysokjan)

## Selected writings

On [[morphisms]] between [[Courant Lie algebroids]]:

* [[Jan Vysoky]], _Hitchhiker's Guide to Courant Algebroid Relations_ ([arXiv:1910.05347](https://arxiv.org/abs/1910.05347))

On [[Riemannian geometry]] on [[Courant algebroids]] and relation to [[supergravity]] [[equations of motion]]:

* [[Branislav Jurčo]], [[Jan Vysoky]], _Courant algebroid connections and string effective actions_, Proceedings of Tohoku Forum for Creativity, Special volume: Noncommutative Geometry and Physics IV ([arXiv:1612.01540](https://arxiv.org/abs/1612.01540))

## related $n$Lab entries

* [[Courant algebroid]]

category: people


Jan Zaanen

Jan Zaanen (1957-2024)

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

* University Leiden News: *[In memoriam Jan Zaanen 1957-2024](https://www.universiteitleiden.nl/en/news/2024/01/in-memoriam-jan-zaanen-1957-2024-the-universe-in-a-speck-of-rusting-copper)*

* [Institute page](https://www.universiteitleiden.nl/en/staffmembers/johannes-zaanen#tab-1)

* [Quantum Matter Theory Group page](https://www.lorentz.leidenuniv.nl/zaanen/wordpress/)

## Selected writings

On [[Abrikosov vortices]] in [[superconductors]] as [[strings]]:

* {#BeekmanZaanen11} [[Aron J. Beekman]], [[Jan Zaanen]], *Electrodynamics of Abrikosov vortices: the field theoretical formulation*, Front. Phys. **6** (2011) 357–369 [[doi:10.1007/s11467-011-0205-0](https://doi.org/10.1007/s11467-011-0205-0)]

On [[topological crystalline insulators]]:

* [[Robert-Jan Slager]], Andrej Mesaros, Vladimir Juricic, [[Jan Zaanen]], *The space group classification of topological band insulators*, Nature Physics **9** 98 (2013) [[doi:10.1038/nphys2513](https://doi.org/10.1038/nphys2513), [arXiv:1209.2610](https://arxiv.org/abs/1209.2610)]

On [[AdS-CFT in condensed matter physics]]:

* [[Jan Zaanen]], [[Yan Liu]], Ya-Wen Sun, [[Koenraad Schalm]], _Holographic Duality in Condensed Matter Physics_, Cambridge University Press 2015 ([doi:10.1017/CBO9781139942492]( https://doi.org/10.1017/CBO9781139942492))

On [[strange metals]], high $T_c$-[[superconductors]] and [[AdS/CMT duality]]:

* [[Jan Zaanen]], _Planckian dissipation, minimal viscosity and the transport in cuprate strange metals_, SciPost Phys. 6, 061 (2019) ([arXiv:1807.10951](https://arxiv.org/abs/1807.10951))

* {#Zaanen21} [[Jan Zaanen]], *Lectures on quantum supreme matter* ([arXiv:2110.00961](https://arxiv.org/abs/2110.00961))

* Chuan-Yin Xia, Hua-Bi Zeng, Yu Tian, Chiang-Mei Chen, [[Jan Zaanen]],
*Holographic Abrikosov lattice: vortex matter from black hole* ([arXiv:2111.07718](https://arxiv.org/abs/2111.07718))

category: people

[[!redirects Johannes Zaanen]]


Jan Zwanenburg

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=116648)

## Selected writings

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

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

category: people


Jan Łopuszański

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)

## Selected writings

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

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

and historical remarks on its conception:

* [[Jan Łopuszański]], *The story of central charges*, p. 16-17 in: [[Steven Duplij]], [[Warren Siegel]], [[Jonathan Bagger]] (eds.), *Concise encyclopedia of supersymmetry -- and noncommutative structures in mathematics and physics*, Springer 2005 ([doi:10.1007/1-4020-4522-0](https://doi.org/10.1007/1-4020-4522-0) [pdf](https://ivv5hpp.uni-muenster.de/u/douplii/susy/SUSYEnc_Story.pdf))

## Related $n$Lab entries

* [[Haag–Łopuszański–Sohnius theorem]]

category: people

[[!redirects Jan Lopuszanski]]


Jan-Erik Roos

Jan-Erik Roos (16 October 1935 - 15 December 2017) was a Swedish mathematician whose research interests were in [[abelian category|abelian]] [[category theory]], [[homological algebra]] and related areas.

* [Wikipedia entry](https://sv.wikipedia.org/wiki/Jan-Erik_Roos)

## Selected writings

On [[distributivity of limits over colimits]]:

* {#Roos64} [[Jan-Erik Roos]], *Introduction à l'étude de la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos)*, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 259 (1964), 969–972 ([[roos-distributivity.pdf:file]])

See also:

* [[Jan-Erik Roos]], _Locally Noetherian categories and generalized strictly linearly compact rings. Applications_, In: Category Theory, Homology Theory and their Applications II. Lecture Notes in Mathematics __92__, Springer 1969 [doi](https://doi.org/10.1007/BFb0080772)

category:people

[[!redirects Jan Erik Roos]]


Jan-Willem van Holten

* [webpage](http://www.nikhef.nl/~t32/)

## Selected writings

On [[spinning particles]] in [[background fields]]:

* [[Jan-Willem van Holten]], *Relativistic Dynamics of Spin in Strong External Fields* [[arXiv:hep-th/9303124](http://arxiv.org/abs/hep-th/9303124)]

## Related entries

* [[string theory]]

* [[M-theory super Lie algebra]]

category: people


Jana Dunfield

[[!redirects Joshua Dunfield]]
* [webpage](https://research.cs.queensu.ca/home/jana/)

## Related $n$Lab pages

* [[type refinement]]

* [[bidirectional type checking]]

category: people


Jana Musilová

* [webpage](http://www.physics.muni.cz/~janam/index.php)

## related $n$Lab entries

* [[variational calculus]]

* [[jet bundle]]

* [[Lepage form]]

category: people

[[!redirects Jana Musilova]]


Janez Mrčun

**Janez Mrčun** is a mathematician from [Ljubljana](http://en.wikipedia.org/wiki/Ljubljana); he was a student of [[Ieke Moerdijk]] with whom he wrote the monograph

* *Introduction to foliations and Lie groupoids*, Cambridge Studies in Advanced Mathematics, 91. Cambridge University Press, Cambridge, 2003. x+173 pp. ISBN: 0-521-83197-0 ([doi:10.1017/CBO9780511615450](https://doi.org/10.1017/CBO9780511615450))

His interests include [[Lie groupoids]], [[Lie algebroids]], Hopf algebroids, [[orbifolds]] and geometry of foliations.

category: people

[[!redirects Janez Mrcun]]


Janusz E. Jacak

* [InSpire page](https://inspirehep.net/authors/1959330)

## Selected writings

On [[Laughlin wavefunctions]] as [[braid representations]]:

* [[Janusz E. Jacak]], *Formal derivation of the Laughlin function and its generalization for other topological phases of FQHE*, Sci Rep **12** 616 (2022) [[doi:10.1038/s41598-021-04672-z](https://doi.org/10.1038/s41598-021-04672-z)]

category: people

[[!redirects Janusz Jacak]]


Janusz Grabowski

* [webpage](https://www.impan.pl/~jagrab/)

## Selected writings

On [[embedding of smooth manifolds into formal duals of R-algebras]]:

* [[Janusz Grabowski]], _Isomorphisms and ideals of the Lie algebras of vector fields_, Inventiones mathematicae volume 50, pages 13–33 (1978) ([doi:10.1007/BF01406466](https://doi.org/10.1007/BF01406466))

* [[Janusz Grabowski]], _Isomorphisms of algebras of smooth functions revisited_, Arch. Math. 85 (2005), 190-196 ([arXiv:math/0310295](https://arxiv.org/abs/math/0310295))

categoru: people


Jarah Evslin

__Jarah Evslin__ is an American [[nlab:string theory|string theorist]].

* [webpage](http://tpcsf.ihep.ac.cn/english/jarah.html)

## Selected writings

On [[Dirac charge quantization]] of [[D-brane charge]] in [[topological K-theory]]:

* [[Jarah Evslin]], _What Does(n't) K-theory Classify?_ ([arXiv:hep-th/0610328](https://arxiv.org/abs/hep-th/0610328))

On the suggestion that an actual [[E8]]-[[principal bundle]] on 11d spacetime plays a role in the [[shifted C-field flux quantization]]:

* Allan Adams, [[Jarah Evslin]], _The loop group of $E_8$ and K-theory from 11d_, JHEP 02 (2003) ([arXiv:hep-th/0203218](https://arxiv.org/abs/hep-th/0203218))

## Related $n$Lab entries

* [[D-brane]]

* [[topological T-duality]]

* [[twisted K-theory]]

* [[black holes in string theory]]

category: people


Jared Weinstein

* [webpage](http://math.bu.edu/people/jsweinst/)

## related $n$Lab entries

* [[Rapoport-Zink space]]

* [[shtuka]]

category: people


Jarl G. Taxerås Flaten

* [personal page](https://publish.uwo.ca/~jtaxers/)

## Selected writings and talks

Formalization of [[abelian category|abelian]] ([[Grothendieck category|Grothendieck]]-) [[univalent categories]] of [[ring]]-[[modules]], in [[homotopy type theory]] ([[univalent foundations of mathematics]]):

* [[Jarl G. Taxerås Flaten]], *Univalent categories of modules* [[arXiv:2207.03261](https://arxiv.org/abs/2207.03261)]

On [[delooping]], [[H-spaces]] and [[Eilenberg-MacLane spaces]] in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

* [[Jarl G. Taxerås Flaten]], *Central H-spaces and banded types*, [[Homotopy Type Theory Electronic Seminar Talks]], 17 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Flaten-2022-10-17-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=-7PG2bzseJI)).

category: people

[[!redirects Jarl Flaten]]
[[!redirects J. G. T. Flaten]]


Jarosław Adam Miszczak

* [personal page](https://miszczak.eu/)

## Selected writings

On [[quantum programming languages]] ([[quantum circuit diagrams]], [[QML]], ...):

* [[Jarosław Adam Miszczak]], *Models of quantum computation and quantum programming languages*, Bull. Pol. Acad. Sci.-Tech. Sci., Vol. 59, No. 3 (2011), pp. 305-324 ([arXiv:1012.6035](https://arxiv.org/abs/1012.6035))

* [[Jarosław Adam Miszczak]], *High-level Structures for Quantum Computing*, Morgan&Claypool 2012 ([doi:10.2200/S00422ED1V01Y201205QMC006](https://doi.org/10.2200/S00422ED1V01Y201205QMC006), [pdf](https://www.morganclaypool.com/doi/pdf/10.2200/S00422ED1V01Y201205QMC006))

category: people

[[!redirects Jarosław Miszczak]]

[[!redirects Jaroslaw Adam Miszczak]]

[[!redirects Jaroslaw Miszczak]]
[[!redirects Jarek Miszczak]]


Jason Alicea

* [research group page](https://iqim.caltech.edu/profile/jason-alicea/)

## Selected writings

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

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

Introducing [[Majorana dimers]] for [[Ising anyons]]:

* Brayden Ware, Jun Ho Son, [[Meng Cheng]], Ryan V. Mishmash, [[Jason Alicea]], and Bela Bauer, *Ising anyons in frustration-free Majorana-dimer models*, Phys. Rev. B 94, 115127 2016 ([arXiv:10.1103/PhysRevB.94.115127](https://doi.org/10.1103/PhysRevB.94.115127))

On [[parafermions]] and [[topological order]]:

* David J. Clarke, [[Jason Alicea]], Kirill Shtengel . *Exotic non-Abelian anyons from conventional fractional quantum Hall states*, Nature Communications **4** 1348 (2013) $[$[doi: 10.1038/ncomms2340 (2013)](https://www.nature.com/articles/ncomms2340)$]$

* [[Jason Alicea]], [[Paul Fendley]], *Topological phases with parafermions: theory and blueprints*, Annual Review of Condensed Matter Physics **7** (2016) 119-139 $[$[doi:10.1146/annurev-conmatphys-031115-011336](https://doi.org/10.1146/annurev-conmatphys-031115-011336)$]$

category: people


Jason Dusek

I'm a software engineer in San Francisco, CA.

+-- {: .query}
Welcome!  I reversed your edit to [[exponential object]], since we don\'t need (and don\'t have, in $Set$) additive inverses, but it did inspire me to write [[2-rig]], so it was helpful.  ---[[Toby Bartels]]

PS:  If you join us, you might find it helpful to look at, and log your work to, the [Latest Changes](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5) forum.
=--

category: people


Jason Gross

* [MO user page](https://esp.mit.edu/teach/teachers/JasonSGross/bio.html)

* [webpage](https://people.csail.mit.edu/jgross/)

## Selected writings

On [[internal categories in homotopy type theory]]:

* [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], *Experience Implementing a Performant Category-Theory Library in Coq*,  In:  *Interactive Theorem Proving. ITP 2014* Lecture Notes in Computer Science, **8558** Springer (2014)  $[$[arXiv:1401.7694](http://arxiv.org/abs/1401.7694), [doi:10.1007/978-3-319-08970-6_18](https://doi.org/10.1007/978-3-319-08970-6_18)$]$

On [[cryptography]]:

* Andres Erbsen, Jade Philipoom, [[Jason Gross]], Robert Sloan, [[Adam Chlipala]], *Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises*, [2019 IEEE Symposium on Security and Privacy (SP)](https://ieeexplore.ieee.org/xpl/conhome/8826229/proceeding) $[$[doi:10.1109/SP.2019.00005](https://doi.org/10.1109/SP.2019.00005)$]$

category: people


Jason Lotay

* [webpage](http://people.maths.ox.ac.uk/lotay/)

## Selected writings

On [[associative submanifolds]] of the [[7-sphere]]:

* [[Jason Lotay]], _Associative Submanifolds of the 7-Sphere_, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 ([arXiv:1006.0361](https://arxiv.org/abs/1006.0361))

## Related $n$Lab entries

* [[7-sphere]]

* [[G2-structure]]

category: people


Jason Parker

* [Homepage](https://www.jasonparkermath.com/)

## Selected writings

On [[exponential objects|exponentiability]] in categories of [[relational structures]]:

* [[Jason Parker]], *Exponentiability in categories of relational structures*  (2022) [[arXiv:2208:07350](https://arxiv.org/abs/2208.07350)]

category: people


Jason Pollack

* [Simons Foundation page](https://www.simonsfoundation.org/people/jason-pollack/)

## Selected writings

On the [[Ryu-Takayanagi formula]]:

* [[Scott Aaronson]], [[Jason Pollack]], *Discrete Bulk Reconstruction* [[arXiv:2210.15601](https://arxiv.org/abs/2210.15601)]

category: people


Jasper Hugunin

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

## Selected writings

On [[higher inductive-inductive types]] in [[cubical type theory]]:

* {#Hugunin19} [[Jasper Hugunin]], _Constructing Inductive-Inductive Types in Cubical Type Theory_, in *Foundations of Software Science and Computation Structures. FoSSaCS 2019*, Lecture Notes in Computer Science **11425** (2019) [[doi:10.1007/978-3-030-17127-8_17](https://link.springer.com/chapter/10.1007/978-3-030-17127-8_17)]

On  [[W-types]]:

* [[Jasper Hugunin]], *Why Not W?*, Leibniz International Proceedings in Informatics (LIPIcs) **188** (2021) [[doi:10.4230/LIPIcs.TYPES.2020.8](https://doi.org/10.4230/LIPIcs.TYPES.2020.8), [pdf](https://jashug.github.io/papers/whynotw.pdf)]

* [[Jasper Hugunin]], _IWTypes_, <https://github.com/jashug/IWTypes>

category:  people


Jaume Garriga

* [webpage](http://www.ffn.ub.es/gcg/personal/jaume.html)

## Selected writings

On [[false vacuum eternal inflation]]:

* {#GarrigaGuthVilenkin06} [[Jaume Garriga]], [[Alan Guth]], [[Alexander Vilenkin]], _Eternal inflation, bubble collisions, and the persistence of memory_, Phys. Rev. D76:123512, 2007 ([arXiv:hep-th/0612242](https://arxiv.org/abs/hep-th/0612242), [doi:10.1103/PhysRevD.76.123512](https://doi.org/10.1103/PhysRevD.76.123512))

category: people


Jaume Gomis

* [webpage](http://www.perimeterinstitute.ca/people/jaume-gomis)

## Selected writings

On the [[mass-deformed ABJM model]] and the [[Myers effect]] for [[M2 branes]] ([[M2-M5 brane bound states]]):

* [[Jaume Gomis]], Diego Rodriguez-Gomez, [[Mark Van Raamsdonk]], [[Herman Verlinde]], _A Massive Study of M2-brane Proposals_, JHEP 0809:113, 2008 ([arXiv:0807.1074](https://arxiv.org/abs/0807.1074))

category: people


Javad Shabani

* [research lab page](https://wp.nyu.edu/shabanilab/)

* [institute page](https://as.nyu.edu/faculty/javad-shabani.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=hg_9iHcAAAAJ&hl=de&oi=ao)

## Selected writings

On [[topological superconductors]] possibly with [[Majorana zero modes]] for [[topological quantum computation]]:

* {#ShabaniEtAl15} [[Javad Shabani]] et al., *Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks*, Phys. Rev. B **93** 155402 (2016) \[doi:10.1103/PhysRevB.93.155402, [arXiv:1511.01127](https://arxiv.org/abs/1511.01127)\]

* {#ShabaniEtAl17} [[Javad Shabani]] et al., *Zero-Energy Modes from Coalescing Andreev States in a Two-Dimensional Semiconductor-Superconductor Hybrid Platform*, Phys. Rev. Lett. **119** (2017) 176805 [[doi:10.1103/PhysRevLett.119.176805](https://doi.org/10.1103/PhysRevLett.119.176805), [arXiv:1703.03699](https://arxiv.org/abs/1703.03699)]

* {#ShabaniEtAl22} [[Javad Shabani]] et al., *Fusion of Majorana Bound States with Mini-Gate Control in Two-Dimensional Systems*,  	Nature Communications **13** (2022) 1738-1747 \[doi:10.1038/s41467-022-29463-6, [arXiv:2101.09272](https://arxiv.org/abs/2101.09272)\]

* {#ShabaniEtAl23} [[Javad Shabani]] et al., *Quasiparticle dynamics in epitaxial Al-InAs planar Josephson junctions*, PRX Quantum **4** 030339 (2023) \[doi:10.1103/PRXQuantum.4.030339, [arXiv:2303.04784](https://arxiv.org/abs/2303.04784)\]

caqtegory: people


Javier Fuentes-Martin

## Selected writings

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

* {#CFMVV17} Alejandro Celis, [[Javier Fuentes-Martin]], Avelino Vicente, [[Javier Virto]], _Gauge-invariant implications of the LHCb measurements on Lepton-Flavour Non-Universality_, Phys. Rev. D 96, 035026 (2017) ([arXiv:1704.05672](https://arxiv.org/abs/1704.05672))

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

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

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

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

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

and in relation to [[axions]]:

* [[Javier Fuentes-Martin]], Mario Reig, Avelino Vicente, _4321... axion!_ ([arXiv:1907.02550](https://arxiv.org/abs/1907.02550))

On [[flavour anomalies]] via [[leptoquarks]] in [[Pati-Salam GUT]]$\,$[[intersecting D-brane model|on branes]]:

* [[Javier Fuentes-Martin]], [[Gino Isidori]], Julie Pagès, Ben A. Stefanek,
*Flavor Non-universal Pati-Salam Unification and Neutrino Masses*, Physics Letters B Volume 820, 10 September 2021, 136484 ([arXiv:2012.10492](https://arxiv.org/abs/2012.10492), [talk recording](https://www.youtube.com/watch?v=ko4R8S8XYV4&t=2s))

category: people


Javier Gutiérrez

* [webpage](http://www.math.ru.nl/~gutierrez/)

## Selected writings

On generalizing the notion of [[solid rings]] to [[monoids in a monoidal category]]:

* {#Gutierrez13} [[Javier J. Gutiérrez]], *On solid and rigid monoids in monoidal categories*, Applied Categorical Structures 23, no. 2 (2015), 575-589 ([arXiv:1303.5265](https://arxiv.org/abs/1303.5265))

On [[Hurewicz cofibrations]]:

* [[Javier Gutiérrez]], *Cofibrations*, ([pdf](https://www.math.ru.nl/~gutierrez/files/Lecture08.pdf), [[Gutierrez_Cofibrations.pdf:file]])

## Related $n$Lab entries

* [[dendroidal set]]

category: people

[[!redirects Javier Gutierrez]]

[[!redirects Javier J. Gutiérrez]]
[[!redirects Javier J. Gutierrez]]


Javier Virto

* [Institute page](http://icc.ub.edu/people/600)

* [spire page](https://inspirehep.net/authors/1032745)

## Selected writings

On [[flavour anomalies]]:

* {#CCDGMV17} Bernat Capdevila, [[Andreas Crivellin]], Sébastien Descotes-Genon, Joaquim Matias, [[Javier Virto]], _Patterns of New Physics in $b \to s \ell^+ \ell^-$ transitions in the light of recent data_, JHEP 1801 (2018) 093 ([arXiv:1704.05340](https://arxiv.org/abs/1704.05340))

* {#CFMVV17} Alejandro Celis, Javier Fuentes-Martin, Avelino Vicente, [[Javier Virto]], _Gauge-invariant implications of the LHCb measurements on Lepton-Flavour Non-Universality_, Phys. Rev. D 96, 035026 (2017) ([arXiv:1704.05672](https://arxiv.org/abs/1704.05672))

* Marcel Algueró, Bernat Capdevila, [[Andreas Crivellin]], Sébastien Descotes-Genon, Pere Masjuan, Joaquim Matias, [[Javier Virto]], _Addendum: "Patterns of New Physics in $b \to s \ell^+ \ell^-$ transitions in the light of recent data"_ ([arXiv:1903.09578](https://arxiv.org/abs/1903.09578))

* {#Virto21} [[Javier Virto]], _Anomalies in $b \to s \ell \ell$ transitions and Global Fits_ ([arXiv:2103.01106](https://arxiv.org/abs/2103.01106))

with emphasis on [[hadron|hadronic]] [[form factors]]:

* {#DGKV19} Sébastien Descotes-Genon, Alexander Khodjamirian, [[Javier Virto]], _Light-Cone Sum Rules for $B \to K \pi$ Form Factors and Applications to Rare Decays_ ([arXiv:1908.02267](https://arxiv.org/abs/1908.02267))

category: people


Jay McCarthy

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

## related $n$Lab entry

* [[function monad]], [[random variable]]

category: people


Jay Sau

* [personal page](http://www2.physics.umd.edu/~jaydsau/)

* [institute page](https://umdphysics.umd.edu/people/faculty/current/item/454-jaydsau.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=oACq8zIAAAAJ&hl=en)

## Selected writings

On [[topological quantum computation]]:

* [[Jay Sau]], *A Roadmap for a Scalable Topological Quantum Computer*, [Physics **10** 68 (2017)](https://physics.aps.org/articles/v10/68)

category: people


Jay Shah

* [personal page](http://math.mit.edu/~jshah/)

## Selected writings

On [[equivariant homotopy theory]], [[G-CW-complexes]] and the [[equivariant Whitehead theorem]]:

* [[Jay Shah]], _Equivariant algebraic topology_, 2010 ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Shah.pdf), [[ShahEquivariantAlgebraicTopology.pdf:file]])

category: people


JBW-algebraic quantum mechanics

# $JBW$-algebras in quantum mechanics
* table of contents
{: toc}

## Idea

It can be nice to describe the [[kinematics]] of a [[quantum system]] using a [[JBW-algebra]].  (Then some extra structure is needed for [[dynamics]]; see [below] (#dynamics).)

## Motivation

$JBW$-algebras are arrived at by the confluence of several lines of motivation.

To begin with, the algebraic approach to quantum mechanics (which is also the basis of [[AQFT]]) is a common generalization of [[classical mechanics]] using [[phase spaces]] (or even [[Poisson manifolds]] or even [[Poisson algebras]]) and [[quantum mechanics]] using [[Hilbert spaces]].  In this approach, we have an [[algebra of observables]], of which special cases are the [[commutative algebra]] underlying a Poisson algebra (or its [[complexification]]) and the [[algebra of operators]] on a Hilbert space.  This also allows one to apply [[superselection rules]] at a fundamental level by restricting to a [[subalgebra]].

More particularly, one may restrict to _bounded_ observables, that is those for which there is a largest possible absolute value that could be observed.  There is a philosophical reason for this, that no measuring device in reality is unbounded; but it also makes the algebra more tractable, and an arbitrary observable $O$ can still be described by a [[sequence]] of bounded observables with increasing cut-offs.

At this point, it\'s natural to expect the algebra of observables to be a [[Banach space]] (and so, being an algebra, some kind of [[Banach algebra]]), with the [[norm]] giving this largest possible absolute value.  In fact, [[complex number|complex]] $C^*$-[[C*-algebra|algebras]] are usually used; both the algebra of bounded [[continuous functions]] on a Poisson manifold (or indeed any [[local compactum]]) and the algebra of [[bounded operators]] on a Hilbert space are $C^*$-algebras.

Then we may further refine from $C^*$-algebras to $W^*$-[[W*-algebra|algebras]].  This actually amounts to generalizing the notion of observable.  Whereas a typical commutative $C^*$-algebra is an algebra of [[continuous functions]], a typical commutative $W^*$-algebra is an algebra of [[measurable functions]].  This is necessary if we want to include [[characteristic functions]], reflecting an observation of whether the system is in some [[measurable subset]] of phase space.  (If it is on the boundary, then we won\'t be able to tell; this is reflected in that the $W^*$-algebra of functions identifies functions that are [[almost equality|equal almost everywhere]].)  The [[functional calculus]] of $W^*$-algebras also allows for [[composition]] of an observable with a measurable function on its [[spectrum]], allowing us to (for example) turn continuous observables into less-precise discrete ones.  Of course, the algebra of bounded operators on a Hilbert space is also a $W^*$-algebra.  Finally, the algebra of observables that respect a class of superselection rules will also typically form a $W^*$-algebra.

There is, however, an objection to all of these algebras of observables, which is that they have many elements that are not observable.  Only the [[self-adjoint elements]] of these algebras are considered to be observable.  (Even if we allow for complex-valued observables, only the [[normal element]]s can be observables.)  By themselves, the self-adjoint operators form a [[Jordan algebra]].  Instead of recovering the classical case when the algebra of operators is [[commutative algebra|commutative]], we recover it when the Jordan algebra of self-adjoint operators is [[associative algebra|associative]] (and specifically we get the classical algebra of real-valued observables immediately).  Thus, the algebra of observables is a Jordan algebra, even if it is viewed as embedded within a larger algebra.

The analytic structure on a Jordan algebra that makes it the algebra of self-adjoint elements in a $W^*$-algebra is captured in the definition of a [[JW-algebra]] (Jordan $W$-algebra, which may be read as a Jordan $W^*$-algebra with trivial [[involution]]) or [[JBW-algebra]] (Jordan--Banach $W$-algebra, which allows exceptional cases such as the [[Albert algebra]]).  This is a loss of [[extra structure|structure]] from the full $W^*$-algebra; in fact, a complex $W^*$-algebra can be recovered from a $JBW$-algebra by giving it an additional compatible [[Lie algebra]] structure, making it into a [[JLBW-algebra]], which (as explained there) is equivalent to a complex $W^*$-algebra.  However, the $JBW$-algebra already identifies what the observables and states are, so it is a sufficient place to start from.

## Description

>This is hastily copied from elsewhere and minimally edited. More work should be done to spell this out.

We have:

* a [[Banach space]] $A$ whose elements are thought of as (bounded, real-valued) [[observables]];
* a [[Banach space]] $A_*$, some of whose norm-$1$ elements (those to be identified later as 'positive') are thought of ([[normal state|normal]], [[mixed state|mixed]]) [[states]];
* with $A$ expressed as the [[dual Banach space|dual]] of $A_*$;
* with a commutative (but [[nonassociative algebra|nonassociative]]) multiplication operation on $A$;
* with the resulting nonassociative [[Banach algebra]] $A$ satisfying a few technical conditions:
  * [[power-associative algebra|power-associativity]]: all ways of parenthesizing $x^n$, for $x$ an element of $A$ and $n$ a [[natural number]], are equal;
  * positivity: ${\|x^2\|} \leq {\|x^2 + y^2\|}$; and
  * the [[B-identity]]: ${\|x^2\|} = {\|x\|^2}$.

In particular, the positivity allows us to define a [[partial order]] on $A$ according to which an element of $A$ is (weakly) _positive_ if it is a limit of sums of squares, and then an element $\psi$ of $A_*$ is _positive_ (and hence a _normal state_ if it has norm $1$) if every positive element $x$ of $A$ takes $\psi$ to a nonnegative real number.

If the multiplication on $A$ happens to be [[associative algebra|associative]], then there exists a [[measure space]] $X$ such that $A_*$ is the [[Lebesgue space]] $L^1(X)$ and $A$ is $L^\infty(X)$.  It is always possible to take $X$ to be [[localizable measure space|localizable]], in which case the [[Radon–Nikodym theorem]] applies and we can identify $L^1(X)$ with the space of [[absolutely continuous measures]] on $X$.  At this point, the measure on $X$ is irrelevant except for its specification of [[full set|full]] (or [[null set|null]]) subsets, so we may treat $X$ as simply a [[localizable measurable space]] (which includes this data).  Then the choice of $X$ is actually [[the|essentially unique]].

In this way, the associative case reduces to [[probability theory]].  We have a measurable space $X$, an observable is an [[essentially bounded function]], and a state is a [[probability measure]].  It is true that we only have localizable spaces, but these are the only ones that satisfy the nice theorems (such as Radon--Nikodym, aforementioned) anyway.  (Similarly, the observables are defined only up to [[almost equality]], and the normal states must be [[absolutely continuous measures]].)

In the general (nonassociative) case, we still think of the elements of $A$ (the observables) as essentially bounded functions, but now on a sort of [[noncommutative geometry|noncommutative]] (or rather nonassociative) space, and we may still think of the elements of $A_*$ (the states) as probability measures on that space.  (This is particularly appropriate in the [[Bayesian interpretation of quantum mechanics]].)  Because $A$ is a Jordan algebra instead of an associative algebra, *all* of its elements may be thought of as observables, but this is more of a convenience than an essential feature.

We might also want to consider generalized (non-normal) [[state on an operator algebra|states]], that is norm-$1$ [[positive operator|positive]] [[linear functionals]] on $A$.  (Every state in $A_*$ may be so interpreted, using the natural map $A_* \to (A_*)^{**} = A^*$).  One may go further and consider [[quasistate]]s on $A$, if any exist (besides the states themselves).

In particular, if $A$ is the space of [[bounded operators|bounded]] [[self-adjoint operators]] on some [[Hilbert space]] $H$, then $A_*$ is the space of [[trace-class operator|trace-class]] self-adjoint operators on $H$.  Then a normal state is a [[density matrix]], and the pure states (those that cannot be written as nontrivial [[convex combinations]] of other states) are those of the form ${|\psi\rangle \langle\psi|}$ for $\psi$ a unit vector in $H$.

## Dynamics {#dynamics}

We can add [[dynamics]] by giving the $JBW$-algebra the structure of a [[pointed object|pointed]] [[JLBW-algebra]], that is by defining a kind of [[commutator]] (the 'L') and identifying one of the operators in the algebra as the [[Hamiltonian]] (the 'pointed').  This is mathematically equivalent to a [[complex numbers|complex]] $W^*$-[[W*-algebra|algebra]] equipped with a [[self-adjoint element]].

## Related concepts

* [[interpretation of quantum mechanics]]

* [[finite quantum mechanics in terms of dagger-compact categories]]

* [[AQFT]]

[[!redirects JBW-algebras in quantum mechanics]]
[[!redirects JBW-algebras in quantum physics]]
[[!redirects JBW-algebraic approach to quantum mechanics]]
[[!redirects JBW-algebraic approach to quantum physics]][[!redirects JBW-algebraic quantum mechanics]]
[[!redirects JBW-algebraic quantum physics]]


JCMcKeown

[[!redirects J C McKeown]]

jesse dot mckeown at mail dot utoronto dot ca

* [webpage](https://jessecmckeown.tumblr.com/)

Around here causes a small amount of trouble on occasion, which leads to smarter people improving the general quality and coverage of the whole.

## related $n$Lab entries

* [[MU]]

category: people

[[!redirects Jesse McKeown]]


Jean Bénabou

Jean Bénabou (1932-2022) was a French mathematician working in [[category theory]]. He studied under the supervision of [[Charles Ehresmann]] and received his Thèse d'État  from the Université de Paris in 1966 on the topic of what are now called *[[monoidal categories]]*.

* [Wikipedia entry](https://fr.wikipedia.org/wiki/Jean_B%C3%A9nabou)

* [[Francis Borceux]]: *JEAN BÉNABOU (1932–2022): The man and the mathematician*, _[[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques_, **LXIII-3** (2022) [[pdf](http://cahierstgdc.com/wp-content/uploads/2022/07/F.-BORCEUX-LXIII-3.pdf)]

## Selected writings
 {#SelectedWritings}

Introducing the notion of [[monoidal categories]] (then under the name "categories with multiplication" or "multiplicative categories"):

* [[Jean Bénabou]], _Catégories avec multiplication_ , C. R. Acad. Sci. Paris **256** (1963) 1887-1890 [[gallica](http://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1965.image)]

* [[Jean Bénabou]], _Algèbre élémentaire dans les catégories avec multiplication_ , C. R. Acad. Sci. Paris **258** (1964) 771-774 [[gallica](gallica.bnf.fr/ark:/12148/bpt6k40102/f817.image)]

* [[Jean Bénabou]], *Les catégories multiplicatives*, Séminaire de mathématique pure **27**, Université de Louvain (1972) [[[Benabou-CategoriesMultiplicatives.pdf:file]]]

Introducing the notion of [[enriched categories]] (and of [[strict 2-categories]], as an example):

* [[Jean Bénabou]], *Catégories relatives*, C. R. Acad. Sci. Paris, 260 (1965), pp. 3824-3827 ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37.item))

* {#Bénabou68} [[Jean Bénabou]], *Structures algébriques dans les catégories*, Cahiers de topologie et géométrie différentielle, **10** 1 (1968) 1-126 [[doi:CTGDC_1968__10_1_1_0](http://www.numdam.org/item/?id=CTGDC_1968__10_1_1_0)]

On [[bicategories]] (and introducing the terminology *[[monad]]*):

* Jean Bénabou, *Introduction to Bicategories*, Lecture Notes in Mathematics **47** Springer (1967) 1-77 [[doi:10.1007/BFb0074299](http://dx.doi.org/10.1007/BFb0074299)]

On [[monadic descent]] via the [[Beck-Chevalley condition]], the [[monadicity theorem]] and introducing the [[Bénabou-Roubaud theorem]]:

* {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270**  (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]],   English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)]

On [[monads]]:

* [[Jean Bénabou]], *Les Triples*, Séminaire de mathématiquepure pure **26**, Université de Louvain (1972) [[[Benabou-LesTriples.pdf:file]]]

On [[toposes]]:

* [[Jean Bénabou]], *Problèmes dans les topos*, Séminaire de mathématiquepure pure **34**, Université de Louvain (1973) [[[Benabou-ProblemesDansLesTopos.pdf:file]]]

On [[profunctors]] ("[[distributors]]", including for [[enriched categories|enriched]] and [[internal categories]]):

* {#Benabou73} [[Jean Bénabou]], *Les distributeurs*, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée,  rapport **33** (1973) [[pdf](http://www.entretemps.asso.fr/maths/DistributeursLouvain.pdf), [[Benabou-LesDistributeurs.pdf:file]]]

On [[fibered categories]]:

* {#Bénabou1975} Jean Bénabou, _Fibrations petites et localement petites_, C. R. Acad. Sci. Paris **281** Série A (1975) 897-900 [[gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f171.image#)]

* {#Bénabou1985} [[Jean Bénabou]], *Fibered Categories and the Foundations of Naive Category Theory*,  The Journal of Symbolic Logic, Vol. **50** 1 (1985) 10-37 [[doi:10.2307/2273784](http://dx.doi.org/10.2307/2273784)]

See also:

* {#Streicher18} [[Thomas Streicher]], _Fibred Categories à la Jean Bénabou_,  [[arXiv:1801.02927](https://arxiv.org/abs/1801.02927)]

On [[foliated categories]]:

* [[Jean Bénabou]], _Cartesian functors and foliated categories_, talk at Oxford (1 May 2012) [[YouTube](https://www.youtube.com/watch?v=Y1CeHmjNcuk)]

* [[Jean Bénabou]], _Foncteurs cartésiens et catégories feuilletées_, talk at *Journée Guitart*, Paris (9 November 2012) [[YouTube](https://www.youtube.com/watch?v=p3vcZDyZ4Yo), [slides](https://docs.google.com/presentation/d/17E0epsZWcDTYlwAMZ4PFCLIf4dyigS-WUA0siUtU-wY/edit?hl=fr#slide=id.gce595186_059), [[Benabou-FoliatedCategories.pdf:file]]]

* [[Jean Bénabou]], _Du vieux et du neuf sur la construction de Grothendieck_, talk at Paris-Diderot (March 2019) [[YouTube](https://youtu.be/NC0I18B45kE)]

  > (the material on foliated categories -- called simply foliations --- starts at 1:01:00).

See also:

* J. Bénabou, _Théories relatives à un corpus_, C. R. Acad. Sci. Paris **281** Série A (1975) 831-834. [[gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f105.image#)]

## Related entries

* [[bicategory]]

* [[Bénabou-Roubaud theorem]]

* [[internal language]] of a [[topos]]
  ([[Mitchell-Bénabou language]])

* [[distributor]]

* [[Grothendieck fibration]]

* [[Bénabou cosmos]]

* [[foliated category]]

* [[displayed category]]

category: people

[[!redirects Jean Benabou]]
[[!redirects Jean Bénabou]]
[[!redirects J. Bénabou]]
[[!redirects Benabou]]
[[!redirects Bénabou]]


Jean Celeyrette

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

## Selected writings

On [[internal categories]] and their [[Grothendieck fibrations]] and on [[Gelfand-Fuks cohomology]]:

* [[Jean Celeyrette]], *Catégories internes et fibrations* & *Cohomologie de Gel'fand-Fuks*, PhD thesis, Paris (1975) [[pdf](/nlab/files/celeyrette-thesis-1975.pdf)]

category: people


Jean Dieudonné

_Jean Alexandre Dieudonné_ was a French mathematician, one of the most prominent members of the group Bourbaki. He helped Grothendieck in writing [[EGA]] and, together with [[Laurent Schwartz]], advised Grothendieck in his early (pre-thesis) research in analysis. Apart from Bourbaki activities, Dieudonné was prolific in expository writing, having a rare "photographic" memory and high working habits (writing steadily at least 5 pages per day in the best period), immense erudition and strong attitude on all topics. Comparatively he was much less productive in original  research - however most important is his development of the study of (Cartier-)[[Dieudonné module]]s arising in theory of formal groups. There is a theorem in [[general topology]] having his name.
He is also a noted historian of mathematics, producing works on history of algebraic/differential topology, algebraic geometry, and functional analysis, also some entries in DSB (notably Poincare and Cartan).

* [wikipedia](http://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9)

## Selected writings

Introducing the [[Dieudonné determinant]]:

* [[Jean Dieudonné]], _Les déterminants sur un corps non commutatif_,  Bulletin de la Société Mathématique de France, Volume 71  (1943), p. 27-45  ([numdam:BSMF_1943__71__27_0](http://www.numdam.org/item/?id=BSMF_1943__71__27_0))

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

* [[Jean Dieudonné]], _A History of Algebraic and Differential Topology, 1900 - 1960_, Modern Birkhauser Classics 2009 ([ISBN:978-0-8176-4907-4](https://www.springer.com/de/book/9780817649067))

category:people

[[!redirects Jean Alexandre Dieudonné]]
[[!redirects Jean Dieudonne]]

[[!redirects Dieudonné]]
[[!redirects Dieudonne]]


Jean Fasel

* [personal page](https://www-fourier.ujf-grenoble.fr/~faselj/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=90942)

## Selected writings

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

category: people


Jean Frenkel

## Selected writings

On [[nonabelian cohomology|non-abelian]] [[Čech cohomology]] and [[fiber bundles]]:

* [[Jean Frenkel]], *Cohomologie à valeurs dans un faisceau non abélien*, C. R. Acad. Se., t. 240 (1955) 2368-2370

* [[Jean Frenkel]], *Cohomologie non abélienne et espaces fibrés*, Bulletin de la Société Mathématique de France, **85** (1957) 135-220 [[numdam:BSMF_1957__85__135_0](http://www.numdam.org/item/BSMF_1957__85__135_0)]

category: people


Jean Gallier

* [webpage](http://www.cis.upenn.edu/~jean/home.html)

## Selected writings

On [[differential geometry]] and [[Lie groups]]:

* {#GallierQuaintance20a} [[Jean Gallier]], [[Jocelyn Quaintance]],  *Differential Geometry and Lie Groups -- A computational perspective*, Geometry and Computing **12**, Springer (2020) [[doi:10.1007/978-3-030-46040-2](https://doi.org/10.1007/978-3-030-46040-2), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)]

* {#GallierQuaintance20b} [[Jean Gallier]], [[Jocelyn Quaintance]],  *Differential Geometry and Lie Groups -- A second course*, Geometry and Computing **13**, Springer (2020) [[doi:10.1007/978-3-030-46047-1](https://doi.org/10.1007/978-3-030-46047-1), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)]

On [[ordinary homology]], [[ordinary cohomology]] and [[abelian sheaf cohomology]] in [[algebraic topology]], [[algebraic geometry]] and [[differential geometry]]:

* [[Jean Gallier]], [[Jocelyn Quaintance]], *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) [[doi:10.1142/12495](https://doi.org/10.1142/12495), [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)]

## Related entries

* [[Clifford algebra]]

category: people


Jean Giraud

[[!redirects J. Giraud]]

Jean Giraud (1936 – March 28, 2007) was a French mathematician, who worked on the theory of toposes and also, extensively, on non-abelian cohomology.

* [Wikipedia entry](http://en.wikipedia.org/wiki/Jean_Giraud_%28mathematician%29)

## Selected writings

Introducing [[stacks]] ([[2-sheaves]]) and [[non-abelian cohomology]]:

* [[Jean Giraud]], _Cohomologie non abélienne, Grundlehren 179, Springer Verlag (1971) ([doi:10.1007/978-3-662-62103-5](https://www.springer.com/gp/book/9783540053071))

* [[Jean Giraud]], _Classifying topos_, in: [[William Lawvere]] (ed.) _Toposes, Algebraic Geometry and Logic_, Lecture Notes in Mathematics, vol 274. Springer (1972) ([doi:10.1007/BFb0073964](https://doi.org/10.1007/BFb0073964))

## Related entries

* [[Giraud theorem]]

* [[nonabelian cohomology]]

* [[gerbe]]

* [[2-sheaf]]

category: people


Jean Goubault-Larrecq

* [personal page](http://www.lsv.fr/~goubault/)

* [GoogleScholar page](https://scholar.google.fr/citations?user=-18V_JUAAAAJ)

## Selected writings

On [[S4 modal logic]]:

* [[Jean Goubault-Larrecq]], [[Éric Goubault]], _On the geometry of intuitionistic S4 proofs_, Homology, Homotopy and Applications **5** 2 (2003) 137-209 [[eudml:50340](https://eudml.org/doc/50340), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-5/issue-2/On-the-geometry-of-intuitionistic-S4-proofs/hha/1088453324.full)]

On [[monads in computer science]] (as [[strong monads]] and [[commutative monad|commutative]] [[monoidal monads]]):

* {#GLLN08} [[Jean Goubault-Larrecq]], [[Slawomir Lasota]], [[David Nowak]], *Logical Relations for Monadic Types*, Mathematical Structures in Computer Science, **18** 6 (2008) 1169-1217 [[arXiv:cs/0511006](https://arxiv.org/abs/cs/0511006), [doi:10.1017/S0960129508007172](https://doi.org/10.1017/S0960129508007172)]

category: people


Jean Leray

Jean Leray was a mathematician at Collège de France.
He got his PhD degree in 1933 from École Normale Supérieure de Lyon, supervised by Henri Villat.

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

## Selected writings

The paper that introduced [[sheaves]]:

* [[Jean Leray]], _L’anneau d’homologie d’une représentation_. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946), 1366–1368. [[Leray-0.pdf:file]]

The paper that introduced [[spectral sequences]]:

* [[Jean Leray]], _Structure de l’anneau d’homologie d’une représentation_, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946) 1419–1422.

category: people