J. Barkley Rosser

__J. Barkley Rosser__ (1907-1989) was an American mathematical logician, a former PhD student of [[Alonzo Church]].

* wikipedia [J. Barkley Rosser](https://en.wikipedia.org/wiki/J._Barkley_Rosser)

He is known for Church-Rosser theorem ([wikipedia](https://en.wikipedia.org/wiki/Church%E2%80%93Rosser_theorem)) in $\lambda$-[[lambda calculus|calculus]], Rosser sieve in number theory, Rosser's trick (a strengthening of the Goedel's incompleteness theorem, [wikipedia](https://en.wikipedia.org/wiki/Rosser%27s_trick)) and the  Kleene-Rosser paradox  ([wikipedia](https://en.wikipedia.org/wiki/Kleene%E2%80%93Rosser_paradox)).

He wrote the monograph

* J. Barkley Rosser, _Logic for mathematicians_, 1954

where we can also find a formalization of [[rule C]].

category: people


J. C. Hurtubise

## Selected writings

On [[homotopy type of spaces of rational maps]]:

* [[Charles P. Boyer]], [[B. M. Mann]], [[J. C. Hurtubise]], [[R. J. Milgram]], *The topology of the space of rational maps into generalized flag manifolds*, Acta Mathematica. 1994;173(1):61-101 ([doi:10.1007/BF02392569](https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full))

* [[J. C. Hurtubise]], *Holomorphic maps of a Riemann surface into a flag manifold*, J. Differential Geom. 43(1): 99-118 (1996) ([doi:10.4310/jdg/1214457899](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-43/issue-1/Holomorphic-maps-of-a-Riemann-surface-into-a-flag-manifold/10.4310/jdg/1214457899.full))

category: people


J. Daniel Christensen

[[!redirects Dan Christensen]]

Dan Christensen is a mathematician at the University of Western Ontario.

* [website](http://jdc.math.uwo.ca/)

## Selected writings

On [[model structures on chain complexes]] in [[homological algebra]]:

* [[Dan Christensen]], [[Mark Hovey]], _Quillen model structures for relative homological algebra_, Math. Proc. Cambridge Philos. Soc. **133** 2 (2002) 261-293 [[arXiv:math/0011216](https://arxiv.org/abs/math/0011216), [doi:10.1017/S0305004102006126](https://doi.org/10.1017/S0305004102006126)]

On [[Cayley-Dickson algebras]]:

* Daniel K. Biss, [[J. Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras II_, 	Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 ([arxiv:math/0702075](https://arxiv.org/abs/math/0702075))

* Daniel K. Biss, [[J. Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Eigentheory of Cayley-Dickson algebras_,
Forum Mathematicum 21(5) (2009), 833-851 ([arxiv:0905.2987](https://arxiv.org/abs/0905.2987))

On the [[D-topology]] on [[diffeological spaces]]:

* {#CSW13} [[J. Daniel Christensen]], [[Gord Sinnamon]], [[Enxin Wu]], _The $D$-topology for diffeological spaces_, Pacific Journal of Mathematics 272 (1), 87-110, 2014 ([arXiv:1302.2935](http://arxiv.org/abs/1302.2935), [doi:10.2140/pjm.2014.272.87](https://msp.org/pjm/2014/272-1/p04.xhtml))

On the [[homotopy theory]] of [[diffeological spaces]] via their [[cohesion|cohesive]] [[shape modality|shape]]:

* [[J. Daniel Christensen]],  [[Enxin Wu]], _The homotopy theory of diffeological spaces_, New York J. Math. 20
(2014), 1269-1303 ([arXiv:1311.6394](http://arxiv.org/abs/1311.6394))

On [[localization of a space|localization]] (in [[algebraic topology]]) formulated in [[modal homotopy type theory|modal]] [[homotopy type theory]]:

* {#CORS20} [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _[[Localization in Homotopy Type Theory]]_, Higher Structures, 4(1) (2020), 1-32 ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

Proof of the [[Hurewicz theorem]] in [[homotopy type theory]], hence in general [[(∞,1)-toposes]]:

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

On [[modal homotopy type theory]]:

* {#ChristensenRijke20} [[J. Daniel Christensen]], [[Egbert Rijke]], _Characterizations of modalities and lex modalities_ ([arXiv:2008.03538](https://arxiv.org/abs/2008.03538))

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)]

## Related $n$Lab entries

* [[model category]]

* [[model structure on chain complexes]]

category: people

[[!redirects Daniel Christensen]]

[[!redirects John Daniel Christensen]]


J. Roger Hindley

* [personal page](http://www.users.waitrose.com/~hindley/)

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

## Selected writings

On [[lambda-calculus]] and [[combinator logic]]:

* [[J. Roger Hindley]], [[Jonathan P. Seldin]], *Lambda-Calculus and Combinators -- An Introduction*, Cambridge University Press (2008) [[doi:10.1017/CBO9780511809835](https://doi.org/10.1017/CBO9780511809835)]

On [[simple type theory]]:

* [[J. Roger Hindley]], *Basic Simple Type Theory*, Cambridge University Press (1997, 2009) [[doi:10.1017/CBO9780511608865](https://doi.org/10.1017/CBO9780511608865)]

category: people

[[!redirects Roger Hindley]]


J. W. S. Cassels

* [Wikipedia entry](https://en.wikipedia.org/wiki/J._W._S._Cassels)

## Selected writings

On [[algebraic number theory]]:

* {#FroehlichCassels67} [[J. W. S. Cassels]], [[Albrecht Fröhlich]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

and [[local fields]]:

* {#Cassels86} [[J. W. S. Cassels]], *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))

## Related $n$Lab entries

* [[valuation]]

* [[algebraic integer]]

category: people

[[!redirects J.W.S Cassels]]
[[!redirects John Cassels]]
[[!redirects John W. S. Cassels]]
[[!redirects John William Scott Cassels]]


J.H.C. Whitehead

John Henry Constantine Whitehead (11 November 1904–8 May 1960), was one of the founders of modern [[algebraic topology]].  Also one of the founders of the influential journal _Topology_.

* [Wikipedia entry](http://en.wikipedia.org/wiki/J._H._C._Whitehead)

* [MacTutor biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Whitehead_Henry.html)

> Not to be confused with [[George Whitehead]], who also worked in [[algebraic topology]].

## Contributions

His contributions to [[mathematics]] include the following:

* [[CW complexes]]; these were introduced in his papers on

* [[combinatorial homotopy]], which in turn were one prime  source of inspiration for [[Ronnie Brown|Brown]], [[Higgins]] and Sivera's, [[Nonabelian Algebraic Topology]] and also for Baues' extensive work on, for instance, [[Combinatorial Homotopy and 4-Dimensional Complexes]];

* [[simple homotopy theory]];

* the [[Whitehead group]] which is a precursor of constructions in algebraic K-theory;

* with [[Saunders Mac Lane]], first characterisation of [[homotopy 2-types]].

* founder of the [[algebraic homotopy]] programme;

* [[Whitehead product]]

* [[Whitehead theorem]]

* he introduced [[crossed modules]]

## Selected writings

On the [[triangulation theorem]] for [[smooth manifolds]]:

* {#Whitehead40} [[J. H. C. Whitehead]], *On $C^1$-complexes*, Ann. of Math. (2) 41 (1940), 809–824 ([doi:10.2307/1968861](https://doi.org/10.2307/1968861), [jstor:1968861](https://www.jstor.org/stable/1968861))

On the [[Hopf invariant]]:

* [[J. H. C. Whitehead]], _An expression of Hopf ’s invariant as an integral_, Proc. Nat. Acad. Sci. U. S. A.33 (1947), 117–123 ([jstor:87688](https://www.jstor.org/stable/87688))

Introducing [[crossed modules]] (and [[crossed homomorphisms]]) as models for [[homotopy types]]:

* [[J. H. C. Whitehead]], _Combinatorial homotopy I_ , Bull. Amer. Math. Soc, 55, (1949), 213–245.

* {#Whitehead49} [[J. H. C. Whitehead]], _Combinatorial Homotopy II_, Bull. Amer. Math. Soc., 55, (1949), 453 – 496 ([euclid:1183513797](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-5/Combinatorial-homotopy-II/bams/1183513797.full))

On [[homotopy theory]]:

* [[J. H. C. Whitehead]], _Algebraic Homotopy Theory_, in Proc. Int. Cong. of Mathematics, Harvard, volume 2, 354 – 357. 51 (1950)

On [[simple homotopy type]]:

* [[J. H. C. Whitehead]], Simple homotopy type, Amer. J. Math., 72, (1950), 1 - 57.

category: people

[[!redirects JHC Whitehead]]
[[!redirects J H C Whitehead]]
[[!redirects J. H. C. Whitehead]]

[[!redirects John Henry Constantine Whitehead]]
[[!redirects John Henry C. Whitehead]]
[[!redirects John Henry Whitehead]]
[[!redirects John Whitehead]]
[[!redirects Henry Whitehead]]


J.M. Ellis McTaggart

John McTaggart Ellis McTaggart[^name] (1866--1925) was a British [[metaphysics|metaphysician]]. He was part of British [[idealism]], that had originated in German idealism with its key figures [[Kant]], [[Hegel]], [[Fichte]], and [[Schelling]]. He was influential on the young [[Bertrand Russell]].

## related $n$Lab entries

* [[mysticism]]

## References

* [Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/mctaggart/)
* [Wikipedia](https://de.wikipedia.org/wiki/John_McTaggart_Ellis_McTaggart)

[^name]: _John McTaggart_ is really his first name and _Ellis McTaggart_ his last name


Jaap Kalkman

* [webpage](https://rocketreach.co/jaap-kalkman-email_8291006)

## Selected writings

On [[equivariant de Rham cohomology]] and the [[BRST complex]]:

* {#Kalkman93} [[Jaap Kalkman]], _BRST model applied to symplectic geometry_, Ph.D. Thesis, Utrecht, 1993 ([arXiv:hep-th/9308132 (broken)](https://arxiv.org/abs/hep-th/9308132), [cds:9308132](http://cds.cern.ch/record/568522), [pdf](http://cds.cern.ch/record/568522/files/9308132.pdf))

* [[Jaap Kalkman]], _BRST Model for Equivariant Cohomology and  Representatives for the Equivariant Thom Class_, Comm. Math. Phys. Volume 153, Number 3 (1993), 447-463. ([euclid:1104252784](http://projecteuclid.org/euclid.cmp/1104252784))

category: people


Jaap van Oosten

* [website](http://www.staff.science.uu.nl/~ooste110/)

## Selected writings

On [[realizability]]:

* {#vanOosten08} [[Jaap van Oosten]], _Realizability: an introduction to its categorical side_, Studies in Logic and the Foundations of Mathematics, vol. 152, Elsevier, 2008 ([preface pdf](http://www.staff.science.uu.nl/~ooste110/boekbegin.pdf))

On [[Lifschitz realizability]]:

* [[Jaap van Oosten]], *Lifschitz realizability*. The Journal of Symbolic Logic, Volume 55, Issue 2 (1990), pp. 805-821. [[doi:10.2307/2274666](https://doi.org/10.2307/2274666)]

* [[Jaap van Oosten]], *Two remarks on the Lifschitz realizability topos*. The Journal of Symbolic Logic, Volume 61, Issue 1, March 1996, pp. 70-79. [[doi:10.2307/2275598](https://doi.org/10.2307/2275598)]

On [[topos theory]]:

* [[Jaap van Oosten]], *Topos Theory* (2018) [[pdf](https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf), [[vanOosten-ToposTheory.pdf:file]]]

## Related entries

* [[realizability]]
* [[Lifschitz realizability]]

category: people


Jacek Brodzki

* [webpage](http://www.personal.soton.ac.uk/jb32/Jacek_Brodzki/Home.html)

category: people


Jack Hall

* [webpage](http://maths-people.anu.edu.au/~hallj/)

## Related entries

* [[GAGA]]

category: people


Jack Mitchell

* [Institute page](https://www.southampton.ac.uk/people/5xvynb/mr-jack-mitchell)

## Selected writings

On [[domain walls]] and [[holographic QCD]]:

* [[Nick Evans]], [[Jack Mitchell]], *Domain Wall AdS/QCD*, Phys. Rev. D **104** (2021) 094018 [[arXiv:2108.12152](https://arxiv.org/abs/2108.12152), [doi:10.1103/PhysRevD.104.094018](https://doi.org/10.1103/PhysRevD.104.094018)]

* [[Nick Evans]], [[Jack Mitchell]], *Thermal Transitions in Domain Wall AdS/QCD* [[arXiv:2207.10374](https://arxiv.org/abs/2207.10374)]

category: people


Jack Morava

* [Webpage](http://mathnt.mat.jhu.edu/mathnew/Faculty/HomePages/~morava.htm)

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

* [[Michael Hopkins]]: _Talk on Jack Morava's work at Jack's 60th birthday conference_ ([web](http://www.mathematics.jhu.edu/new/moravavideo.htm))

## Selected writings

On [[cobordism cohomology theory]], [[algebraic topology]] and [[chromatic homotopy theory]]:

* [[Jack Morava]], _Complex cobordism and algebraic topology_ ([arXiv:0707.3216](http://arxiv.org/abs/0707.3216))

On [[stable cohomotopy]] as the [[algebraic K-theory]] of [[FinSet]]:

* {#MoravaSomeBackground} [[Jack Morava]], _Some background on Manin's theorem $K(\mathbb{F}_1) \sim \mathbb{S}$_ ([pdf](http://www.alainconnes.org/docs/Morava.pdf), [[MoravaSomeBackground.pdf:file]])

Comments on the analogy between [[power operations]] in [[homotopy theory]] and [[Lambda ring]] structure in [[Borger's absolute geometry]]:

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](http://arxiv.org/abs/math/0612327))

* {#MoravaSanthanam} [[Jack Morava]], Rekha Santhanam, _Power operations and absolute geometry_, 2012 ([pdf](http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf))

* {#MoravaSanthanam12} [[Jack Morava]], Rakha Santhanam, _Power operations and Absolute geometry_, 2012 ([pdf](http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf))

On [[configuration spaces of points]] and a possible [[Galois group]] over the [[sphere spectrum]]:

* {#MoravaBeardsly17} [[Jack Morava]], [[Jonathan Beardsley]], _Toward a Galois theory of the integers over the sphere spectrum_,  Journal of Geometry and Physics Volume 131, September 2018, Pages 41-51 ([arXiv:1710.05992](https://arxiv.org/abs/1710.05992), [doi:10.1016/j.geomphys.2018.04.011](https://doi.org/10.1016/j.geomphys.2018.04.011))

On [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]:

* [[Jack Morava]], _Toward a fundamental groupoid for the stable homotopy category_ ([arXiv:math/0509001](http://arxiv.org/abs/math/0509001))

On the [[cosmic Galois group]]:

* [[Jack Morava]], _The motivic Thom isomorphism_ ([web](http://www.math.uiuc.edu/K-theory/0654/),  [pdf](http://www.math.uiuc.edu/K-theory/0654/gamma2.pdf))

* [[Jack Morava]], _The Cosmic Galois group as Koszul dual to Waldhausen's A(pt)_ ([arXiv:1108.4627](http://arxiv.org/abs/1108.4627))

* {#KitchlooMorava12} [[Nitu Kitchloo]], [[Jack Morava]], _The Grothendieck--Teichmüller group and the stable symplectic category_, 2012.  ([arxiv:1212.6905](http://arxiv.org/abs/1212.6905))

On [[orbifolds]] in [[mathematical physics]] and in particular [[string theory]]:

* [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], _[[Orbifolds in Mathematics and Physics]]_, Contemporary Mathematics 310, American Mathematical Society, 2002

On [[prequantum line bundle|prequantized]] (and [[polarization|polarized]]) [[symplectic manifolds]] in the context of [[cobordism rings]] and [[Thom spectra]]:

* {#Morava99} [[Jack Morava]], _Cobordism of symplectic manifolds and asymptotic expansions_, talk at the conference in honor of S.P. Novikov's 60th birthday ([arXiv:9908070](http://arxiv.org/abs/math/9908070))

On the [[Weinstein symplectic category]]:

* [[Nitu Kitchloo]], [[Jack Morava]], _The Grothendieck--Teichmüller group and the stable symplectic category_, 2012.  ([arxiv:1212.6905](http://arxiv.org/abs/1212.6905))

On [[quotient stacks]]:

* _Theories of anything_ ([arXiv:1202.0684](http://arxiv.org/abs/1202.0684))

On the [[canonical formula of myth]]:

* {#Morava03}[[Jack Morava]], _On the Canonical Formula of C. Lévi-Strauss_ , arXiv.0306174v2 (2003). ([abstract](http://arxiv.org/abs/math/0306174))

* {#Morava04}[[Jack Morava]], _Une interprétation mathématique de la formule canonique de Claude Lévi-Strauss_ , Cahiers de L'Herne **88** (2004) pp.216-218.

* {#Morava05}[[Jack Morava]], _From Lévi-Strauss to Chaos and Complexity_ ,  in Mosko, Damon (eds.), _On the Order of Chaos_ , Berghahn Oxford 2005.

## Related $n$Lab entries

* [[Morava K-theory]]

* [[Morava E-theory]]

category: people


Jack polynomial

__Jack polynomials__ (or Jack symmetric functions) form a class of [[symmetric function|symmetric functions]] $J_\lambda^\alpha$ labelled by a partition $\lambda$ and a parameter $\alpha$; by a result of I. G. Macdonald, they form a family of orthogonal polynomials. Jack polynomials can be defined as eigenfunctions of certain Laplace/Beltrami type operator coming in the theory of [[Calogero integrable system|Calogero integrable systems]] and in [[random matrix theory]]. If $\alpha = 1$ the Jack polynomials become [[Schur polynomial|Schur polynomials]].

* Henry Jack, _A class of symmetric polynomials with a parameter_, Proc. Roy. Soc. Edinburgh Sec. A: Math. Phys. Sci. 69, 1-18, 1969-70, [MR289462](http://www.ams.org/mathscinet-getitem?mr=289462); reprinted in: Contemp. Math. __417__, Jack, Hall-Littlewood and Macdonald polynomials, 57–74, Amer. Math. Soc. 2006
* H. Jack, _A class of polynomials in search of a definition, or the symmetric group parametrized_, in: Jack, Hall-Littlewood and Macdonald polynomials, 75–106, AMS 2006
* Wolframwolrd: [Jack polynomial](http://mathworld.wolfram.com/JackPolynomial.html)
* I. G. Macdonald, _Symmetric functions and Hall polynomials_, Oxford Math. Monographs, 2nd enlarged ed. 1995
* Richard P. Stanley, _Some combinatorial properties of Jack symmetric functions_, Adv. Math. __77__ (1989), no. 1, 76–115, [MR90g:05020](http://www.ams.org/mathscinet-getitem?mr=1014073) doi

[[!redirects Jack polynomials]]


Jack Segal

__Jack Segal__ is working in geometric topology and shape theory. With [[Sibe Mardešić]] he developed an ANR-approach to [[shape theory]] and wrote a monograph on shape theory.

* [webpage](http://www.math.washington.edu/People/fac_individ.php?mathid=segal)

## Selected writings

On [[shape theory]]:

* [[Sibe Mardešić]], [[Jack Segal]], _Shapes of compacta and ANR-systems_, Fund. Math. 72 (1971) 41-59

* {#DydakSegal78} [[Jerzy Dydak]], [[Jack Segal]], _Shape Theory_, Lecture Notes in Mathematics, Springer 1978 ([doi:10.1007/BFb0067572](https://link.springer.com/book/10.1007/BFb0067572))

* [[Sibe Mardešić]], [[Jack Segal]], _Shape theory_, North-Holland 1982

category: people

[[!redirects J. Segal]]


Jackiw-Teitelboim gravity

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

#Contents#
* table of contents
{:toc}

## Idea

_JT-gravity_ ([Teitelboim 83](#Teitelboim83), [Jackiw 85](#Jackiw85)) is [[gravity]] [[2d gravity|in 1+1 dimensions]] with a [[dilaton]].

## Properties

### Relation to near-extremal black holes
 {#RelationToNearExtremalBlackHoles}

JT-gravity gives a good approximation to the [[anti de Sitter spacetime|AdS]]-factor in the [[near horizon geometry]] $AdS_2\times S^{d-2}$ of near-[[extremal black holes]] in $d$-dimensional spacetime
([NSST18](#NSST18), [MTV18](#MTV18)).

Via the [[AdS/CFT]]-dual of JT-gravity ([Almheiri-Polchinski 14](#AlmheiriPolchinski14)) given by [[random matrix theory]] ([Saad-Shenker-Stanford 19](#SaadShenkerStanford19), [Stanford-Witten 19](#StanfordWitten19)) (or [[SYK model]]) this allows to compute genuine [[quantum gravity]]-aspects of near-[[extremal black holes]], such as notable their microscopic [[black hole entropy]]. Computations are now under way...

Notice that near-[[extremal black holes]] have been observed in nature, by the Chandra telescope [see eg here](https://scitechdaily.com/chandra-views-spinning-black-holes-across-cosmic-sea/).

## Related concepts

* [[SYK model]]

* [[Liouville theory]]

## References

### General

The theory is due to

* {#Teitelboim83} [[Claudio Teitelboim]], _Gravitation and hamiltonian structure in two spacetime dimensions_, Physics Letters B Volume 126, Issues 1–2, 23 June 1983, Pages 41-45 (doi:10.1016/0370-2693(83)90012-6

* {#Jackiw85} [[Roman Jackiw]], _Lower dimensional gravity_, Nuclear Physics B
Volume 252, 1985, Pages 343-356 (doi:10.1016/0550-3213(85)90448-1)

See also

* Wikipedia, _[Jackiw-Teitelboim gravity](https://en.wikipedia.org/wiki/Jackiw%E2%80%93Teitelboim_gravity)_

Further development:

* Andreas Blommaert, Thomas G. Mertens, Henri Verschelde, _Eigenbranes in Jackiw-Teitelboim gravity_ ([arXiv:1911.11603](https://arxiv.org/abs/1911.11603))

Via [[twistors]]:

* Wolfgang Wieland, _Twistor representation of Jackiw-Teitelboim gravity_ ([arXiv:2003.13887](https://arxiv.org/abs/2003.13887))

On [[supergravity]]-versions of JT-gravity:

* Gustavo J. Turiaci, [[Edward Witten]], *$\mathcal{N} = 2$ JT Supergravity and Matrix Models* [[arXiv:2305.19438](https://arxiv.org/abs/2305.19438)]

[[!include AdS2-CFT1 -- references]]

### Application to near-extremal near-horizons

Application to [[near horizon geometry]] of near-[[extremal black holes]]:

* {#NSST18} Pranjal Nayak, Ashish Shukla, Ronak M Soni, [[Sandip Trivedi]], V. Vishal, _On the Dynamics of Near-Extremal Black Holes_ ([arXiv:1802.09547](https://arxiv.org/abs/1802.09547))

* {#MTV18} Upamanyu Moitra, [[Sandip Trivedi]], V. Vishal, _Near-Extremal Near-Horizons_ ([arXiv:1808.08239](https://arxiv.org/abs/1808.08239))

### Lift to string/M-theory

Realization of [[JT-gravity]] as [[Kaluza-Klein reduction]] of [[D=6 supergravity]] on the [[worldvolume]] of [[D1-D5 brane bound states]] or [[M2-M5 brane bound states]]:

* Yue-Zhou Li, Shou-Long Li, H. Lu, _Exact Embeddings of JT Gravity in Strings and M-theory_, Eur. Phys. J. C (2018) 78: 791 ([arXiv:1804.09742](https://arxiv.org/abs/1804.09742))

* Iosif Bena, Pierre Heidmann, David Turton, _$AdS_2$ Holography: Mind the Cap_, JHEP 1812 (2018) 028 ([arXiv:1806.02834](https://arxiv.org/abs/1806.02834))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_2$ solutions and their massive IIA origin_, JHEP 05 (2019) 107 ([arXiv:1811.11572](https://arxiv.org/abs/1811.11572))

* Junho Hong, [[Niall Macpherson]], Leopoldo A. Pando Zayas, _Aspects of $AdS_2$ classification in M-theory: Solutions with mesonic and baryonic charges_, JHEP 11 (2019) 127 ([arXiv:1908.08518](https://arxiv.org/abs/1908.08518))

### In terms of weight systems on chord diagrams

Discussion of ([[Lie algebra weight system|Lie algebra]]-)[[weight systems]] on [[chord diagrams]] encoding [[JT gravity]] [[observables]]

(for more see at [[weight systems on chord diagrams in physics]]):

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _The Schwarzian Theory - A Wilson Line Perspective_, JHEP 1812 (2018) 022 ([arXiv:1806.07765](https://arxiv.org/abs/1806.07765))

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _Fine Structure of Jackiw-Teitelboim Quantum Gravity_, JHEP 1909 (2019) 066 ([arXiv:1812.00918](https://arxiv.org/abs/1812.00918))

and similarly, under [[AdS2/CFT1]], as [[correlators]] in the [[SYK model]]:

* Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, _Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$_, JHEP 04 (2018) 146 ([arXiv:1801.02696](https://arxiv.org/abs/1801.02696))

* [[Micha Berkooz]], Prithvi Narayan, [[Joan Simón]], _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

following:

* László Erdős, Dominik Schröder, _Phase Transition in the Density of States of Quantum Spin Glasses_, D. Math Phys Anal Geom (2014) 17: 9164 ([arXiv:1407.1552](https://arxiv.org/abs/1407.1552))

which in turn follows

* Philippe Flajolet, Marc Noy, _Analytic Combinatorics of Chord Diagrams_, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), _Formal Power Series and Algebraic Combinatorics_, Springer 2000 ([doi:10.1007/978-3-662-04166-6_17](https://doi.org/10.1007/978-3-662-04166-6_17))

With emphasis on [[AdS-CFT|holographic content]]:

* [[Micha Berkooz]], Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: _Towards a full solution of the large $N$ double-scaled SYK model_, JHEP 03 (2019) 079 ([arxiv:1811.02584](https://arxiv.org/abs/1811.02584))

* Vladimir Narovlansky, Slide 23 (of 28) of: _Towards a Solution of Large $N$ Double-Scaled SYK_, 2019 ([[NarovlanskySYK19.pdf:file]])

[[!redirects JT-gravity]]

[[!redirects JT gravity]]


Jackson Clarke

* [webpage](http://syymmetries.blogspot.de/)

## related $n$Lab entries

* [[naturalness]]

category: people


Jackson Morrow

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

## related $n$Lab entries

* [[anabelian geometry]]

* [[inter-universal Teichmüller theory]]

category: people


Jaco Ruit

[website](https://sites.google.com/view/jacoruit/home)

## Selected writings

On the [[(∞,1)-Grothendieck construction]] via [[straightening and unstraightening]]:

* [[Fabian Hebestreit]], [[Gijs Heuts]], [[Jaco Ruit]], _A short proof of the straightening theorem_ ([arXiv:2111.00069](https://arxiv.org/abs/2111.00069)).

## See also

* [[straightening functor]]


Jacob Bekenstein

* [webpage](http://www.phys.huji.ac.il/~bekenste/)

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

## Selected writings

On [[general relativity|relativistic]]  [[model (in theoretical physics)|models]] for [[MOND]]:

* {#Bekenstein04} [[Jacob Bekenstein]], *Relativistic gravitation theory for the modified Newtonian dynamics paradigm*, Physical Review D **70** 8 (2004) 083509, [[arXiv:astro-ph/0403694](https://arxiv.org/abs/astro-ph/0403694), [doi:10.1103/PhysRevD.70.083509](https://doi.org/10.1103/PhysRevD.70.083509)]

## Related entries

* [[Bekenstein-Hawking entropy]]

category: people


Jacob Biamonte

## About ##

* Welcome to the nLab page for [Jacob Biamonte](http://qubit.org/jacob-biamonte.html).

***

## nLab pages ##

* My personal nLab page: [http://ncatlab.org/nlab/show/Jacob+Biamonte](http://ncatlab.org/nlab/show/Jacob+Biamonte)

* Diagrams page: [http://ncatlab.org/johnbaez/show/Diagrams](http://ncatlab.org/johnbaez/show/Diagrams)

* Petri net field theory paper: [http://ncatlab.org/johnbaez/show/Diagrams](http://ncatlab.org/johnbaez/show/Diagrams)

## Links ##

* [My homepage](http://deepquantum.ai)

## Selected writings

On [[tensor networks]]:

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

* [[Jacob Biamonte]], _Lectures on Quantum Tensor Networks_ ([arXiv:1912.10049](https://arxiv.org/abs/1912.10049))

On [[quantum machine learning]]:

* [[Jacob Biamonte]], Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, [[Seth Lloyd]], *Quantum Machine Learning*, Nature **549**  (2017) 195–202 ([doi:10.1038/nature23474](https://doi.org/10.1038/nature23474))

  EurekaAlert, *Quanutum Machine Learning* [14-Sep-2017](https://www.eurekalert.org/pub_releases/2017-09/iiop-qml091417.php)


Jacob Lurie

Jacob Lurie is a mathematician at the Institute for Advanced Study.

* [website](https://www.math.ias.edu/~lurie)

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

* [Institute for Advanced Study entry](https://www.ias.edu/scholars/lurie)

After an early interest in [[formal logic]] (specifically notions of [[computability|computable]] [[surreal numbers]], see [Notices of the AMS vol 43, Number 7](http://www.ams.org/notices/199607/comm-conway.pdf)) Lurie indicated in [his PhD thesis](http://dspace.mit.edu/handle/1721.1/30144) how the [[moduli stack of elliptic curves]] together with the collection of [[elliptic cohomology]] [[spectra]] associated to each [[elliptic curve]] is naturally understood as a geometric object in a [[homotopy theory|homotopy theoretic]] refinement of [[algebraic geometry]] that has come to be known as [[derived algebraic geometry]]. He then embarked on a monumental work laying out detailed foundations of the subjects necessary for this statement, which is [[homotopy theory]] in its modern incarnation as [[higher category theory]], [[higher geometry]] in terms of [[higher topos theory]] and finally [[higher algebra]] in terms of [[(infinity,1)-operad|higher operads]], all in principle  very much along the lines originally developed by [[Alexander Grothendieck]] and his school for ordinary [[algebraic geometry]], but now considerably further refined to the general context of [[homotopy theory]]. While some developments in these topics had been available before, Lurie's comprehensive work has arguably led these subjects to an era of reinvigorated activity with a variety of further spin-offs. Among these most notable is maybe the formalization and proof of the [[cobordism hypothesis]], which lays [[monoidal (infinity,n)-category|higher monoidal category theoretic]] foundations for ([[extended topological quantum field theory|local, topological]]) [[quantum field theory]]. In 2014 Lurie was awarded a [MacArthur Genius Grant](http://www.macfound.org/fellows/921/) and the Breakthrough Prize in Mathematics.

#Contents#
* table of contents
{:toc}

## References (partly) indexed on the $n$Lab ##

### Higher category theory

*  _[[(∞,2)-Categories and the Goodwillie Calculus]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf), [arXiv](http://arxiv.org/abs/0905.0462))

   on [[(∞,1)-category|(∞,1)-categories]] of [[(∞,n)-category|(∞,n)-categories]], specifically on [[internal (∞,1)-categories]], and on [[Goodwillie calculus]]

  on [[higher category theory]], [[internal (∞,1)-categories]]

* [[On the Classification of Topological Field Theories]] ([arXiv](http://arxiv.org/abs/0905.0465))

  on [[FQFT|functorial]] [[extended topological quantum field theory|extended]] [[TQFT|topological]] [[quantum field theory]] classified via the [[cobordism theorem]] by [[symmetric monoidal (∞,n)-categories]]

### Higher topos theory

* _[[Higher Topos Theory]]_ ([arXiv](http://arxiv.org/abs/math.CT/0608040), [published version](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf))

  on [[(∞,1)-category theory]] and [[higher topos theory]]

following:

* {#Lurie03} [[Jacob Lurie]], Section 2.2 of: *On $\infty$-Topoi* ([arXiv:math/0306109](https://arxiv.org/abs/math/0306109))

### Higher algebra

* _[[Higher Algebra]]_

  on [[higher algebra]]

subsuming

* [[Stable ∞-Categories]] ([arXiv](http://arxiv.org/abs/math/0608228))

  on [[stable (∞,1)-categories]] [[stable homotopy theory]] and [[homological algebra]]

* [[higher algebra|Noncommutative Algebra]] ([arXiv](http://arxiv.org/abs/math/0702299))

  and

  [[higher algebra|Commutative Algebra]] ([arXiv](http://arxiv.org/abs/math/0703204), [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf))

* _$\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf))

  on [[little cubes operad]]

* [[Deformation Theory]] ([arXiv](http://arxiv.org/abs/0709.3091))

  on the notions of [[Kähler differential]] and [[cotangent complex]]
  and their generalization from the ordinary context of [[algebra]] to that of [[higher algebra]].

* _[[Moduli Problems and DG-Lie Algebras]]_

  on [[∞-Lie algebra]]s as formal neighbourhoods of point in [[∞-stack]]s.

### Higher geometry

The foundations of [[higher geometry]]:

* [[Structured Spaces]] ([arXiv](http://arxiv.org/abs/0905.0459))

  * on [[structured (∞,1)-topos]]es and [[generalized schemes]]

Survey on the general program

* PhD thesis, [[Derived Algebraic Geometry]]

  on [[derived algebraic geometry]]

A volume on [[E-∞ geometry]] ([[spectral algebraic geometry]])

* [[Jacob Lurie]], _[[Spectral Algebraic Geometry]]_

The basic definitions are in

* [[Jacob Lurie]], _[[Spectral Schemes]]_
 {#Lurie}

Fundamental properties of $E_\infty$-geometry are discussed in

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

* [[Jacob Lurie]], _[[Closed Immersions]]_

* [[Jacob Lurie]], _[[Descent Theorems]]_

* [[Jacob Lurie]], _[[Proper Morphisms, Completions,
and the Grothendieck Existence Theorem]]_

* [[Jacob Lurie]], _[[Representability Theorems]]_

* [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

Application to [[moduli stack of elliptic curves]]:

* [[A Survey of Elliptic Cohomology]] ([pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf))

  on the [[generalized (Eilenberg-Steenrod) cohomology]] theory [[tmf]], the gluing of all [[elliptic cohomology]] theories

### Elliptic cohomology

* _Elliptic cohomology I: Spectral abelian varieties_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf))

* _Elliptic cohomology II: Orientations_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-II.pdf))

* _Elliptic cohomology III: Tempered Cohomology_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf))

* _Elliptic cohomology IV: Equivariant elliptic cohomology_, (announced)

### Publications before thesis

* M H Freedman, A Kitaev, J Lurie, _Diameters of homogeneous spaces_, Math. res. lett. 10:1, 11-20 (2003)

* J. Lurie, _Anti-admissible sets_, J Symb Logic 64:2, 408-435 (1999)

* J. Lurie, _On a conjecture of Conway_, Illinois J. Math.  46:2, 497-506 (2002)

* J. Lurie, _On simply laced Lie algebras and their minuscule representations_, Comment. Math. Helv. 76:3, 515-575 (2001)
[doi](http://dx.doi.org/10.1007/PL00013217)

##Other material

* [Informal notes](http://www.math.harvard.edu/~lurie/FF.html) on the [[Fargues-Fontaine curve]] with [[Dennis Gaitsgory]].

* [[Kerodon]] is an online textbook on [[Categorical Homotopy Theory|categorical homotopy theory]] and related mathematics.

category: people

[[!redirects J. Lurie]]
[[!redirects Lurie]]


Jacob McNamara

* [spire page](https://inspirehep.net/authors/1804692)

## Selected writings

Introducing the [[swampland cobordism conjecture]]:

* {#McNamaVafa17} [[Jacob McNamara]], [[Cumrun Vafa]], _Cobordism Classes and the Swampland_ ([arXiv:1909.10355](https://arxiv.org/abs/1909.10355), [spire:1755441](https://inspirehep.net/literature/1755441))

category: people


Jacob Mostovoy

* [webpage](https://www.math.cinvestav.mx/~mostovoy/)

## Selected writings

On [[homotopy of rational maps]]:

* {#Mostovoy01} [[Jacob Mostovoy]], *Spaces of Rational Loops on a Real Projective Space*, Transactions of the American Mathematical Society, Vol. 353, No. 5 (May, 2001), pp. 1959-1970 ([jstor:221802](https://www.jstor.org/stable/221802))

* [[Jacob Mostovoy]], *Spaces of rational maps and the Stone–Weierstrass theorem*, Topology Volume 45, Issue 2, March 2006, Pages 281-293 ([doi:10.1016/j.top.2005.08.003](https://doi.org/10.1016/j.top.2005.08.003))

* [[Jacob Mostovoy]], Erendira Munguia-Villanueva, *Spaces of morphisms from a projective space to a toric variety*, Revista Colombiana de Matematicas **48** 1 (2014) ([arXiv:1210.2795](https://arxiv.org/abs/1210.2795), [published pdf](http://scm.org.co/archivos/revista/Articulos/1129.pdf))

On [[Vassiliev invariants]] ([[chord diagrams]], [[Jacobi diagrams]], [[weight systems]], [[knot invariants]] etc.)

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

category: people


Jacob Neumann

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

## Selected writings and talks

* [[Jacob Neumann]], *(Co)ends and (Co)structure*, [[Homotopy Type Theory Electronic Seminar Talks]], 1 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Neumann-2022-12-01-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=X4v5HnnF2-o))

* [[Jacob Neumann]], *Paranatural Category Theory*, 2023, [arxiv:2307.09289](https://arxiv.org/abs/2307.09289)

category:people


Jacob Sonnenschein

* [webpage](https://physics.tau.ac.il/profile/cobi)

## Selected writings

On [[Seiberg-Witten curves]] understood as transversal factors of [[M5-brane]]-[[worldvolumes]]:

* A. Brandhuber, [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], *M Theory And Seiberg-Witten Curves: Orthogonal and Symplectic Groups*, Nucl. Phys. B **504** (1997) 175-188 [[arXiv:hep-th/9705232](https://arxiv.org/abs/hep-th/9705232), doi:10.1016/S0550-3213(97)00531-2]

On [[supergravity]], [[D-branes]] and the [[large N limit]] of [[super Yang-Mills theories]]:

* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

On [[heterotic M-theory on ADE-orbifolds]]:

* [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$_, Nuclear Physics B **590** 1–2 (2000) 123-160 Nuclear Physics B [[arXiv:hep-th/9912144](https://arxiv.org/abs/hep-th/9912144), doi:10.1016/S0550-3213(00)00460-0]

* {#GKSTY02} E. Gorbatov, [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015 (2002) [[arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135)]

On [[baryons]] and [[mesons]] in [[holographic QCD]]:

* {#BISY98} A. Brandhuber, N. Itzhaki, [[Jacob Sonnenschein]], [[Shimon Yankielowicz]],  _Baryons from Supergravity_, JHEP 9807:020 (1998) [[arxiv:hep-th/9806158](https://arxiv.org/abs/hep-th/9806158)]

* {#SonnenscheinWeissman18} [[Jacob Sonnenschein]], Dorin Weissman, _Excited mesons, baryons, glueballs and tetraquarks: Predictions of the Holography Inspired Stringy Hadron model_, ([arXiv:1812.01619](https://arxiv.org/abs/1812.01619))

On [[non-perturbative quantum field theory]] (from [[2d CFT]] to [[QCD]]):

* [[Yitzhak Frishman]], [[Jacob Sonnenschein]], *Non-Perturbative Field Theory -- From Two Dimensional Conformal Field Theory to QCD in Four Dimensions*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511770838](https://doi.org/10.1017/CBO9780511770838), summary: [arXiv:1004.4859](https://arxiv.org/abs/1004.4859)] open access (2023) [[doi:10.1017/9781009401654](https://doi.org/10.1017/9781009401654)]

On [[sigma-mesons]] and [[omega-mesons]] mediating [[baryon]] [[interaction]], discussed in [[holographic QCD]] via [[D3-D7 brane intersections]]:

* Anatoly Dymarsky, Dmitry Melnikov, [[Jacob Sonnenschein]], _Attractive Holographic Baryons_, JHEP 06 (2011) 145 ([arXiv:1012.1616](https://arxiv.org/abs/1012.1616))

On [[sigma-mesons]] and attractive [[nuclear forces]] between [[baryons]] in [[holographic QCD]] ([[Witten-Sakai-Sugimoto model]] on [[D4-D8 brane intersections]]):

* [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], _Searching for an Attractive Force in Holographic Nuclear Physics_, JHEP 05 (2011) 058 ([arXiv:1003.2621](https://arxiv.org/abs/1003.2621))

On a [[TT-deformation]] of the [[Skyrme model]] applied to [[nucleon]] [[scattering]]:

* [[Horatiu Nastase]], [[Jacob Sonnenschein]], _A $T \bar T$-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering_ ([arXiv:2101.08232](https://arxiv.org/abs/2101.08232))

category: people


Jacob White

Jacob White is currently a visiting assistant professor at Texas A&M University. He is currently engaged in an ongoing research project with Marcelo Aguiar and Swapneel Mahajan involving the study of near-rings in duoidal categories and in colax distributive monoidal categories.


Jacobi curve

## References

* Wikipedia, _[Jacobi curve](http://en.wikipedia.org/wiki/Jacobian_curve)_

[[!redirects Jacobi curves]]

[[!redirects Jacobi quartic]]
[[!redirects Jacobi quartics]]


Jacobi diagram

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A closed _Jacobi diagram_ is a connected undirected [[graph]] with oriented trivalent vertices and with an embedded oriented [[circle]], regarded modulo cyclic identifications, if any.

Here is a picture of a typical Jacobi diagram:

Here the internal lines need not form a [[tree]]; the following is also a Jacobi diagram:

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

If all the [[vertices]] sit on the circle, Jacobi diagrams specialize to [[chord diagrams]].

Half the number of vertices of a Jacobi diagram is called its _order_

$$
  order(\Gamma) \;\coloneqq\; \#Vertices(\Gamma)/2
$$

For example, these are the two closed Jacobi diagrams with 2 vertices (order 1):

and these are are the 10 closed Jacobi diagrams with 4 vertices (order 2):

> graphics grabbed from [Chmutov-Duzhin-Mostovoy 11](#ChmutovDuzhinMostovoy11)

In applications to [[Vassiliev invariants]] in [[knot theory]], Jacobi diagrams play the role of connected [[Feynman diagrams]] for [[Chern-Simons theory]] in the presence of one [[Wilson line]] (the circle is then the knot/Wilson line).

**Terminology.** Jacobi diagrams have originally been called "chinese character diagrams" ([Bar-Natan 95, Def. 1.8](#BarNatan95))
or "[[Feynman diagrams]] for [[Chern-Simons theory]]" ([Kricker-Spence-Aitchison 97](#KrickerSpenceAitchison97))
or "circle diagrams" ([Kneissler 97](#Kneissler97))
or "round diagrams" ([Willerton 99](#Willerton99)).
The term "Jacobi diagram" ([Chmutov-Duzhin-Mostovoy 11, Chapter 5](#ChmutovDuzhinMostovoy11), [Jackson-Moffat 19, Section 13](#JacksonMoffat19)) alludes to the [[Jacobi identity]] -- here also called the [[STU relation]] -- via the fact that these diagrams are naturally labelled by a [[Lie algebra]] equipped with a [[Lie algebra representation]] (which is really just the [[interaction]]-aspect of their interpretation as [[Chern-Simons theory]] [[Feynman diagrams]]...).

## Definition

### Jacobi diagrams

(...)

### STU-relations and weight systems
 {#STURelationsAndWeightSystems}

For $R \in $ [[CRing]] a [[commutative ring]], let $R\langle \mathcal{D}^t \rangle$ denote the $R$-[[linear span]] of the [[set]] $\mathcal{D}^t$ of [[Jacobi diagrams]].

Then one traditionally writes

\[
  \label{QuotientSpaceBySTU}
  \mathcal{A}^t
    \;\coloneqq\;
  R\langle \mathcal{D}^c \rangle/STU
\]

for the [[quotient spaces]] of the [[linear span]] of [[Jacobi diagrams]] by the [[STU relations]]:

Hence:

A [[linear function]]

$$
  w \;\colon\; \mathcal{A}^t \longrightarrow R
  \,,
$$

-- hence a plain $R$-valued [[function]] on the [[set]] $\mathcal{D}^t$ of [[Jacobi diagrams]] which is [[invariant]] under the [[STU relations]] --

is called a _[[framed weight system]]_. See there for more.

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

## Properties

### Relation to chord diagrams

[[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]:

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

### Relation to Adams operations

On the [[vector space]] $\mathcal{A}$ of [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU-relations]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] [[chord diagrams]] [[quotient vector space|modulo]] [[4T relations]])  there is a system of [[linear maps]] (for $q \in \mathbb{Z}$, $q \neq 0$)

$$
  \psi^q
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathcal{A}
$$

which respect the [[coalgebra]] [[mathematical structure|structure]] and satisfy

$$
  \psi^{q_2} \circ \psi^q_1
  \;=\;
  \psi^{q_1 \cdot q_2}
$$

and as such are (dually) analogous to the [[Adams operations]] on [[topological K-theory]].

([Bar-Natan 95, Def. 3.11 & Theorem 7](#BarNatan95))

In fact, when evaluated in [[Lie algebra weight systems]] $w_{\mathbf{N}}$ and under the identification (see [here](equivariant+K-theory#RelationToRepresentationTheory)) of the [[representation ring]] of a [[compact Lie group]] $G$ with the $G$-[[equivariant K-theory]] of the point, these Adams operations on [[Jacobi diagrams]] correspond to the [[Adams operations]] on [[equivariant K-theory]]:

$$
  w_{\mathbf{N}}(\psi^q D)
  \;=\;
  w_{\psi^q \mathbf{N}}(D)
  \,.
$$

([Bar-Natan 95, Exc. 6.24](#BarNatan95)
)

For more see at _[[Adams operation on Jacobi diagrams]]_.

## Related concepts

[[!include chord diagrams and weight systems -- table]]

## References

### General

Original articles

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* {#KrickerSpenceAitchison97} A. Kricker, B. Spence and I. Aitchison, _Cabling the Vassiliev invariants_, J. Knot Theory Ramifications 6 (1997) 327–358

* {#Kneissler97} [[Jan Kneissler]], _The number of primitive Vassiliev invariants up to degree 12_ ([arXiv:q-alg/9706022](https://arxiv.org/abs/q-alg/9706022))

* {#Willerton99} [[Simon Willerton]], _The Kontsevich integral and algebraic structures on the space of diagrams_ in: Knots in Hellas ’98. Series on Knots and Everything, vol. 24, World Scientific, 2000, 530–546 ([arXiv:math/9909151](https://arxiv.org/abs/math/9909151))

Lecture notes

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))

Textbook accounts

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 5 of:  _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* {#JacksonMoffat19} [[David Jackson]], [[Iain Moffat]], Section 13 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

[[!include weight systems on chord diagrams in physics]]

[[!redirects Jacobi diagrams]]


Jacobi form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Jacobi forms are [[power series]] of two variables which in one variable behave like a [[modular form]] and in the other have an "elliptic" nature. They arise naturally as the characteristic series of the [[elliptic genus]]/[[Witten genus]] ([Zagier 86, pages 8-9](#Zagier86)).

## Definition

For $k, n \in \mathbb{Z}$, a _Jacobi form_ of _weight_ $k$ and _index_ $n$ is a [[function]] of the form

$$
  \phi \;\colon\; H \times \mathbb{C} \longrightarrow \mathbb{C}
$$

hence from the [[product]] of the [[upper half plane]] with the full [[complex plane]] which transforms under

$$
  \left(
    \array{
       a & b
       \\
       c & d
    }
  \right)
  \in
  SL_2(\mathbb{Z})
$$

as

$$
  \phi
  \left(
     \frac{a \tau + b}{c \tau + d},
     \frac{z}{c \tau + d}
  \right)
   =
  (c \tau+ d)^k
  \exp(2 \pi i n c z^2 / (c \tau + d))
  \phi(\tau, z)
  \,.
$$

## Examples

### Jacobi theta-functions
 {#JacobiThetaFunctions}

The most important examples are the  [[Jacobi theta-functions]].
The four Jacobi $\theta$-functions are (with $q = e^{2\pi i \tau}$)

$$
  \theta(z,\tau)
    \coloneqq
  2 q^{1/8} sin(\pi z)
  \prod_{j = 1}^{\infty}
  \left(
    \left(
      1 - q^{j}
    \right)
    \left(
      1 - e^{2\pi i z} q^{j}
    \right)
    \left(
      1 - e^{-2 \pi i z} q^{j}
    \right)
  \right)
$$

$$
  \theta_1(z,\tau)
    \coloneqq
  2 q^{1/8} cos(\pi z)
  \prod_{j = 1}^{\infty}
  \left(
    \left(
      1 - q^{j}
    \right)
    \left(
      1 + e^{2\pi i z} q^{j}
    \right)
    \left(
      1 + e^{-2 \pi i z} q^{j}
    \right)
  \right)
$$

$$
  \theta_2(z,\tau)
    \coloneqq
   \;\;\;\;\;\;\;\;\;
  \prod_{j = 1}^{\infty}
  \left(
    \left(
      1 - q^{j}
    \right)
    \left(
      1 - e^{2\pi i z} q^{j - 1/2}
    \right)
    \left(
      1 - e^{-2 \pi i z} q^{j - 1/2}
    \right)
  \right)
$$

$$
  \theta_3(z,\tau)
    \coloneqq
   \;\;\;\;\;\;\;\;\;
  \prod_{j = 1}^{\infty}
  \left(
    \left(
      1 - q^{j}
    \right)
    \left(
      1 + e^{2\pi i z} q^{j - 1/2}
    \right)
    \left(
      1 + e^{-2 \pi i z} q^{j - 1/2}
    \right)
  \right)
$$

See for instance ([KL 95, section 2.4](#KL95), [Chen-Han-Zhang 10, section 2](#ChenHanZhang10)) for a review in the context of [[elliptic genera]].

As part of this, the [[Kac-Weyl character]] of an integral highest-weight [[loop group representation]] is a Jacobi form ([KL 95, section 2.2](#KL95)).

The **Jacobi identity** (see at _[[Jacobi triple product]]_) asserts that these are related by

$$
  \theta'(0,\tau) \coloneqq \frac{\partial}{\partial z}\theta(0,\tau)
  =
  \pi \theta_1(0,\tau) \theta_2(0,\tau) \theta_3(0,\tau)
  \,.
$$

### Weierstrass function

(...)

## References

The original canonical account is

* Martin Eichler, [[Don Zagier]], _The theory of Jacobi forms_, Progress in Mathematics 55, Boston, MA: Birkhäuser Boston (1985), ISBN 978-0-8176-3180-2, MR 781735

Discussion of Jacobi forms as coefficients of the [[elliptic genus]]/[[Witten genus]] includes

* {#Zagier86} [[Don Zagier]], pages 8,9 of _Note on the Landweber-Stong elliptic genus_ 1986 ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0078047/fulltext.pdf))

* {#KL95} [[Kefeng Liu]], _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 ([EUCLID](http://projecteuclid.org/euclid.jdg/1214456221), [pdf](http://www.math.ucla.edu/~liu/Research/loja.pdf))

* {#AndoFrenchGanter08} [[Matthew Ando]], Christopher French, [[Nora Ganter]], _The Jacobi orientation and the two-variable elliptic genus_, Algebraic and Geometric Topology 8 (2008) p. 493-539 ([pdf](http://www.msp.warwick.ac.uk/agt/2008/08-01/agt-2008-08-016s.pdf))

* {#ChenHanZhang10} Qingtao Chen, [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))

See also

* Wikipedia, _[Jacobi form](http://en.wikipedia.org/wiki/Jacobi_form)_

[[!redirects Jacobi forms]]

[[!redirects Jacobi theta-function]]


Jacobi identity

> This entry is about the identity in [[Lie theory]]. For another identity named after [[Carl Jacobi]], see at _[[Jacobi theta-function]]_ and _[[Jacobi form]]_.

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## Idea

The Jacobi identity is an important equational identity that holds in [[Lie algebras]] and is also of interest in other [[nonassociative algebra|algebras]].  It can be generalised to [[higher algebra]]s.

There is a non-additive non-abelian form of the Jacobi identity that occurs in considering certain examples in [[identities among relations]] in the presentation of certain groups. There it is also called the Hall-Witt identity.

## For $1$-algebras

### Definition

Given a [[nonassociative algebra]] $\mathfrak{g}$ over a field or a ring $k$, whose bilinear product is denoted by bracket $[,]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$, the __Jacobi identity__ holds for a triple of elements $x,y,z \in \mathfrak{g}$ if

$$
  [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0
  \,.
$$

The principal example is that of a [[Lie algebra]]: here the Jacobi identity by definition holds for all triples of elements (and the bracket is skew-symmetric).

### Relation to the Leibniz identity

If the bracket $[-,-]$ is skew-symmetric the Jacobi identity for all triples is equivalent to the **[[Leibniz identity]]** that for all $x,a,b$,
the linear map $ad_x = [x,-] : \mathfrak{g} \to \mathfrak{g}$ is a [[derivation]] with respect to the Lie bracket:

$$
  \left[x,\left[a,b\right]\right] = \left[\left[x,a\right],b\right]  + \left[a,\left[x,b\right]\right]
  \,.
$$

One can also consider right derivations (right Leibniz identity), what is again equivalent in the presence of antisymmetry:

$$
 \left[\left[a,b\right],x\right] = \left[\left[a,x\right],b\right]  + \left[a,\left[b,x\right]\right]
  \,.
$$

Left and right [[Leibniz algebras]] generalize Lie algebras by having a left or right Leibniz identity, but not necessarily antisymmetry of the bracket. In particular, Jacobi identity, and left and right Leibniz identities do not need to coincide without antisymmetry.

### In terms of Chevalley--Eilenberg algebras

A useful way to think of Jacobi identity for finite-dimensional Lie algebras, is dually in terms of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ (see there for details). In terms of this [[dg-algebra]] $(\wedge^\bullet \mathfrak{g}^*, d_{\mathfrak{g}})$, the Lie bracket is encoded in the [[differential]]

$$
  d_{\mathfrak{g}}|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*
$$

and the Jacobi identity is equivalent to the statement that this differential squares to $0$

$$
  d \circ d = 0
  \,.
$$

### In terms of string diagrams / Jacobi diagrams
 {#InTermsOfStringDiagrams}

In [[string diagram]]-notation for [[Lie algebra objects]] [[internalization|interal]] to [[tensor categories]], the [[Lie action property]] looks as follows:

$$
  \begin{aligned}
    \Leftrightarrow
    &
    \;\;\;\;\;
    \rho(f(x,y),z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    \\
    \underset{
      {f = [-,-]}
      \atop
      {Lie\;bracket}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Lie\;action\;property}
    }{
    \underbrace{
      \rho([x,y],z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    }
    }
    \\
    \underset{
      {\rho = -[-,-]}
      \atop
      {adjoint\;action}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Jacobi\;identity}
    }{
    \underbrace{
      [[x,y],z] \;=\; - [y,[x,z]] + [x,[y,z]]
    }
    }
  \end{aligned}
$$

where the last line shows the equivalence to the [[Jacobi identity]] on the [[Lie algebra object]] itself in the case that the Lie action is the [[adjoint action]].

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

In the language of [[Jacobi diagrams]] this is called the _[[STU-relation]]_.
and is the reason behind the existence of [[Lie algebra weight systems]] in [[knot theory]].

## For $L_\infty$-algebras

When Lie algebras are generalized to [[∞-Lie algebra]]s, the Jacobi identity in terms of the binary bracket is relaxed to hold only up to a [[natural isomorphism]] called the __jacobiator__, $[-,-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_1$,

$$
  \left[x,\left[y,z\right]\right] + \left[y,\left[z,x\right]\right] + \left[z,\left[x,y\right]\right]  =
  \delta \left[x,y,z\right]
  \,,
$$

where $\delta$ is the differential.

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
On the other hand, in terms of the [[Chevalley-Eilenberg algebra]] this is still encoded in just $d \circ d  = 0$ (see there for details).

## Related entries

* [[STU relation]]

* [[identity among the relations]]

* [[Leibniz algebra]]

[[!redirects Jacobi identity]]
[[!redirects Jacobi identities]]
[[!redirects jacobiator]]
[[!redirects jacobiators]]
[[!redirects Jacobiator]]
[[!redirects Jacobiators]]


Jacobi theta function

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## Idea

The fundamental example of [[theta functions]] is the _Jacobi theta function_ given by

$$
  \vartheta(z,\tau)
  =
  \underoverset{n = - \infty}{\infty}{\sum}
   \exp(\pi i n^2 \tau + 2\pi i n z)
 \,.
$$

As a variable of two arguments, this is actually a _[[Jacobi form]]_. These are the local [[coordinate]] expressions of the the [[covariant derivative|covariantly constant]] [[sections]] of the [[Hitchin connection]] (for [[circle group|circle]] [[gauge group]]) on the [[moduli space of elliptic curves]] ([Hitchin 90, remark 4.12](#Hitchin90)). See there for more and see at _[[theta function]]_ for the general idea.

At $z =0$ the function

$$
  \vartheta(0,\tau)
  =
  \underoverset{n = - \infty}{\infty}{\sum}
   \exp(\pi i n^2 \tau)
 \,.
$$

is what in [[number theory]] is often just called "the theta function". This is the one whose [[Mellin transform]] is the [[Riemann zeta function]], see at _[Riemann zeta function -- Relation to Jacobi theta function](http://ncatlab.org/nlab/show/Riemann+zeta+function#RelationToThetaFunctions)

## Properties

### Functional equation and Reciprocity
 {#FunctionalEquation}

By the [[Poisson summation formula]] the number-theoretic theta function $\theta(0,z)$ satisfies the following [[functional equation]]:

$$
  \theta(0,\tau) = \frac{1}{\sqrt{\tau}} \theta(0,\frac{1}{\tau})
  \,.
$$

Under the [[Mellin transform]] this implies the functional equation of the [[Riemann zeta function]], see at _[Riemann zeta function -- Functional equation](Riemann+zeta+function#functional_equation)_.

It also provides an analytic proof of the [[Landsberg-Schaar relation]]

$$\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi i n^2 q}{p}\right)=\frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi i n^2 p}{2q}\right)$$

where $p$ and $q$ are arbitrary positive integers. To prove it, apply theta reciprocity to $\tau=2iq/p+\epsilon$, $\epsilon \gt 0$, and then let $\epsilon\to 0$.

This reduces to the formula for the quadratic Gauss sum when $q=1$:

$$\sum_{n=0}^{p-1} e^{2 \pi i n^2 / p} =
\left\{
\array{
\sqrt{p} & {if } \; p\equiv 1\mod 4 \\
i\sqrt{p} & {if } \; p\equiv 3\mod 4
}
\right.
$$

(where $p$ is an odd prime). From this, it's not hard to deduce Gauss's "golden theorem".

[[quadratic reciprocity]]: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)(q-1)/4}$ for odd primes $p$ and $q$.

See e.g. ([Karlsson](#Karlsson)).

> some of this material from  [this MO discussion](http://mathoverflow.net/q/120067/381)

## Related concepts

* [[Poisson summation formula]], [[functional equation]]

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

Due to [[Carl Jacobi]].

Review includes

* Wikipedia, _[Jacobi theta-function](http://en.wikipedia.org/wiki/Theta_function#Jacobi_theta_function)_

* section 9 in _Analytic theory of modular forms_ ([pdf](https://web.archive.org/web/20150326115730/http://www.math.harvard.edu/~jbland/ma259x_notes.pdf))

* {#Karlsson} Anders Karlsson, _Applications of heat kernels on abelian groups: $\zeta(2n)$, quadratic reciprocity, Bessel integrals_  ([psd](http://www.math.kth.se/~akarl/langmemorial.pdf))

* {#Hitchin90} [[Nigel Hitchin]], _Flat connections and geometric quantization_, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. ([Euclid](http://projecteuclid.org/euclid.cmp/1104200841))

[[!redirects Jacobi theta functions]]

[[!redirects Jacobi theta-function]]
[[!redirects Jacobi theta-functions]]


Jacobi triple product

## Statetement

The [[equation]]

$$
  \underoverset{n=1}{\infty}{\prod}
  \left(
    1- x^{2n}
  \right)
  \left(
    1- x^{2n-1} y^2
  \right)
  \left(
    1- x^{2n-1} y^{-2}
  \right)
  =
  \underoverset{n = -\infty}{\infty}{\sum}
  x^{2n} y^{2n}
$$

Equivalently this is a relation between the four [[Jacobi theta functions]] (see there and at _[[Jacobi form]]_).

## References

Due to

* [[Carl Jacobi]],  _Fundamenta nova theoriae functionum ellipticarum_ (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012

Review includes

* Wikipedia, _[Jacobi triple product](http://en.wikipedia.org/wiki/Jacobi_triple_product)_

A large collection of identities between the various [[Jacobi theta functions]] related is at

* WolframResearch, _[Jacobi Thetas](http://functions.wolfram.com/EllipticFunctions/EllipticThetaPrime4/introductions/JacobiThetas/ShowAll.html)_


Jacobian

> This entry is about the concept in [[differential geometry]]. For the concept of [[Jacobian variety]] see there.

***

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## Definition

If $f : \mathbb{R}^n \to \mathbb{R}^m$ is a $C^1$-[[differentiable map]], between [[Cartesian space]]s, its **Jacobian matrix** is the $(m \times n)$ [[matrix]]

$$
  J(f) \in Mat_{m \times n}(C^0(\mathbb{R}, \mathbb{R}))
$$

of [[partial derivative]]s

$$
  J(f)^i_j := \frac{\partial f^i}{\partial x^j},\,\,\,\,\,\,\,i=1,\ldots,m; j = 1,\ldots,n,
$$
where $x = (x^1,\ldots,x^n)$. Here the convention is that the upper index is a row index and the lower index is the column index; in particular $\mathbf{R}^n$ is the space of real column vectors of length $n$.

In more general situation, if $f = (f^1(x),\ldots,f^m(x))$ is differentiable at a point $x$ (and possibly defined only in a [[neighborhood]] of $x$), we define the Jacobian $J_p f$ of map $f$ at point $x$ as a matrix with real values $(J_p f)^i_j = \frac{\partial f^i}{\partial x^j}|_x$.

That is, the Jacobian is the matrix which describes the [[total derivative]].

If $n=m$ the Jacobian matrix is a square matrix, hence its [[determinant]] $det(J(f))$ is defined and called __the Jacobian of $f$__ (possibly only at a point). Sometimes one refers to Jacobian matrix rather ambigously by Jacobian as well.

## Properties

The [[chain rule]] may be phrased by saying that the Jacobian matrix of the composition $\mathbf{R}^n\stackrel{f}\to\mathbf{R}^m\stackrel{g}\to\mathbf{R}^r$ is the [[matrix product]] of the Jacobian matrices of $g$ and of $f$ (at appropriate points).

If $g:M\to N$ is a $C^1$-map of $C^1$-manifolds, then the [[tangent bundle|tangent map]] $T g: T M\to T N$ defined point by point abstractly by $(T_p g)(X_p)(f) = X_p(f\circ g)$, for $p\in M$, can in local [[coordinates]] be represented by a Jacobian matrix. Namely, if $(U,\phi)\ni p$ and $(V,\psi)\ni g(p)$ are [[charts]] and $X_p = \sum X^i\frac{\partial}{\partial x^i}|_p$ (i.e. $X_p(f)  = \sum_i X^i_p \frac{\partial (f\circ \phi^{-1})}{\partial x^i}|_{\phi(p)}$ for all germs $f\in \mathcal{F}_p$), then
$$
(T_p g)(X_p) = \sum_{i,j} J_p(\psi \circ g\circ\phi^{-1})_i^j X^i_p \frac{\partial}{\partial y^j}|_{g(p)}
$$

## Related concepts

* [[differentiation]], [[Hessian matrix]], [[extremum]]

* [[pullback of a distribution]]

[[!redirects Jacobians]]

[[!redirects Jacobian matrix]]
[[!redirects Jacobian determinant]]

[[!redirects Jacobian matrices]]
[[!redirects Jacobian determinants]]


Jacobian algebra

Given a [[quiver]] $Q$ [[quiver with potential|with potential]] which is an element in a (possibly completed) path algebra ${\mathbf C} Q$ of $Q$, one can consider cyclic derivations $\partial_i:HH_0({\mathbf C} Q)\to {\mathbf C} Q$ and the "Jacobian" ideal in ${\mathbf C} Q$ generated by all of them. The quotient algebra is called Jacobi algebra or a Jacobian algebra of the quiver. It is a noncommutative analogue of the
[[Milnor algebra]] of a [[hypersurface singularity]].

Its role in [[noncommutative algebraic geometry]] analoguous to describing some properties of hypersurface singularities is partial achieved in

* Zheng Hua, Guisong Zhou, _Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras_, Math. Ann. (2022) [doi](https://doi.org/10.1007/s00208-022-02435-3); _Quasi-homogeneity of potentials_, J. Noncommut. Geom. 15 (2021) 399--422
[doi](https://doi.org/10.4171/JNCG/415)
* Zheng Hua, [[Bernhard Keller]], _Quivers with analytic potentials_, [arXiv:1909.13517](https://arxiv.org/abs/1909.13517)

It is called derivation-quotient algebra in

* Raf Bocklandt, [[Travis Schedler]], Michael Wemyss, _Superpotentials and higher order derivations_, J. Pure and Appl. Alg. __214__:9 (2010) 1501-1522 [doi](https://doi.org/10.1016/j.jpaa.2009.07.013)

Other literature

* Daniel Labardini-Fragoso, Bea Schumann, _Landau-Ginzburg potentials via projective representations_, [arXiv:2208.00028](https://arxiv.org/abs/2208.00028)

> We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of [[cluster variety|cluster varieties]] as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.

* C. Geiß, D. Labardini-Fragoso, J. Schröer, _The representation type of Jacobian algebras_, Adv. Math. 290 (2016) 364--452

* M. Pressland, _Mutation of frozen Jacobian algebras_, J. Algebra 546 (2020) 236--273

category: algebra
[[!redirects Jacobian ring]]


Jacobian conjecture

**Jacobian conjecture**: Let $k$ be an [[algebraically closed field]] of characteristics zero, $n\geq 2$ and $\phi: k^n\to k^n$ a (regular) [[endomorphism]] of $k^n$ with constant Jacobian (the determinant of the Jacobian matrix, which is in this polynomial case algebraically defined). Then $\phi$ is a regular [[automorphism]], i.e. has a polynomially defined inverse.

The conjecture is open still stated by Keller in 1939. There were many failed attempts to prove the Jacobian conjecture, especially for $n = 2$; there are also some reductions to special cases. For example, it is known that the Jacobian conjecture holds iff it holds for $\phi$ a polynomial map of degree 3. The Jacobian conjecture is also known to hold at least for those $\phi$ which have a rational inverse.

* A. van den Essen, [Jacobian conjecture](http://eom.springer.de/J/j120010.htm), Springer Online Encyclopedia of Mathematics

* A. van den Essen, _Polynomial automorphisms and the Jacobian conjecture_, [pdf](http://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_55-81.pdf), Algèbre non commutative, groupes quantiques et invariants (Reims, 1995),  55--81, Sémin. Congr., 2, Soc. Math. France, Paris, 1997.

* Arno van den Essen, _Polynomial automorphisms and the Jacobian conjecture_, Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. xviii+329 pp. ISBN: 3-7643-6350-9

The Jacobian conjecture is also equivalent to the Dixmier conjecture: every endomorphism of the $r$-th [[Weyl algebra]] $A_{r,k}$ over $k$ is an automorphism for all $r$. This is a statement of the article

* A. Belov-Kanel, M. [[Kontsevich]], _The Jacobian conjecture is stably equivalent to the Dixmier conjecture_, Mosc. Math. J., 7:2 (2007), 209–218; [math.RA/0512171](http://arxiv.org/abs/math/0512171)

which does contain an error in the proof, which has been later amended by others. It is actually known that each endomorphism of the $r$-th Weyl algebra is *injective*, and not known wheather it is *surjective*.
A shorter algebraic proof is given in

* V. Bavula, _The ${Jacobian Conjecture}_{2n}$ implies the ${Dixmier Problem}_n$_, [math.AG/0512250](http://arxiv.org/abs/math/0512250)

There is a recent proof of related Kontsevich's statement on automorphisms of Weyl algebra

* Alexei Kanel-Belov, Andrey Elishev, Jie-Tai Yu, _Automorphisms of Weyl Algebra and a Conjecture of Kontsevich_, [arxiv/1802.01225](https://arxiv.org/abs/1802.01225)

There is an interesting blog discussion, from the point of view of [[algebraic geometry]]:

* David Speyer at _secret blogging seminar_: [how not to prove the Jacobian conjecture](http://sbseminar.wordpress.com/2009/05/27/how-not-to-prove-the-jacobian-conjecture)


Jacobian variety

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## Idea

To every nonsingular [[algebraic curve]] $C$ (over the [[complex numbers]]) of [[genus of a surface|genus]] $g$ one associates the _Jacobian variety_ or simply _Jacobian_ $J(C)$, either via [[differential 1-forms]] or equivalently via [[line bundles]]: the Jacobian is the [[moduli space]] of [[degree of a coherent sheaf|degree]]-$0$ [[line bundles]] over $C$, i.e. the [[connected component]]

$$
  Jac(X) = Pic_0(X)
$$

of the [[neutral element]] of the [[Picard scheme]] of $C$. See also at _[intermediate Jacobian -- Examples -- Jacobian](intermediate+Jacobian#ExamplePicard)_.

Jacobian varieties are the most important class of [[abelian varieties]].

## Properties

### Abel-Jacobi map

The __[[Abel-Jacobi map]]__ $C\to J(C)$ is defined with help of [[periods]].

### Line bundles and theta functions

Over the [[complex numbers]], [[line bundles]] on a Jacobian variety over a given [[Riemann surface]] are naturally encoded by [[Riemann theta functions]].

## Related concepts

* [[Albanese variety]]

* [[Prym variety]], [[Prym-Tyurin variety]]

[[!include moduli of higher lines -- table]]

## References

* Wikipedia, _[Jacobian variety](http://en.wikipedia.org/wiki/Jacobian_variety)_, _[Abel-Jacobi map](http://en.wikipedia.org/wiki/Abel%E2%80%93Jacobi_map)_

* P. Griffiths, J. Harris, _Principles of algebraic geometry_

* A. Beauville, _Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables_, Acta Math. __164__ (1990), 211-235.

A generalizatioin of Abel-Jacobi map to the setting of formal deformation theory is in

* [[Domenico Fiorenza]], [[Marco Manetti]], _Formal Abel-Jacobi maps_, [arxiv/1610.08684](https://arxiv.org/abs/1610.08684)

Review for [[Riemann surfaces]] includes

* [[Razvan Gelca]], [[Alejandro Uribe]], section 2 of _From classical theta functions to topological quantum field theory_  ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))

[[!redirects Jacobian varieties]]
[[!redirects Jacobi variety]]


Jacobson Blomquist

* [webpage](https://math.osu.edu/people/blomquist.9)

## related $n$Lab entries

* [[integral chains homotopy theory]]

category: people


Jacobson radical

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## Definition

Given a unital (typically noncommutative) [[ring]] $R$, the __Jacobson radical__ $J(R)$ is defined as the set of elements $r\in R$ satisfying the following equivalent properties:

1. For each [[simple object|simple]] left $R$-[[module]] $M$, $r M =0$.
2. Each [[maximal ideal|maximal]] left [[ideal]] of $R$ contains $r$.
3. For all $x\in R$, $1 - r x$ is left [[invertible element|invertible]] in $R$.

Alternatively,

4. $J(R)$ is the intersection of all maximal left ideals of $R$.
5. $J(R)$ is the intersection of all maximal right ideals of $R$.

The properties required remain the same if one interchanges left and right (modules, invertibility etc.) i.e. $J(R)=J(R^{op})$.

$J(R)$ is a $2$-sided [[ideal]] in $R$. The rings for which $J(R)=0$ are called __semiprimitive rings__. In other words, for each nonzero element $r$ in a semiprimitive ring, by the definition, there is a simple module **not** left annihilated by $r$.
Given any ring $R$, the [[quotient object|quotient]] $R/J(R)$ is semiprimitive.

Some authors occasionally say Jacobson ideal.

## Examples

+-- {: .num_example #JacobsonRadicalOfFormalPowerSeriesAlgebra}
###### Example
**(Jacobson radical of [[formal power series algebra]])**

The Jacobson radical of a [[formal power series algebra]] consists of those formal power series whose constant term vanishes.

=--

+-- {: .num_example #JacobsonRadicalOfLocalRing}
###### Example
**(Jacobson radical of [[local ring]])**

The Jacobson radical of a [[local ring]] is the set of non-invertible elements.

=--

+-- {: .num_example #JacobsonRadicalOfLocalPrefieldRing}
###### Example
**(Jacobson radical of local [[prefield ring]])**

The Jacobson radical of a local [[prefield ring]] is the set of [[zero divisors]].

=--

+-- {: .num_example #JacobsonRadicalOfPossiblyTrivialLocalRing}
###### Example
**(Jacobson radical of a possibly trivial [[local ring]])**

The Jacobson radical of a possibly trivial [[local ring]] $R$ is the set of elements $x \in R$ such that $x$ being invertible implies that $0 = 1$.

=--

## Related concepts

* [[radical]]

* [[scale]]

## References

* EoM: [Jacobson radical](https://encyclopediaofmath.org/wiki/Jacobson_radical)

* Wikipedia, _[Jacobson radical](https://en.wikipedia.org/wiki/Jacobson_radical#Examples)_

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) [[doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf)]

[[!redirects Jacobson ideal]]

[[!redirects Jacobson radicals]]

[[!redirects semiprimitive ring]]
[[!redirects semiprimitive rings]]


Jacobson ring

A __Jacobson ring__ is a commutative (unital) [[ring]] whose every prime ideal is an intersection of the maximal ideals which contain it.

Equivalently, every radical ideal is an intersection of maximal ideals. Equivalently, the [[Zariski topology]] on its [[spectrum of a ring|spectrum]] is a [[Jacobson topology]].

## References

* Stacks project 1.10.35 tag:[00FZ](https://stacks.math.columbia.edu/tag/00FZ)
* Wikipedia, _[Jacobson ring](https://en.wikipedia.org/wiki/Jacobson_ring)_

[[!redirects Jacobson rings]]

category: algebra


Jacobson-Morozov theorem

+-- {: .rightHandSide}
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###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

If $\mathfrak{g}$ is a [[semisimple Lie algebra]] over [[ground field]] the [[complex numbers]], then for every [[nilpotent element]] $x \in \mathfrak{g}$ (i.e. $\underset{n \in \mathbb{N}}{\exists} [x,-]^n = 0$) there is as [[homomorphism]] of [[Lie algebras]]

$$
  \mathfrak{sl}(2,\mathbb{C})
  \longrightarrow
  \mathfrak{g}
$$

from [[sl(2)]], such that $x$ is the [[image]] of a [[nilpotent element]] of $\mathfrak{sl}(2,\mathbb{C})$.

## References

Lecture notes:

* Ana Balibanu, _The Jacobson-Morozov theorem_, Section 5 of" _Geometry of semisimple Lie algebras_ ([pdf](http://www.math.harvard.edu/~ana/part1.pdf))

Original articles:

* {#Kostant59} [[Bertram Kostant]], _The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group_, Amer. J. Math. 81 (1959), 973–1032 ([jstor:2372999](https://www.jstor.org/stable/2372999))

See also

* Wikipedia, _[sl2-triple](https://en.wikipedia.org/wiki/Sl2-triple)_


Jacopo Emmenegger

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

## Selected writings

On [[fibered category|fibrations]] of [[enriched groupoids]] as [[categorical semantics]] for [[identity types]] in [[homotopy type theory]]:

* [[Jacopo Emmenegger]], [[Fabio Pasquali]], [[Giuseppe Rosolini]], §3 in: *Elementary fibrations of enriched groupoids*, Mathematical Structures in Computer Science **31** 9 (2021) 958-978 [[doi:10.1017/S096012952100030X](https://doi.org/10.1017/S096012952100030X)]

A [[2-comonad]] characterizing [[Grothendieck fibrations]]:

* [[Jacopo Emmenegger]], [[Luca Mesiti]], [[Giuseppe Rosolini]], [[Thomas Streicher]], *A comonad for Grothendieck fibrations* [[arXiv:2305.01474](https://arxiv.org/abs/2305.01474)]

category: people


Jacques Carette

* [website](http://www.cas.mcmaster.ca/~carette/)

category: people


Jacques Distler

+-- {: .rightHandSide}
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###Contents### {: .clickToReveal}
###Contents### {: .clickToHide tabindex="0"}
+--{: .hide}
[[!include contents]]
=--
=--
=--

I know it seems silly, but here's a [link to my blog](http://golem.ph.utexas.edu/~distler/blog "Musings").

I'm the maintainer of [Instiki](http://golem.ph.utexas.edu/instiki/show/HomePage).

category: people


Jacques Dixmier

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

## Selected writings

On [[C-star algebras|$C^\ast$-algebras]]:

* {#Dixmier77} [[Jacques Dixmier]], *$C^\ast$-algebras*, North Holland (1977)

## Related entries

* [[Dixmier-Douady class]]

category: people


Jacques Hadamard

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

## Quotes

> {#TheShortestPath} Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.

> ([source](http://homepage.math.uiowa.edu/~jorgen/hadamardquotesource.html))

## Some works

* _Psychology of invention in the mathematical field_, Princeton University Press 1945

## Related pages

* [[Hadamard lemma]]

* [[Hadamard's formula]]

category: people

[[!redirects Hadamard]]
[[!redirects J. Hadamard]]


Jacques Hurtubise

* Wikipedia entry

* [Institute page](https://www.mcgill.ca/mathstat/jacques-claude-hurtubise)

## Selected writings

On the relation between the [[moduli space of Yang-Mills monopoles]] and "[[rational maps]]" as concerned with "scattering data":

* [[Jacques Hurtubise]], *Monopoles and rational maps: a note on a theorem of Donaldson*,  Comm. Math. Phys. 100(2): 191-196 (1985) ([euclid:cmp/1103943443](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-100/issue-2/Monopoles-and-rational-maps--a-note-on-a-theorem/cmp/1103943443.full))

On identifying the [[moduli space of Yang-Mills monopoles]] for [[gauge group]] any [[classical Lie group]] with a space of [[rational maps]], namely [[holomorphic maps]] from the [[Riemann sphere]] to a [[flag variety]]:

* [[Jacques Hurtubise]], [[Michael K. Murray]], *On the construction of monopoles for the classical groups*,  Comm. Math. Phys. 122(1): 35-89 (1989) ([euclid:cmp/1104178316](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-122/issue-1/On-the-construction-of-monopoles-for-the-classical-groups/cmp/1104178316.full))

* [[Jacques Hurtubise]], *The classification of monopoles for the classical groups*, Commun. Math. Phys. 120, 613–641 (1989) ([doi:10.1007/BF01260389](https://doi.org/10.1007/BF01260389))

* [[Jacques Hurtubise]], [[Michael K. Murray]], *Monopoles and their spectral data*, Comm. Math. Phys. 133(3): 487-508 (1990) ([euclid:cmp/1104201504](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-133/issue-3/Monopoles-and-their-spectral-data/cmp/1104201504.full))

category: people


Jacques Penon

Jacques Penon is a category theorist who has worked on [[synthetic differential geometry]] and higher categories. He introduced the concept of a [[quasitopos]] in ([Penon 1973](#Penono73)) and gave a definition of weak n-category that was later proved to be too strict by Cheng and Makkai. His proposal was updated to take this particular objection into account.

* [Publications in NUMDAM](http://www.numdam.org/search/%22Penon,%20Jacques%22-c/)

## Related entries

* [[n-category]]

* [[quasitopos]]

## selected writings

* {#Penon73}J. Penon, _Quasi-topos_, C.R. Acad. Sci. Paris **276** Série A (1973) pp.237–240. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6217213f/f251.image))

* J. Penon, _Catégories localement internes_,  C.R. Acad. Sci. Paris **278** Série A (1974) pp.1577–1580. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f355.image))

* {#Penon81} [[Jacques Penon]], _Infinitésimaux et intuitionnisme_,  Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 22 (1981) no. 1, p. 67-72 ([numdam:CTGDC_1981__22_1_67_0](http://www.numdam.org/item/?id=CTGDC_1981__22_1_67_0))

  (on the [[logical topology]] in [[synthetic differential topology]])

* J. Penon, _Approche polygraphique des $\infty$-catégories non strictes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 40 no. 1 (1999), p. 31-80 [NUMDAM](http://www.numdam.org/item?id=CTGDC_1999__40_1_31_0)

## References

* [[Eugenia Cheng]], [[Michael Makkai]], _A note on the Penon definition of $n$-category_ Cahiers de Topologie et Geometrie Differentielle Categoriques 50 (2009) p. 83-101, [arXiv:0907.3961](http://arxiv.org/abs/0907.3961)

category: people
[[!redirects Penon]]


Jacques Roubaud

## Selected writings

On [[monadic descent]] via the [[Beck-Chevalley condition]], the [[monadicity theorem]] and proving 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)]

## Related $n$Lab entries

* [[Benabou-Roubaud theorem]]

category: people


Jacques Tits

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

category: people


Jade Master

Jade Master is a grad student at [University of California, Riverside](https://mathdept.ucr.edu/). Her advisor is [[John Baez]].

[website](https://sites.google.com/view/jadeedenstarmaster/home)

## Selected writings

On categorical approaches to [[Petri nets]]:

* [[John C. Baez]] and [[Jade Master]], _Open Petri nets_, 2018. [arxiv:1808.05415](https://arxiv.org/abs/1808.05415)

* [[Jade Master]], _Generalized Petri nets_, 2019. [arxiv:1904.09091](https://arxiv.org/abs/1904.09091)

On [[DisCoCat]]:

* [[Tai-Danae Bradley]], [[Martha Lewis]], [[Jade Master]], and [[Brad Theilman]]
_Translating and Evolving: Towards a Model of Language Change in DisCoCat_ [arXiv:1811.11041](https://arxiv.org/abs/1811.11041)

category: people


Jaemo Park

* [webpage](http://www-ph.postech.ac.kr/people/Faculty/jaemo.html)

## Selected writings

On [[orientifolds]] and [[discrete torsion]]:

* [[Atish Dabholkar]], [[Jaemo Park]], _Strings on Orientifolds_, Nucl. Phys. B477 (1996) 701-714 ([arXiv:hep-th/9604178](https://arxiv.org/abs/hep-th/9604178))

On [[Green-Schwarz action functionals]] for the [[D-branes]] (including the [[DBI action]]):

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

On the [[Green-Schwarz sigma-model]] of the [[M5-brane]] including the [[self-dual higher gauge field]] and [[double dimensional reduction]] to the [[D4-brane]]:

* [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _World-Volume Action of the M Theory Five-Brane_ ([arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166))

On [[Green-Schwarz sigma models]] for [[D-branes]]:

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

On [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]:

* {#ParkUranga98} [[Jaemo Park]], [[Angel Uranga]], _A Note on Superconformal $\mathcal{N}=2$ theories and Orientifolds_, Nucl. Phys. B542:139-156, 1999 ([arXiv:hep-th/9808161](https://arxiv.org/abs/hep-th/9808161))

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

* [[Jaemo Park]], Soo-Jong Reym, _Supertwistor Orbifolds: Gauge Theory Amplitudes and Topological Strings_, JHEP 0412:017, 2004 ([arXiv:hep-th/0411123](https://arxiv.org/abs/hep-th/0411123))

On the [[3d-3d correspondence]]:

* [[Dongmin Gang]], Eunkyung Koh, Sangmin Lee, [[Jaemo Park]], _Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus_, JHEP01(2014)063 ([arXiv:1305.0937](https://arxiv.org/abs/1305.0937))

On [[3d superconformal gauge field theory]] ([[ABJM model]]):

* {#HLLLP08} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds_, JHEP 0809:002, 2008 ([arXiv:0806.4977](https://arxiv.org/abs/0806.4977))

* {#HLLLP08b} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=4$ Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets_, JHEP 0807:091,2008 ([arXiv:0805.3662](https://arxiv.org/abs/0805.3662))

On [[D4-D8 brane intersections]] with application to [[geometric engineering of QFTs|geometric engineering]] of [[QCD]] ("[[holographic QCD]]", such as the [[Witten-Sakai-Sugimoto model]]):

* Chiung Hwang, Joonho Kim, Seok Kim, [[Jaemo Park]], _General instanton counting and 5d SCFT_ ([arxiv:1406.6793](https://arxiv.org/abs/1406.6793))

On [[E-strings]]:

* {#KKLPV14} Joonho Kim, Seok Kim, Kimyeong Lee, [[Jaemo Park]], [[Cumrun Vafa]], _Elliptic Genus of E-strings_, JHEP 1709 (2017) 098 ([arXiv:1411.2324](https://arxiv.org/abs/1411.2324))

category: people


Jagdish Mehra

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

## Selected writings

On the [[interpretation of quantum mechanics]]

* [[Jagdish Mehra]], *The quantum principle: Its interpretation and epistemology*, Dialectica **27** 2 (1973) 75-157 [[jstor:42968519](https://www.jstor.org/stable/42968519)]

and on Wigner's writings on the matter:

* [[Eugene P. Wigner]], *Epistemology of Quantum Mechanics*, Part I in: [[Jagdish Mehra]] (ed.), *Philosophical Reflections and Syntheses* Vol B.VI in: *The Collected Works of Eugene Wigner*, Springer (1973) [[doi:10.1007/978-3-642-78374-6_5](https://doi.org/10.1007/978-3-642-78374-6_5)]

On the history of [[quantum mechanics]]

with focus on [[quantum probability]] ([[Born rule]]):

* [[Jagdish Mehra]], Helmut Rechenberg, *The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932*, Part 1 in:  *The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941*, Springer (2001)   [[ISBN:978-0-387-98971-6](https://link.springer.com/book/9780387989716)]

category: people


Jake Bian

* [webpage](-)

category: people


Jake Chandler

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

## related $n$Lab entries:

* [[necessity and possibility]]

category: people


Jakob Böhme

+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jakob_B%C3%B6hme)

* [[Hegel]], section _[Bacon and Böhme](Lectures+on+the+History+of+Philosophy#BaconAndBoehme)_ in _[[Lectures on the History of Philosophy]]_

## related $n$Lab entries

* [[mysticism]]

* [[Spinoza's system]]

category: people

[[!redirects Jakob Boehme]]
[[!redirects Böhme]]
[[!redirects Boehme]]


Jakob Nielsen

_Jakob Nielsen_ was a Danish mathematician and one of the pioneers of the [[combinatorial group theory]]. [[Nielsen-Schreier theorem]], [[HNN-extension]]s, Nielsen–Thurston classification, Nielsen fixed point theorem, [[Nielsen invariant]], [[Fenchel-Nielson coordinates]] on Teichmüller space and Nielsen's method of small cancellations were named after him.

* wikipedia: [Jakob Nielsen (mathematician)](http://en.wikipedia.org/wiki/Jakob_Nielsen_%28mathematician%29), [Nielsen–Thurston classification](http://en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston_classification), [Nielsen theory](http://en.wikipedia.org/wiki/Nielsen_theory)

category: people


Jakob Palmkvist

* [webpage](https://www.chalmers.se/en/staff/Pages/jakob-palmkvist.aspx)

## Selected writings

On [[M2-brane 3-algebras]]:

* [[Jakob Palmkvist]], _Three-algebras, triple systems and 3-graded Lie superalgebras_, J. Phys. A43:015205, 2010 ([arXiv:0905.2468](https://arxiv.org/abs/0905.2468))

On [[tensor hierarchies]] in [[gauged supergravity]]:

* Jesper Greitz, [[Paul Howe]], [[Jakob Palmkvist]], _The tensor hierarchy simplified_, Class. Quantum Grav. 31 (2014) 087001 ([arXiv:1308.4972](https://arxiv.org/abs/1308.4972))

On relation of [[Borcherds algebras]] to [[tensor hierarchies]] in [[gauged supergravity]]:

* {#Palmkvist12} [[Jakob Palmkvist]], _Tensor hierarchies, Borcherds algebras and $E_{11}$_, JHEP 1202 (2012) 066 ([arXiv:1110.4892](http://arxiv.org/abs/1110.4892))

* [[Jakob Palmkvist]], _The tensor hierarchy algebra_, J. Math. Phys. 55, 011701 (2014) ([arXiv:1305.0018](https://arxiv.org/abs/1305.0018))

and to [[exceptional generalized geometry]]:

* [[Jakob Palmkvist]], _Exceptional geometry and Borcherds superalgebras_ ([arXiv:1507.08828](http://arxiv.org/abs/1507.08828))

On the [[Kantor-Koecher-Tits construction]]:

* [[Jakob Palmkvist]], _A generalization of the Kantor-Koecher-Tits construction_, J. Gen. Lie Theory Appl. 2, (2008) 226 ([arXiv:1308.3761](https://arxiv.org/abs/1308.3761))

A proposal for understanding [[tensor hierarchies]] as [[dg-Lie algebra]]/[[L-infinity algebra]]-refinenents of [[Leibniz algebras]]:

* [[Sylvain Lavau]], [[Jakob Palmkvist]], _Infinity-enhancing of Leibniz algebras_ ([arXiv:1907.05752](https://arxiv.org/abs/1907.05752))

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

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

category: people


Jakob Scholbach

* [personal page](https://ivv5hpp.uni-muenster.de/u/jscho_04/)

## Selected writings

On [[rectification]] of [[symmetric operad|symmetric]] [[colored operads]]:

* {#PavlovScholbach18} [[Dmitri Pavlov]], [[Jakob Scholbach]], *Admissibility and rectification of colored symmetric operads*, Journal of Topology **11** 3 (2018) 559-601 $[$[doi:10.1112/topo.12008](https://doi.org/10.1112/topo.12008), [arXiv:1410.5675](https://arxiv.org/abs/1410.5675)$]$

category: people


Jakob Schwichtenberg

* [personal page](http://jakobschwichtenberg.com/)

## Selected writings

Exposition of the [[asymptotic series|asymptotic]] nature of the [[Feynman perturbation series]] in [[perturbative quantum field theory]]:

* [[Jakob Schwichtenberg]], *[divergence-perturbation-series-qft](http://jakobschwichtenberg.com/divergence-perturbation-series-qft)*

category: people


Jakob Stix

* [webpage](http://www.math.uni-frankfurt.de/~stix/publikationen.html)

category: people


Jakob von Raumer

* [website](http://von-raumer.de/academic/)

## Publications

* [[Jakob von Raumer]], [[Nicolai Kraus]], *Path Spaces of Higher Inductive Types in Homotopy Type Theory*,  34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Vancouver, Canada, 2019 ([arXiv:1901.06022](https://arxiv.org/abs/1901.06022))

## Talks

* [[Jakob von Raumer]], *Coherence via Well-Foundedness*, [[Homotopy Type Theory Electronic Seminar Talks]], 24 September 2020 ([video](https://www.youtube.com/watch?v=5iAQ-JVdlmc), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/von-Raumer-2020-09-24-HoTTEST.pdf))

category: people


Jakob Yngvason

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

## Selected writings

On [[self-adjoint extensions]] for [[local nets of observables]] in [[AQFT]]:

*  [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *Local nets and self-adjoint extensions of quantum field operators*, Letters in Mathematical Physics, **21** (1991) 151–155 [[doi:10.1007/BF00401649](https://doi.org/10.1007/BF00401649)]

On the [[PCT theorem]] for [[local observables]] in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *On the PCT--Theorem in the Theory of Local Observables*, in: *Mathematical physics in mathematics and physics: Quantum and operator algebraic aspects*, Fields Inst. Commun. **30** (2001) 39-64  [[arXiv:math-ph/0012020](http://arxiv.org/abs/math-ph/0012020), [spire:538524](https://inspirehep.net/literature/538524)]

## Related entries

* [[von Neumann algebra factor]]

category: people


James Alvey

* [personal page](https://james-alvey-42.github.io/)

## Selected writings

On [[primordial nucleosynthesis]]:

* [[James Alvey]], *What can you measure with BBN in 2021?*, Cosmology Talks 2021 ([[Alvey_BBN2021.pdf:file]])

category: people


James Arthur

* [webpage](http://www.math.toronto.edu/arthur/)

## Selected writings

On the work of [[Robert Langlands]]:

* [[James Arthur]], _The work of Robert Langlands_ ([arXiv:2307.02571](https://arxiv.org/abs/2307.02571))

## Related entries

* [[Arthur-Selberg trace formula]]

category:people


James Becker

* [mathematics genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=5541)

## Related entries

* [[dual object]]

* [[Becker-Gottlieb transfer]]

category: people


James Bjorken

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

## Selected writing

In the history of the [[standard model of particle physics]]:

* {#Bjorken85} [[James Bjorken]], _The November Revolution: A Theorist Reminisces_, 1985 ([spire:214067](http://inspirehep.net/record/214067))

> The big international conference of $[$1974$]$ in
London was a turning point
 $[$...$]$ Ellis' catalog well reflected the state  of theoretical confusion and general disarray in trying to  interpret the $e^+ e^-$ data. But in the midst of all of this was a
talk by [[John Iliopoulos]] (I think I was there too). With
passionate zealotry, he laid out with great accuracy what we call
the [[standard model of particle physics|standard model]]. Everything was there: [[proton decay]], [[charm quark|charm]],  the GIM mechanism of course, [[QCD]], the $SU(2)\times U(1)$ [[electroweak  theory]], $SU(5)$ [[GUT|grand unification]], [[Higgs field|Higgs]], etc. It was all presented with absolute conviction and sounded at the time just a little mad, at least to me (I am a conservative).

category: people


James Borger

* [webpage](http://maths-people.anu.edu.au/~borger/)

## related $n$Lab entries

* [[F1]]
* [[Witt vector]]
* [[lambda-ring]]
* [[arithmetic jet space]]

## Papers and talks

* _Witt vectors, lambda-rings, and arithmetic jet spaces_, [lecture series](http://maths-people.anu.edu.au/~borger/classes/copenhagen-2016/index.html), April-June 2016.

* _Generalized symmetries and arithmetic applications_, [slides](http://www.mat.uc.pt/~ct2018/slides/J_Borger.pdf), CT2018, Azores, July 2018

category: people


James Cannon

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

category: people


James Chapman

* [personal page](https://jmchapman.io/index.html)

## Selected writings

Introducing [[relative monad|relative]] [[monads in computer science]]:

* [[Thorsten Altenkirch]], [[James Chapman]], [[Tarmo Uustalu]], *Monads need not be endofunctors*, Logical Methods in Computer Science **11** 1:3 (2015) 1–40 [[arXiv:1412.7148](https://arxiv.org/abs/1412.7148), [pdf](http://www.cs.nott.ac.uk/~txa/publ/jrelmon.pdf), doi:10.2168/LMCS-11(1:3)2015]

category: people


James Clerk Maxwell

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

## Selected writings

Introducing [[Maxwell's equations]] for [[electromagnetism]]:

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

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

Historical account of a poem Maxwell addressed to [[Peter Tait]] concerning the theory developed jointly with [[Lord Kelvin]] on atoms as knotted vortices of ether (cf. *[[On Vortex Atoms]]*)

* Daniel S. Silver, _The Last Poem of James Clerk Maxwell_, Notices of the AMS 55(10), 1266-70 ([pdf](https://www.ams.org/notices/200810/tx081001266p.pdf))

## Related entries

* [[electromagnetism]]

* [[Maxwell's equations]]

category: people

[[!redirects James Maxwell]]