Peter Landweber

* [webpage](http://www.math.rutgers.edu/people/index.php?type=emeritus&id=210)

## Selected writings

The [[Landweber exact functor theorem]]:

* {#Landweber76} [[Peter Landweber]], _Homological properties of comodules over $MU^\ast(MU)$ and $BP^\ast(BP)$_, American Journal of Mathematics (1976): 591-610 ([jstor:2373808](https://www.jstor.org/stable/2373808))

On [[bordism homology theory]]/[[cobordism cohomology theory]]:

* [[Peter Landweber]], _A survey of bordism and cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 2 September 1986 , pp. 207-223 ([doi:10.1017/S0305004100066032](https://doi.org/10.1017/S0305004100066032))

Introducing [[elliptic cohomology]]:

* {#Landweber88} [[Peter Landweber]], *Elliptic Cohomology and Modular Forms*, in: _Elliptic Curves and Modular Forms in Algebraic Topology_, Lecture Notes in Mathematics **1326** (1988) 55-68 [[doi:10.1007/BFb0078038](https://doi.org/10.1007/BFb0078038)]

  > (see also *[[modular forms]]*)

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et al (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

* {#Landweber88} [[Peter Landweber]], _Elliptic genera: An introductory overview_ In: P. Landweber (eds.) _Elliptic Curves and Modular Forms in Algebraic Topology_, Lecture Notes in Mathematics, vol 1326. Springer  (1988) ([doi:10.1007/BFb0078036](https://doi.org/10.1007/BFb0078036))

## Related $n$Lab entries

* [[Landweber exact functor theorem]]

* [[elliptic genus]]

category: people


Peter Lax

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

## Selected writings

Introducing what came to be called the *[[Lax equation]]* for [[integrable systems]]:

* [[Peter Lax]], *Integrals of nonlinear equation of evolution and solitary waves*, Commun. on Pure and Applied Mathematics __21__ 5 (1968) 467-490 [[doi:10.1002/cpa.3160210503](http://dx.doi.org/10.1002/cpa.3160210503)]

category: people


Peter LeFanu Lumsdaine

* Webpage: [http://peterlefanulumsdaine.com](http://peterlefanulumsdaine.com)

## Selected writings

On the [[relation between type theory and category theory|categorical semantics]] of [[homotopy type theory]] in [[simplicial sets]]/[[infinity-groupoids|$\ionfty$-groupoids]]:

* {#KapulkinLumsdaine12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The Simplicial Model of Univalent Foundations (after Voevodsky)*, Journal of the European Mathematical Society **23** (2021) 2071–2126  $[$[arXiv:1211.2851](https://arxiv.org/abs/1211.2851), [doi:10.4171/jems/1050](https://doi.org/10.4171/jems/1050)$]$

On [[higher inductive types]] in [[homotopy type theory]]:

* [[Peter LeFanu Lumsdaine]], *Higher inductive types, a tour of the menagerie* (2011) [[blog post](http://homotopytypetheory.org/2011/04/24/higher-inductive-types-a-tour-of-the-menagerie/)]

* [[Peter LeFanu Lumsdaine]], *Higher Inductive Types: The circle and friends, axiomatically* (2011) [[pdf](http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS2011/Lumsdaine_slides.pdf), [[Lumsdaine-HITs.pdf:file]]]

and on their [[categorical semantics]]:

* {#LumsdaineShulman17} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], *Semantics of higher inductive types*, Math. Proc. Camb. Phil. Soc. **169** (2020) 159-208 [[arXiv:1705.07088](https://arxiv.org/abs/1705.07088), talk slides [pdf](http://home.sandiego.edu/~shulman/papers/cellcxs.pdf), [doi:10.1017/S030500411900015X](https://doi.org/10.1017/S030500411900015X)]

with precursors in:

* {#ShulmanLumsdaine12} [[Mike Shulman]], [[Peter LeFanu Lumsdaine]], *Semantics of higher inductive types* (2012) [[[ShulmanLumsdaine-HITs2012.pdf:file]]]

* {#ShulmanLumsdaine16} [[Mike Shulman]], [[Peter LeFanu Lumsdaine]], _Semantics and syntax of higher inductive types_ (2016) [[slides](http://home.sandiego.edu/~shulman/papers/stthits.pdf)]

On [[synthetic homotopy theory]] in [[homotopy type theory]]:

* [[Guillaume Brunerie]],  [[Dan Licata]], [[Peter LeFanu Lumsdaine]]: _Homotopy theory in type theory_, 2013 ([pdf slides](http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf), [[Licata-HomotopyInTypeTheory.pdf:file]], [blog entry 1](https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction), [blog entry 2](https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/))

Introducing the [[functional programming language|functional]] [[quantum programming language]] [[Quipper]]:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))

On the [[categorical semantics of dependent type theory]] via [[comprehension categories]]:

* {#LumsdaineWarren13} [[Peter LeFanu Lumsdaine]], [[Michael Warren]], *The local universes model: An overlooked coherence construction for dependent type theory*, ACM Trans. Comput. Log., **16** 3 no 23 (2015) 1-31 [[arXiv:1411.1736](https://arxiv.org/abs/1411.1736), [doi:10.1145/2754931](https://doi.org/10.1145/2754931)]

Formalization of (the proof of) the [[Blakers-Massey theorem]] in [[homotopy type theory]]:

* {#FFLL16} [[Kuen-Bang Hou]] ([[Favonia]]), [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], *A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory*, LICS '16 (2016) 565–574  [[arXiv.1605.03227](https://arxiv.org/abs/1605.03227), [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)]

On [[dependent type theory]]:

* {#BauerHaselwarterLumsdaine20} [[Andrej Bauer]], [[Philipp G. Haselwarter]], [[Peter LeFanu Lumsdaine]], *A general definition of dependent type theories* [[arXiv:2009.05539](https://arxiv.org/abs/2009.05539)]

## Related entries

* [[initiality conjecture]]

* [[univalent foundations of mathematics]]

* [[UniMath project]]

category: people

[[!redirects Peter Lumsdaine]]


Peter May

J. Peter May is a [[homotopy theory|homotopy theorist]] at the University of Chicago, inventor of [[operads]] as a technique for studying [[infinite loop spaces]] and [[spectrum|spectra]].

* [Peter May's webpage](http://www.math.uchicago.edu/~may/).

* [Peter May: Wikipedia entry](http://en.wikipedia.org/wiki/J._Peter_May).

Peter May's work makes extensive use of [[enriched category theory|enriched]]- and [[model category|model]]-[[category theory]] as power tools in [[algebraic topology]]/[[homotopy theory]], notably in discussion of [[highly structured spectra]] in [MMSS00](#MMSS00)'s _[[Model categories of diagram spectra]]_ (for exposition see _[[Introduction to Stable homotopy theory -- 1-2]]_), or in the discussion of [[genuine equivariant spectra]] or [[K-theory of a permutative category|K-theory of permutative categories]], etc.. While he has co-edited a book collection on [[higher category theory]] ([Baez-May 10](#BaezMay10)) and eventually had high praise ([May 16](#MayBanff16)) for [[2-category theory]] as a tool in [[algebraic topology]]/[[higher algebra]], he has vocally warned against seeing abstract [[(∞,1)-category theory]] as a replacement for concrete realizations in [[model category]]-theory ([P. May, MO comment Dec 2013](https://mathoverflow.net/a/83307/381)).

## Selected writings

On [[algebraic topology]]

* [[Peter May]], _[[A concise course in algebraic topology]]_

* [[Peter May]], [[Kate Ponto]], _[[More concise algebraic topology]]_

On [[simplicial objects]] in [[algebraic topology]] ([[simplicial homotopy theory]]):

* [[Peter May]], _Simplicial objects in algebraic topology_, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

On [[infinite loop spaces]]:

* {#May1972} [[Peter May]], _The geometry of iterated loop spaces_, 1972 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))

* [[Peter May]], _Infinite loop space theory_, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. ([Euclid](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183538891))

  _Infinite loop space theory revisited_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/28.pdf))

On [[E-infinity spaces|$E_\infty$-spaces]] and [[E-infinity ring spectra|$E_\infty$-ring sectra]]:

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

On [[equivariant cohomology]] and [[equivariant homotopy theory]]:

* {#May96} [[Peter May]] et al., _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics Volume: 91; 1996 ([ISBN:978-0-8218-0319-6](https://bookstore.ams.org/cbms-91), [pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf), [pdf](https://ncatlab.org/nlab/files/MayEtAlEquivariant96.pdf))

On [[highly structured spectra]]:

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society Volume 82, Issue 2, 2000  ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [doi:10.1112/S0024611501012692]( https://doi.org/10.1112/S0024611501012692))

On [[H-infinity ring spectra|$H_\infty$-ring spectra]]:

* {#BrunerMayMcLureSteinberger86} [[Robert Bruner]], [[Peter May]], [[James McClure]], [[Mark Steinberger]], *$H_\infty$-Ring Spectra and their Applications*, Lecture Notes in Mathematics **1176**, Springer (1986) [[doi:10.1007/BFb0075405](https://link.springer.com/book/10.1007/BFb0075405), [pdf](http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf)]

On [[equivariant bundles]]:

* [[Richard Lashof]], [[Peter May]], _Generalized equivariant bundles_, Bull. Soc. Math. Belg. Sér. A 38, 265-271, 1986 ([pdf](http://math.uchicago.edu/~may/PAPERS/55.pdf), [[LashofMayGeneralizedEquivariantBundles.pdf:file]])

On [[classifying spaces]]/[[universal principal bundles]] for [[equivariant principal bundles]]:

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Théorie de l'homotopie, Astérisque, no. 191 (1990), 15 p. ([numdam:AST_1990__191__239_0](http://www.numdam.org/item/?id=AST_1990__191__239_0))

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]] _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

On [[equivariant bundles]] with [[abelian group|abelian]] [[structure group]]:

* {#LashofMaySegal83} [[Richard Lashof]], [[Peter May]], [[Graeme Segal]], _Equivariant bundles with abelian structural group_, Contemporary Mathematics,  Volume 19, 1983 ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019), [pdf](http://www.math.uchicago.edu/~may/PAPERS/45.pdf), [[LashofMaySegalEquivariantBundles.pdf:file]])

On [[higher algebra]] ([[brave new algebra]]) in [[stable homotopy theory]], i.e. on [[ring spectra]], [[module spectra]] etc.:

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]],
_[[Rings, modules and algebras in stable homotopy theory]]_,
Mathematical Surveys and Monographs Volume 47, AMS 1997 ([ISBN:978-0-8218-4303-1](https://bookstore.ams.org/surv-47-s), [pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

On [[module spectra]]:

* [[Peter May]], _Equivariant and non-equivariant module spectra_, Journal of Pure and Applied Algebra Volume 127, Issue 1, 1 May 1998, Pages 83–97 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/modfinal.pdf))

On [[operads]] and [[motives]]:

* [[Igor Kriz]], [[Peter May]], _Operads, algebras, modules and motives_, Astérisque __233__, Société Mathématique de France (1995).

On [[equivariant stable homotopy theory]]:

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

On [[equivariant complex oriented cohomology theory]]:

* [[Steven Costenoble]], [[Peter May]], [[Stefan Waner]], _Equivariant orientation theory_, Homology Homotopy Appl. Volume 3, Number 2 (2001), 265-339 ([euclid:hha/1139840256](https://projecteuclid.org/euclid.hha/1139840256))

On [[tensor triangulated categories]] and [[traces]]:

* {#May01} [[Peter May]], _The Additivity of Traces in Triangulated Categories_, Advances in Mathematics, Volume 163, Issue 1, 2001, Pages 34-73 ([doi:10.1006/aima.2001.1995](https://doi.org/10.1006/aima.2001.1995))

On the [[Picard infinity-group]] of [[equivariant stable homotopy theory]] and the notion of [[RO(G)-grading]]:

* {#FauskLewisMay01} [[Halvard Fausk]], [[L. Gaunce Lewis]], [[Peter May]],  _The Picard group of equivariant stable  homotopy theory_, Advances in Mathematics Volume 163, Issue 1, 15 October 2001, Pages 17–33  ([pdf](http://www.math.uchicago.edu/~may/PAPERS/FLMJan01.pdf))

On [[six operations]] and [[Wirthmüller contexts]]:

* [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

On [[parametrized stable homotopy theory]]:

* [[Peter May]], [[Johann Sigurdsson]], _Parametrized Homotopy Theory_, Mathematical Surveys and Monographs, vol. 132, AMS 2006  ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf]( http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

On [[enriched model category theory]]:

* [[Bertrand Guillou]], [[Peter May]], _Enriched model categories and presheaf categories_, New York J. Math. 26 (2020) 37–9 ([arXiv:1110.3567](http://arxiv.org/abs/1110.3567), [](https://www.emis.de/journals/NYJM/j/2020/26-3.html), [NYJM:2020/26-3](https://www.emis.de/journals/NYJM/j/2020/26-3.html))

On [[higher category theory]]:

* {#BaezMay10} [[John Baez]], [[Peter May]] (eds.), _[[johnbaez:Towards Higher Categories|Towards Higher Categories]]_, Springer 2010 ([doi:10.1007/978-1-4419-1524-5](https://link.springer.com/book/10.1007/978-1-4419-1524-5))

Specifically on [[2-category theory]] as a tool in [[spectral algebraic geometry]], [[equivariant homotopy theory]] and [[infinite loop space]]-theory:

* {#MayBanff16} [[Peter May]], _Input for derived algebraic geometry:equivariant multiplicativeinfinite loop space theory_, Banff 2016 ([pdf](http://www.math.uchicago.edu/~may/TALKS/Chicago2016.pdf), [[MayBanff2016.pdf:file]])

On [[equivariant homotopy theory]] and [[Elmendorf's theorem]] via [[enriched model categories]]:

* {#GuillouMayRubin13} [[Bertrand Guillou]], [[Peter May]], [[Jonathan Rubin]], _Enriched model categories in equivariant contexts_, Homology, Homotopy and Applications 21 (1), 2019 ([arXiv:1307.4488](https://arxiv.org/abs/1307.4488), [arXiv:10.4310/HHA.2019.v21.n1.a10](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10))

## Related $n$Lab entries

* [[stable homotopy theory]]

* [[equivariant homotopy theory]] ([[Bredon cohomology]], [[equivariant stable homotopy theory]], [[rational equivariant stable homotopy theory]])

* [[higher algebra]]

* [[model category]]

* [[operad]]

* [[infinite loop space]]

category: people

[[!redirects May]]
[[!redirects J. Peter May]]


Peter Mayr

* [webpage](http://www.theorie.physik.uni-muenchen.de/MATH/members/asc/professors/mayr_peter/index.html)

## Selected writings

On [[quantum K-theory rings]]:

* [[Hans Jockers]], [[Peter Mayr]], *Quantum K-Theory of Calabi-Yau Manifolds*, J. High Energ. Phys. **2019** 11 (2019) [[arXiv:1905.03548](https://arxiv.org/abs/1905.03548), doi:10.1007/JHEP11(2019)011]

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

* [[Hans Jockers]], [[Peter Mayr]], *A 3d Gauge Theory/Quantum K-Theory Correspondence*, Advances in Theoretical and Mathematical Physics **24** 2 (2020) [[arXiv:1808.02040](https://arxiv.org/abs/1808.02040), [doi:10.4310/ATMP.2020.v24.n2.a4](https://dx.doi.org/10.4310/ATMP.2020.v24.n2.a4)]

category: people


Peter McCullagh

* [Home page](http://www.stat.uchicago.edu/~pmcc/)

category:people


Peter Michor

* [website](http://www.mat.univie.ac.at/~michor/)

## Selected writings

Characterizing the [[Schouten-Nijenhuis bracket]] as a [[natural operation]]:

* {#Michor87} [[Peter W. Michor]], *Remarks on the Schouten-Nijenhuis bracket*,  In: J. Bureš, [[Vladimir Souček|V. Souček]] (eds.): *Proceedings of the Winter School "Geometry and Physics"* Circolo Matematico di Palermo, Palermo (1987) 207-215 [[dml:701423](https://dml.cz/handle/10338.dmlcz/701423), [pdf](https://dml.cz/bitstream/handle/10338.dmlcz/701423/WSGP_07-1987-1_21.pdf), [[Michor-SchoutenNijenhuisBracket.pdf:file]]]

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

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

A textbook on [[differential geometry]], including an account of [[Schouten–Nijenhuis bracket]], [[Frölicher–Nijenhuis bracket]], and [[Nijenhuis–Richardson bracket]]:

* [[Peter W. Michor]], _Topics in Differential Geometry_, Graduate Studies in Mathematics 93 (2008).  [PDF](https://www.mat.univie.ac.at/~michor/dgbook.pdf).

## Related entries

* [[generalized smooth space]]

* [[generalized smooth algebra|C∞-ring]]

category: people

[[!redirects Peter W. Michor]]


Peter Minkowski

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

## Selected writings

Introducing "SO(10)" ([[Spin(10)]]) [[grand unified theory]]:

* [[Harald Fritzsch]], [[Peter Minkowski]], _Unified interactions of leptons and hadrons_, Annals of Physics Volume 93, Issues 1–2, 5 September 1975, Pages 193-266 (doi:10.1016/0003-4916(75)90211-0)

Introducing the concept of [[glueballs]]:

* [[Harald Fritzsch]], [[Peter Minkowski]], _Ψ-resonances, gluons and the Zweig rule_, Nuov. Cim. A, 1975,30: 393 ([doi:10.1007/BF02730295](https://doi.org/10.1007/BF02730295))

category: people


Peter O'Hearn

* [webpage](http://www0.cs.ucl.ac.uk/staff/p.ohearn/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_O%27Hearn)

## Selected writings

On [[bunched logic]]:

* [[Peter O'Hearn]], [[David J. Pym]], *The Logic of Bunched Implications*, The Bulletin of Symbolic Logic **5** 2 (1999) 215-244  [[pdf](http://www.lsv.ens-cachan.fr/~demri/OHearnPym99.pdf), [doi:10.2307/421090](https://doi.org/10.2307/421090)]

* [[Peter O'Hearn]], *On bunched typing*, Journal of Functional Programming **13** 4 (2003) 747-796 [[doi:10.1017/S0956796802004495](https://doi.org/10.1017/S0956796802004495)]

On [[bireflective subcategories]] with [[ambidextrous adjoints]]:

* {#FOPTST99} [[Peter Freyd]], [[Peter O’Hearn]],  [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], *Bireflectivity*, Theoretical Computer Science **228** 1–2 (1999) 49-76 [doi:10.1016/S0304-3975(98)00354-5]

## Related entries

* [[bunched logic]]
* [[separation logic]]

category: people

category: computer science

[[!redirects Peter O’Hearn]]
[[!redirects P. W. O’Hearn]]


Peter Olver

_Peter J. Olver_ is a mathematician at the University of Minnesota in Minneapolis. His interests include the application of the Lie theory to dynamical systems, Hamiltonian systems, [[variational calculus]], invariant theory...

* [homepage](http://www.math.umn.edu/~olver)

* books by Peter Olver: [complete list](http://www.math.umn.edu/~olver/books.html)

The following two books have much overlap with some topics in $n$lab:

* Applications of Lie groups to differential equations, Springer (few editions);

* Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.

category: people


Peter Orlik

Peter Orlik is a professor emeritus at the Departement of Mathematics of the University of Wisconsin at Madison.

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

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=5509)

* [university page](https://math.wisc.edu/staff/orlik-peter)

## Selected writings

On the ([[holomorphic de Rham complex|holomorphic]]) [[de Rham cohomology]] of [[configuration spaces of points]] (whose "[[twisted de Rham cohomology|twisted]]" generalization leads to the [[hypergeometric construction of KZ solutions]]):

* [[Peter Orlik]], [[Louis Solomon]], *Combinatorics and topology of complements of hyperplanes*, Invent Math **56** (1980) 167–189  [[doi:10.1007/BF01392549](https://doi.org/10.1007/BF01392549)]

On [[local system]]-[[twisted de Rham cohomology]] over [[configuration spaces of points]] (such as for the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]):

* [[Peter Orlik]], *Hypergeometric integrals and arrangements*, Journal of Computational and Applied Mathematics **105** (1999) 417–424 [doi:10.1016/S0377-0427(99)00036-9, [pdf](https://core.ac.uk/download/pdf/82631708.pdf)]

* [[Daniel C. Cohen]], [[Peter Orlik]], *Arrangements and local systems*, Math. Res. Lett. **7** (2000) 299-316 $[$[arXiv:math/9907117](https://arxiv.org/abs/math/9907117), [doi:10.4310/MRL.2000.v7.n3.a5](https://dx.doi.org/10.4310/MRL.2000.v7.n3.a5)$]$

On [[Gauss-Manin connections]] over [[configuration spaces of points]]:

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, I Eigenvalues*, Compositio Math. **136** (2003) 299-316 $[$[arXiv:math/0105063](https://arxiv.org/abs/math/0105063), [doi:10.1023/A:1023262022279](https://doi.org/10.1023/A:1023262022279)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, II Nonresonant weights*, Amer. J. Math. **127** (2005) 569-594 $[$[arXiv:math/0207114](https://arxiv.org/abs/math/0207114), [jstor:40067930](https://www.jstor.org/stable/40067930)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, III Formal connections*, Trans. Amer. Math. Soc. **357** (2005) 3031-3050 $[$[arXiv:math/0307210](https://arxiv.org/abs/math/0307210), [doi:10.1090/S0002-9947-04-03621-9](https://doi.org/10.1090/S0002-9947-04-03621-9)$]$

category: people


Peter Ouyang

* [webpage](https://academictree.org/physics/peopleinfo.php?pid=459280)

## Selected writings

On [[flavor branes]] modeled as [[probe branes]] in the [[AdS/QCD correspondence]]:

* [[Peter Ouyang]], _Holomorphic D7-Branes and Flavored $\mathcal{N}=1$ Gauge Theories_, Nucl. Phys. B699:207-225, 2004 ([arXiv:hep-th/0311084](https://arxiv.org/abs/hep-th/0311084))

* Thomas S. Levi, [[Peter Ouyang]], _Mesons and Flavor on the Conifold_, Phys. Rev. D76:105022, 2007 ([arXiv:hep-th/0506021](https://arxiv.org/abs/hep-th/0506021))

category: people


Peter Sarnak

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

## related $n$Lab entries

* [[zeta function of a Riemann surface]]

category: people


Peter Schauenburg

* [personal page](http://schauenburg.perso.math.cnrs.fr/personnelle.html)

* [personal page](https://www.mathematik.uni-muenchen.de/~schauen/)

* [GoogleScholar page](https://scholar.google.de/citations?user=ajDR-O4AAAAJ&hl=en)

## Selected writings

On [[bialgebras]]:

* [[Peter Schauenburg]], _Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules_, Appl. Cat. Str. **6** (1998) 193–222 [[doi:10.1023/A:1008608028634](https://doi.org/10.1023/A:1008608028634)]

On [[strict monoidal categories]]:

* [[Peter Schauenburg]], *Turning Monoidal Categories into Strict Ones*, New York Journal of Mathematics **7** (2001) 257-265 [[nyjm:j/2001/7-16](http://nyjm.albany.edu/j/2001/7-16.html), [eudml:121925](https://eudml.org/doc/121925)]

## Related entries

* [[Schauenburg bialgebroid]]

category: people


Peter Schneider

* [webpage](http://wwwmath.uni-muenster.de/u/schneider/)

## Selected writings

On [[Deligne cohomology]] and [[Beilinson's conjectures]] on [[special values of L-functions]]:

* [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_  Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)

category: people


Peter Scholze

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

## Selected writings

On [[p-adic Hodge theory]]:

* {#Scholze13} [[Peter Scholze]], _p-adic Hodge theory for rigid-analytic varieties_, Forum of Mathematics, Pi, 1, e1 2013. [arXiv:1205.3463](https://arxiv.org/abs/1205.3463)

* {#BhattMorrowScholze16}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]] *Integral $p$-adic Hodge Theory* [[arXiv:1602.03148](https://arxiv.org/abs/1602.03148)]

* {#BhattMorrowScholze18}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]], *Topological Hochschild homology and integral $p$-adic Hodge theory* [[arXiv:1802.03261](https://arxiv.org/abs/1802.03261)]

* {#BhattScholze19}[[Bhargav Bhatt]], [[Peter Scholze]], _Prisms and prismatic cohomology_,  Ann. of Math. (2) 196(3): 1135-1275 [doi](https://10.4007/annals.2022.196.3.5) arXiv:[1905.08229](https://arxiv.org/abs/1905.08229)

* Bhargav Bhatt, [[Peter Scholze]], _Projectivity of the Witt vector affine Grassmannian_, Invent. math. 209, 329-423 (2017) [doi](https://doi.org/10.1007/s00222-016-0710-4) [arXiv:1507.06490](https://arxiv.org/abs/1507.06490)

On [[cyclotomic spectra]] and [[topological cyclic homology]]:

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_,  Acta Math. **221** 2 (2018) 203-409 [[arXiv:1707.01799](https://arxiv.org/abs/1707.01799), [doi:10.4310/ACTA.2018.v221.n2.a1](https://dx.doi.org/10.4310/ACTA.2018.v221.n2.a1)]

On the [[local Langlands conjecture]]:

* [[Laurent Fargues]], [[Peter Scholze]], _Geometrization of the local Langlands correspondence_ ([pdf](http://www.math.uni-bonn.de/people/scholze/Geometrization.pdf))

On [[condensed mathematics]] and [[analytic geometry]]:

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)

* {#ScholzeLAG} [[Peter Scholze]], _Lectures on analytic geometry_, [pdf](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf)

* [[Dustin Clausen]], [[Peter Scholze]], _Condensed Mathematics and Complex Geometry_, [pdf](https://people.mpim-bonn.mpg.de/scholze/Complex.pdf)

## Related entries

* [[p-adic Hodge theory]]

* [[local Langlands correspondence]]

* [[pro-étale morphism]]

* [[perfectoid space]]

* [[Rapoport-Zink space]]

* [[shtuka]]

* [[prismatic cohomology]]

category: people


Peter Schweitzer

* [Institute page](https://physics.uconn.edu/person/peter-schweitzer/)

# Selected writings

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

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

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

category: people


Peter Selinger

* [personal webpage](http://www.mscs.dal.ca/~selinger/)

* [institute page](https://www.dal.ca/faculty/science/math-stats/faculty-staff/our-faculty/mathematics/peter-selinger.html)

## Selected writings

Early consideration of [[quantum programming languages]] ([[QPL]]):

* {#Selinger04} [[Peter Selinger]], _Towards a quantum programming language_, Mathematical Structures in Computer Science 14(4):527–586, 2004 ([doi:10.1017/S0960129504004256](https://doi.org/10.1017/S0960129504004256), [pdf](https://www.mathstat.dal.ca/~selinger/papers/qpl.pdf),  [web](https://www.mathstat.dal.ca/~selinger/papers.html#qpl))

via [[quantum lambda-calculus]] invoking [[linear type theory|linear types]]:

* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *A lambda calculus for quantum computation with classical control*, Proc. of TLCA 2005 $[$[arXiv:cs/0404056](https://arxiv.org/abs/cs/0404056), [doi:10.1007/11417170_26](https://doi.org/10.1007/11417170_26)$]$

* {#SelingerValiron09} [[Peter Selinger]], [[Benoît Valiron]], *Quantum Lambda Calculus*, in: *Semantic Techniques in Quantum Computation*, Cambridge University Press (2009) 135-172 $[$[doi:10.1017/CBO9781139193313.005](https://doi.org/10.1017/CBO9781139193313.005), [pdf](https://www.mscs.dal.ca/~selinger/papers/qlambdabook.pdf)$]$

> [[Selinger (2016)](Quipper#Selinger16):] When the [QPL workshop series](https://www.mathstat.dal.ca/~selinger/qpl/) was first founded, it was called "*Quantum Programming Languages*". One year I wasn't participating, and while I wasn't looking they changed the name to "*Quantum Physics and Logic*" --- same acronym!

> Back in those days in the early 21st century we were actually trying to do programming languages for quantum computing $[$[Selinger & Valiron 2004](#SelingerValiron04)$]$, but the sad thing is: In those days nobody really cared. $[...]$

> Now it's 15 years later and several of these parameters have changed: There has been a renewed interest, from government agencies and also from companies who are actually building quantum computers. $[...]$.

> So now people are working on quantum programming languages *again*.

then via [[Quipper]]:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))

* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))

* [[Peter Selinger]], _The Quipper Language_ ([web](http://www.mathstat.dal.ca/~selinger/quipper/))

* [[Peter Selinger]], *Introduction to Quipper*, talk at [QPL2016](http://qpl2016.cis.strath.ac.uk) (2016) [[video](https://youtu.be/59frzb__Eqo)]

On [[self-dual objects]] in [[star-autonomous categories]] and their relation to [[inner products]] and to [[dagger-category|dagger-structure]]:

* {#Selinger12} [[Peter Selinger]], _Autonomous categories in which $A \simeq A^\ast$_, talk at QPL 2012 ([[SelingerSelfDual.pdf:file]])

On [[string diagram]]-calculus for [[monoidal categories]]:

* {#Selinger09} [[Peter Selinger]], _A survey of graphical languages for monoidal categories_, in: [[Bob Coecke]] (ed.) _New Structures for Physics_, Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg (2010) ([arXiv:0908.334](http://arxiv.org/abs/0908.3347), [doi:10.1007/978-3-642-12821-9_4](https://doi.org/10.1007/978-3-642-12821-9_4))

On [[finite-dimensional Hilbert spaces]] as a [[dagger-compact category]]:

* {#Selinger12} [[Peter Selinger]], *Finite dimensional Hilbert spaces are complete for dagger compact closed categories*, Logical Methods in Computer Science, **8** 3 (2012) lmcs:1086 [[arXiv:1207.6972](https://arxiv.org/abs/1207.6972), doi:10.2168/LMCS-8(3:6)2012]

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]

On [[categorical semantics]] for [[Quipper]]:

* [[Francisco Rios]], [[Peter Selinger]], *A Categorical Model for a Quantum Circuit Description Language*, EPTCS **266** (2018) 164-178 [[arXiv:1706.02630](https://arxiv.org/abs/1706.02630), [doi:10.4204/EPTCS.266.11](https://doi.org/10.4204/EPTCS.266.11)]

On [[quantum programming]] via [[dependent linear type theory]]:

* {#FKS20} [[Peng Fu]], [[Kohei Kishida]], [[Peter Selinger]], _Linear Dependent Type Theory for Quantum Programming Languages_, LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceJuly 2020 Pages 440–453 ([arXiv:2004.13472](https://arxiv.org/abs/2004.13472), [doi:10.1145/3373718.3394765](https://dl.acm.org/doi/10.1145/3373718.3394765), [pdf](https://depend.cs.uni-saarland.de/lics-icalp/papers/B5.F), [video](https://www.youtube.com/watch?v=GUT8j4V6Nzg))

* [[Peng Fu]], [[Kohei Kishida]], [[Neil Ross]], [[Peter Selinger]],  _A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper_, in I. Lanese, M. Rawski  (eds.) _Reversible Computation_ RC 2020. Lecture Notes in Computer Science, vol 12227 ([arXiv:2005.08396](https://arxiv.org/abs/2005.08396), [doi:10.1007/978-3-030-52482-1_9](https://doi.org/10.1007/978-3-030-52482-1_9))

On the [[Agda]] [[proof assistant]]:

* [[Peter Selinger]], *Lectures on Agda* (2021) [[web](https://www.mathstat.dal.ca/~selinger/agda-lectures/)]

On [[dependent linear type theory]] and [[categorical semantics]] for versions of "[[Quipper]]":

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *A biset-enriched categorical model for Proto-Quipper with dynamic lifting* [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *Proto-Quipper with dynamic lifting* [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

category: people


Peter Slodowy

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

* [German Wikipedia entry](https://de.wikipedia.org/wiki/Peter_Slodowy)

* [Mathematics Genealogy Page](http://www.genealogy.ams.org/id.php?id=18259)

## Related $n$Lab entries

* [[ADE singularity]]

category: people


Peter Stangl

* [institute page](https://www.itp.unibe.ch/about_us/people/people/index_eng.html?id=216)

## Selected writings

On the [[flavour anomalies]]:

* Jason Aebischer, [[Wolfgang Altmannshofer]], Diego Guadagnoli, Meril Reboud, [[Peter Stangl]], David M. Straub, _B-decay discrepancies after Moriond 2019_ ([arXiv:1903.10434](https://arxiv.org/abs/1903.10434))

* {#AltmannshoferStangl21} [[Wolfgang Altmannshofer]], [[Peter Stangl]], _New Physics in Rare B Decays after Moriond 2021_ ([arXiv:2103.13370](https://arxiv.org/abs/2103.13370))

In the context of [[composite Higgs boson]]-models

* [[Peter Stangl]], _Flavour anomalies and (fundamental) partial compositeness_ ([arXiv:1907.05158](https://arxiv.org/abs/1907.05158))

and in the context of [[leptoquark]]-models:

* {#GST21} [[Admir Greljo]], [[Peter Stangl]], Anders Eller Thomsen, _A Model of Muon Anomalies_ ([arXiv:2103.13991](https://arxiv.org/abs/2103.13991))

category: people


Peter Symonds

* [Webpage](https://personalpages.manchester.ac.uk/staff/peter.symonds/)

## selected writings

* {#Symonds91} [[Peter Symonds]], _A splitting principle for group representations_, Comment. Math. Helv. (1991) 66: 169 ([doi:10.1007/BF02566643](https://doi.org/10.1007/BF02566643))

## related $n$Lab entries

* [[representation ring]]

* [[characteristic class of a linear representation]]

* [[splitting principle]]

* [[Brauer induction theorem]]

category:people


Peter Tait

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

## related $n$Lab entries

* [[knot theory]]

* [[On Vortex Atoms]]

category: people


Peter Teichner

* [website Berkeley](http://math.berkeley.edu/people/faculty/peter-teichner)

* [webpage MPIM](http://people.mpim-bonn.mpg.de/teichner/Math/Home.html)

## Selected writings

On relating [[topological K-theory]] and, maybe, [[elliptic cohomology]] (at least the [[Witten genus]]) to [[functorial quantum field theory]]:

* [[Stefan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_

* {#StolzTeichner11} [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_, in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[Mathematical foundations of Quantum field theory and String theory](http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionStolzTeichner)_, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS (2011)

On [[supermanifolds]] form the point of view of [[functorial geometry]]:

* {#HohnholdSolzTeichner11} [[Henning Hohnhold]], [[Stephan Stolz]], [[Peter Teichner]], _Super manifolds: an incomplete survey_, Bulletin of the Manifold Atlas (2011) 1–6 ([pdf](https://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/Survey-Journal.pdf))

* [[Henning Hohnhold]], [[Matthias Kreck]], [[Stephan Stolz]], [[Peter Teichner]], Sections 2-3 of: _Differential forms and 0-dimensional supersymmetric field theories_, Quantum Topology Volume 2, Issue 1 (2011) pp. 1–41 ([doi:10.4171/QT/12](http://dx.doi.org/10.4171/QT/12))

## Related $n$Lab entries

* [[geometric models for tmf]]

category: people


Peter van Nieuwenhuizen

* [webpage](http://insti.physics.sunysb.edu/physics/forms/profilesearch.cgi?lastname=van%20nieuwenhuizen)

One of the co-discoveres of [[supergravity]] ([[4d supergravity]]).

## Selected writings

Introducing [[D=4 supergravity]]:

* {#FreedmanNieuwenhuizenFerrara76} [[Daniel Freedman]], [[Peter van Nieuwenhuizen]], [[Sergio Ferrara]], _Progress toward a theory of supergravity_, Phys. Rev. D13 (1976) 3214 ([doi.org/10.1103/PhysRevD.13.3214](https://doi.org/10.1103/PhysRevD.13.3214))

Review of [[supergravity]]:

* {#Nieuwenhuizen81} [[Peter van Nieuwenhuizen]], _Supergravity_, Physics Reports, **68,** (1981) 189-398 [doi:10.1016/0370-1573(81)90157-5]

Secretly introducing [[super L-∞ algebras]] in their [[formal dual]] incarnation ("[[FDA]]"s in the [[supergravity]] literature):

* {#Nieuwenhuizen82} [[Peter van Nieuwenhuizen]], _Free Graded Differential Superalgebras_, in: *Group Theoretical Methods in Physics*, Lecture Notes in Physics **180**, Springer (1983) 228–247 [[doi:10.1007/3-540-12291-5_29](https://doi.org/10.1007/3-540-12291-5_29), [spire:182644](http://inspirehep.net/record/182644)]

and the resulting [[D'Auria-Fré formulation of supergravity]]:

* {#CFGPN83} [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], *Gauging of $d = 11$ supergravity?*, Annals of Physics **146** 1 (1983) 35-77 [[spire:11998](https://inspirehep.net/literature/11998), doi:10.1016/0003-4916(83)90052-0]

On [[Freund-Rubin compactifications]] of [[11d supergravity]]:

* [[Riccardo D'Auria]], [[Pietro Fre]], [[Peter van Nieuwenhuizen]], _Symmetry Breaking in $d=11$ Supergravity on the Parallelized Seven Sphere_, Phys. Lett. B 122 (1983) 225 ([spire:181634](https://inspirehep.net/literature/181634), doi:10.1016/0370-2693(83)90689-5)

On [[D=7 supergravity]] (as [[gauged supergravity]]):

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], _Gauged Maximally Extended Supergravity in Seven Dimensions_ , Phys. Lett. 143B (1984) 103 ([spire:15064](https://inspirehep.net/literature/15064), doi:10.1016/0370-2693(84)90813-X)

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], [[Nicholas Warner]], _Noncompact Gauging and Critical Points of Maximal Supergravity in Seven Dimensions_, Nucl. Phys. B249 (1985) 381 (doi:10.1016/0550-3213(85)90046-X>doi:10.1016/0550-3213(85)90046-X)

* [[Paul Townsend]], [[Peter van Nieuwenhuizen]], _Gauged Seven-Dimensional Supergravity_, Phys. Lett. 125B (1983) 41 ([spire:189055](http://inspirehep.net/record/189055), doi:10.1016/0370-2693(83)91230-3)

On the early history of [[supergravity]]:

* {#vanNieuwenhuizen1993} van Nieuwenhuizen, _Some personal recollections about the discovery of supergravity_, 1993 ([pdf](http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/30/028/30028372.pdf), [[vanNieuwenhuizen1993.pdf:file]])

Part of this story also appears as

* David Appell, _When supergravity was born_, 2012 ([pdf](http://www.davidappell.com/articles/PWSep12appell-supergravity.pdf))

## Related entries

* [[supergravity]]

category: people


Peter W. Shor

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

## Selected writings

Introducing [[Shor's algorithm]]:

* [[Peter W. Shor]], *Algorithms for quantum computation: discrete logarithms and factoring*,  Proceedings 35th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press: 124-134 (1994) ([doi:10.1109/SFCS.1994.365700](https://doi.org/10.1109/SFCS.1994.365700))

Introducing the notion of [[quantum error correcting codes]]:

* [[Peter W. Shor]], *Scheme for reducing decoherence in quantum computer memory*, Phys. Rev. A 52, R2493(R) 1995 ([doi:10.1103/PhysRevA.52.R2493](https://doi.org/10.1103/PhysRevA.52.R2493))

* [[Robert Calderbank]], [[Peter W. Shor]], *Good Quantum Error-Correcting Codes Exist*, Phys. Rev. A, Vol. 54, No. 2, pp. 1098-1106, 1996 ([doi:10.1103/PhysRevA.54.1098](https://doi.org/10.1103/PhysRevA.54.1098))

Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter W. Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

Introducing [[stabilizer codes]]:

* [[Robert Calderbank]], E. M Rains, [[Peter W. Shor]], N. J. A. Sloane, *Quantum Error Correction and Orthogonal Geometry*, Phys. Rev. Lett. **78** 405-408 (1997) [[arXiv:quant-ph/9605005](https://arxiv.org/abs/quant-ph/9605005)]

On the [[DQC1]] [[quantum complexity theory|quantum complexity]] [[complexity class|class]] ("one clean qbit"):

* {#Shor08} [[Peter W. Shor]], Stephen P. Jordan, *Estimating Jones polynomials is a complete problem for one clean qubit*, Quantum Information and Computation **8** (2008) 681 [[arXiv:0707.2831](https://arxiv.org/abs/0707.2831), [doi:10.5555/2017011.2017012](https://dl.acm.org/doi/10.5555/2017011.2017012)]

On [[quantum computation]]:

* [[Peter Shor]], *Quantum Computation*, Lecture notes (2022) [[web](https://math.mit.edu/~shor/435-LN/)]

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

* Yuanjie Ren, [[Peter Shor]], *Topological quantum computation assisted by phase transitions* \[arXiv:2311.00103\]

category: people

[[!redirects Peter W. Shor]]
[[!redirects Peter Shor]]


Peter Webb

Peter Webb is a professor of Mathematics at the University of Minnesota. His research includes work in representation theory and the cohomology of categories.

* [Home Page](http://www-users.math.umn.edu/~webb/)

category:people


Peter West

Peter West is professor of mathematics at King's college London.

* Wikipedia entry

* [institute page](https://www.kcl.ac.uk/people/peter-west)

* [institute page](https://kclpure.kcl.ac.uk/portal/en/persons/peter.west)

## Selected writings

Early construction of [[type IIA supergravity]]:

* I. C. G. Campbell, [[Peter C. West]] *$N = 2$, $D = 10$ non-chiral supergravity and its spontaneous compactification*, Nuclear Physics B
**243** 1 (1984) 112-124 [doi:10.1016/0550-3213(84)90388-2]

On the [[Green-Schwarz mechanism]] and [[KK-compactification]] of [[heterotic supergravity]]/[[heterotic superstring]] to 6 dimensions (see also at [[D=6 N=(1,0) SCFT]]):

* [[Michael Green]], [[John Schwarz]], [[Pete West]], _Anomaly-free chiral theories in six dimensions_, Nuclear Physics B Volume 254, 1985, Pages 327-348 (doi:10.1016/0550-3213(85)90222-6)

On [[Landau-Ginzburg models]] in the sense of [[string theory]] ([[sigma models]] with certain [[potential energy|potentials]]):

* [[Paul Howe]], [[Peter West]], *Landau-Ginzburg Hamiltonians, string theory and critical phenomena*, in: *Proceedings of 25th International Conference on High Energy Physics*, Singapore, August 2-8, 1990 [[inspire:301119](https://inspirehep.net/literature/301119), [pdf](https://inspirehep.net/files/f9b5ac3ccd39ef8c30ca5c7b69b92fc8)]

On the fundamental [[M5-brane]] [[Green-Schwarz sigma-model]]:

* {#HoweSezginWest97} [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], _Covariant Field Equations of the M Theory Five-Brane_, Phys. Lett. B **399** (1997) 49-59 [[arXiv:hep-th/9702008](https://arxiv.org/abs/hep-th/9702008), doi:10.1016/S0370-2693(97)00257-8]

On obtaining [[massive type IIA supergravity]] from a form o [[D=11 supergravity]]:

* [[Paul S. Howe]], [[Neil D. Lambert]], [[Peter C. West]], *A New Massive Type IIA Supergravity From Compactification*, Phys. Lett. B **416** (1998) 303-308 [[arXiv:hep-th/9707139](https://arxiv.org/abs/hep-th/9707139), doi:10.1016/S0370-2693(97)01199-4]

On the [[3-brane in 6d]] as a [[black brane]] in an [[M5-brane]]/[[NS5-brane]] [[worldvolume]]

* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Threebrane Soliton of the M-Fivebrane_, Phys. Lett. B419 (1998) 79-83 ([arXiv:hep-th/9710033](http://arxiv.org/abs/hep-th/9710033))

and of its relation to [[D=4 N=2 super Yang-Mills theory]]:

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _Classical M-Fivebrane Dynamics and Quantum $N=2$ Yang-Mills_, Phys. Lett. B418 (1998) 85-90 ([arXiv:hep-th/9710034](https://arxiv.org/abs/hep-th/9710034))

* [[Neil Lambert]], [[Peter West]], _Gauge Fields and M-Fivebrane Dynamics_, Nucl. Phys. B524 (1998) 141-158 ([arXiv:hep-th/9712040](https://arxiv.org/abs/hep-th/9712040))

* {#LambertWest98} [[Neil Lambert]], [[Peter West]], _$N=2$ Superfields and the M-Fivebrane_, Phys. Lett. B424 (1998) 281-287 ([arXiv:hep-th/9801104](https://arxiv.org/abs/hep-th/9801104))

* [[Neil Lambert]], [[Peter West]], _Monopole Dynamics from the M-Fivebrane_, Nucl. Phys. B556 (1999) 177-196 ([arXiv:hep-th/9811025](https://arxiv.org/abs/hep-th/9811025))

On [[Dp-D(p+2) brane intersections]] (such as [[D1-D3 brane intersections]]) as spikes/[[BIons]] from the [[M5-brane]]:

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

On deriving the [[equations of motion]] for the [[M5-brane]] via [[superspace]]-methods ([[superembedding approach]]):

* {#HoweSezginWest97} [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], _Covariant Field Equations of the M Theory Five-Brane_, Phys. Lett. B399 (1997) 49-59 ([arXiv:hep-th/9702008](https://arxiv.org/abs/hep-th/9702008))

On [[E11]]-[[exceptional generalized geometry]] for [[D=11 supergravity]]/[[M-theory]]:

* {#West01} [[Peter West]], _$E_{11}$ and M theory_, Class. Quant. Grav., 18:4443–4460, 2001 ([arXiv:hep-th/0104081](http://arxiv.org/abs/hep-th/0104081))

* {#West03} [[Peter West]], _$E_{11}$, $SL(32)$ and Central Charges_, Phys. Lett. B575:333-342, 2003 ([arXiv:hep-th/0307098](http://arxiv.org/abs/hep-th/0307098))

* {#West04} [[Peter West]], _$E_{11}$ origin of Brane charges and U-duality multiplets_, JHEP 0408 (2004) 052 ([arXiv:hep-th/0406150](http://arxiv.org/abs/hep-th/0406150))

* {#West05} [[Peter West]], _$E_{11}$, ten forms and supergravity_, JHEP0603:072,2006 ([arXiv:hep-th/0511153](http://arxiv.org/abs/hep-th/0511153))

* {#West11} [[Peter West]], _Generalised geometry, eleven dimensions and $E_{11}$_, J. High Energ. Phys. (2012) 2012: 18 ([arXiv:1111.1642](http://arxiv.org/abs/1111.1642))

* {#West16} [[Peter West]], _A brief review of E theory_,  International Journal of Modern Physics A Vol. 31, No. 26, 1630043 (2016)  ([arXiv:1609.06863](http://arxiv.org/abs/1609.06863), [doi:10.1142/S0217751X1630043X](https://doi.org/10.1142/S0217751X1630043X))

Discussion of the [[dual graviton]] in terms of [[E11]] [[U-duality]]:

* {#Wess11} [[Peter West]], section N (pp. 19) of _Generalised geometry, eleven dimensions and $E_{11}$_ ([arXiv:1111.1642](http://arxiv.org/abs/1111.1642))

* Keith Glennon, [[Peter West]], _Gravity, Dual Gravity and $A_1^{+++}$_ ([arXiv:2004.03363](https://arxiv.org/abs/2004.03363))

* Nicolas Boulanger, [[Paul P. Cook]], Josh A. O'Connor, [[Peter West]], *Higher dualisations of linearised gravity and the $A_1^{+++}$ algebra* [[arXiv:2208.11501](https://arxiv.org/abs/2208.11501)&rbrack

On basics of [[string theory]]:

* _[[Introduction to Strings and Branes]]_

## Related entries

* [[U-duality]], [[E11]]

category: people

[[!redirects Pete West]]

[[!redirects Peter C. West]]


Peter Whittle

* Wikipedia entry

## related $n$Lab entries

* [[expectation value]]

* [[quantum probability space]]

category: people


Peter Woit

Peter Woit is a mathematician and theoretical physicist at Columbia University.

* [website](https://www.math.columbia.edu/~woit/)

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

## Selected publications

* [[Peter Woit]], *Supersymmetric Quantum Mechanics, Spinors and the Standard Model*, Nuclear Physics B, Volume 303, Issue 2, 27 June 1988, Pages 329-342 [doi:10.1016/0550-3213(88)90185-X]

### Preprints

* [[Peter Woit]], *Quantum Field Theory and Representation Theory: A Sketch* ([arXiv:hep-th/0206135](https://arxiv.org/abs/hep-th/0206135))

* [[Peter Woit]], *Notes on the Twistor $P^1$* ([arXiv:2202.02657](https://arxiv.org/abs/2202.02657))

### Textbooks

* [[Peter Woit]], *Quantum Theory, Groups and Representations: An Introduction*, Springer 2017 [[doi:10.1007/978-3-319-64612-1](https://doi.org/10.1007/978-3-319-64612-1), ISBN:978-3-319-64610-7]

category:people


Peter Zvengrowski

* [Institute page](https://contacts.ucalgary.ca/info/math/profiles/101-152986)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=4844)

## Selected writings

On [[fundamental groups]] of [[3-manifolds]] and the [[group cohomology]] of [[finite rotation groups]]:

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

category: people


Peter-Weyl theorem

#Contents#
* table of contents
{:toc}

## Idea

[[representative function|Representative function]]s on a compact topological group form a dense subspace of the space of all continuous functions.

## References

Textbook account:

* [[Tammo tom Dieck]], [[Theodor Bröcker]], Sections III.3, III.4 in: *Representations of compact Lie groups*, Springer 1985 ([doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0))

Lecture notes:

* [[Linus Kramer]], Section 3 of: *Locally Compact Groups*, 2017 ([pdf](https://www.uni-muenster.de/AGKramer/content/LCManuscript.pdf), [[Kramer_LocallyCompactGroups.pdf:file]])

See also:

* Wikipedia, *[Peter-Weyl theorem](https://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem)*


Peterson space

## Definition

Given a finitely generated [[abelian group]] $A$ and $n\ge 3$,
the $n$th __Peterson space__ $P^n(A)$ of $A$
is the [[simply connected space]]
whose [[reduced cohomology groups]] vanish in dimension $k\ne n$
and the $n$th cohomology group is isomorphic to $A$.

## Existence and uniqueness

The Peterson space exists and is unique up to a [[weak homotopy equivalence]]
given the indicated conditions on $A$ and $n$.

There are counterexamples both to existence and uniqueness without these conditions.

For example, the Peterson space does not exist if $A$ is the abelian group of [[rationals]].

## Functoriality

If $n\ge 4$,
then $P_n$ is a functor from [[abelian groups]] without 2-[[torsion]]
to the [[homotopy category]] of [[pointed spaces]].

In fact, for all $n\ge 4$ the map
$$Hom(P^n(B),P^n(A))\to Hom(A,B)$$
is an isomorphism if $A$ has no 2-[[torsion]].

## Corepresentation of homotopy groups with coefficients

For all $n\ge2$, we have a canonical isomorphism
$$\pi_n(X,A)\cong [P^n(A),X],$$
where the left side denotes [[homotopy groups with coefficients]]
and the right side denotes morphisms in the pointed [[homotopy category]].

## Relation to Moore spaces

[[Moore spaces]] $M_n(A)$ are defined similarly to Peterson spaces,
using [[homology]] instead of [[cohomology]].

We have natural weak equivalences
$$P^n(A) \simeq M_n(Hom(A,\mathbf{Z}))$$
if $A$ is a finitely generated [[free abelian group]]
and
$$P^n(A) \simeq M_{n-1}(Hom(A,\mathbf{Q}/\mathbf{Z}))$$
if $A$ is a finite [[abelian group]].

## Examples

If $A=\mathbf{Z}$, then $P^n(A)=S^n$, so $\pi_n(X,\mathbf{Z})=\pi_n(X)$.

If $A=\mathbf{Z}/k\mathbf{Z}$,
then $P^n(A)$ is obtained by attaching an $n$-cell to an $(n-1)$-sphere
along a map of degree $k$.
Thus, $\pi_n(X,\mathbf{Z}/k\mathbf{Z})$ is defined for all $n\ge 2$.

## Related concepts

* [[Moore space]]

* [[Eilenberg–MacLane space]]

* [[homotopy groups with coefficients]]

## References

* [[Franklin P. Peterson]], _Generalized Cohomotopy Groups_.  American Journal of Mathematics 78:2 (1956), 259–281.  [doi:10.2307/2372515](http://dx.doi.org/10.2307/2372515)

* [[Joseph A. Neisendorfer]], _Homotopy groups with coefficients_,
Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338.  [doi:10.1007/s11784-010-0020-1](http://dx.doi.org/10.1007/s11784-010-0020-1).

[[!redirects Peterson spaces]]


Petr Hořava

* [website](http://www-theory.lbl.gov/~horava/)

## Selected writings

On [[n-plectic geometry|$n$-plectic]] formulation of [[Einstein-Maxwell theory]]:

* [[Petr Hořava]], *Covariant Hamilton-Jacobi Equation for Pure Gravity*, Institute of Physics, Czechoslovak Academy of Sciences, PRA-HEP-90/4 (1990) [[arXiv:2312.14412](https://arxiv.org/abs/2312.14412)]

* [[Petr Hořava]], _On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory_ Classical and Quantum Gravity **8** 11 (1991) 2069 [[doi:10.1088/0264-9381/8/11/016](https://iopscience.iop.org/article/10.1088/0264-9381/8/11/016)]

On [[Chern-Simons theory]] on [[orbifolds]] via [[equivariant bundles]] in the general sense of [tom Dieck 69](equivariant+bundle#tomDieck69) (i.e. with [[action]] of the [[semidirect product]] of the [[gauge group]] with the [[equivariance group]]):

* {#Horava94} [[Petr Hořava]], _Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions_, J. Geom. Phys. 21:1-33, 1996 ([arXiv:hep-th/9404101](https://arxiv.org/abs/hep-th/9404101))

Introducing [[Hořava-Witten theory]]:

* {#HoravaWitten95} [[Petr Hořava]], [[Edward Witten]], _Heterotic and Type I string dynamics from eleven dimensions_, Nucl. Phys. B460 (1996) 506 ([arXiv:hep-th/9510209](http://arxiv.org/abs/hep-th/9510209))

* {#HoravaWitten96} [[Petr Hořava]], [[Edward Witten]],  _Eleven dimensional supergravity on a manifold with boundary_, Nucl. Phys. B475 (1996) 94 ([arXiv:hep-th/9603142](http://arxiv.org/abs/hep-th/9603142))

Understanding [[solitonic brane|solitonic]] (non-[[singular brane|singular]]) [[D-branes]] and their [[T-duality]] in terms of the [[D-brane charge quantization in K-theory]]:

* {#BergmanGimonHořava99} [[Oren Bergman]], [[Eric G. Gimon]], [[Petr Hořava]], *Brane Transfer Operations and T-Duality of Non-BPS States*, JHEP 9904 (1999) 010 \[doi:10.1088/1126-6708/1999/04/010, [arXiv:hep-th/9902160](https://arxiv.org/abs/hep-th/9902160)\]

On [[Ricci flow]] and [[quantum gravity]]:

* Alexander Frenkel, [[Petr Horava]], Stephen Randall, _Topological Quantum Gravity of the Ricci Flow_ ([arXiv:2010.15369](https://arxiv.org/abs/2010.15369))

* Alexander Frenkel, [[Petr Horava]], Stephen Randall, _Perelman's Ricci Flow in Topological Quantum Gravity_ ([arXiv:2011.11914](https://arxiv.org/abs/2011.11914))

## Related entries

* [[Hořava-Witten theory]]

[[!redirects Petr Horava]]

category: people


Petr M. Lavrov

* [Spire page](https://inspirehep.net/authors/1000930?ui-citation-summary=true)

## Selected writings

On [[gauge invariance|gauge invariant]] [[renormalization]] of [[perturbative quantum gravity]]:

* [[Petr M. Lavrov]], [[Ilya Shapiro]], *Gauge invariant renormalizability of quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09271](https://arxiv.org/abs/2210.09271)]

category: people

[[!redirects Petr Lavrov]]


Petr Vopěnka

Petr Vopěnka was a Czech mathematician.

He got his Diploma in 1958 (advised by [[Eduard Čech]]) and PhD degree in 1962 from Charles University in Prague.

His PhD students include [[Tomáš Jech]].

He is known for [[Vopěnka's principle]].


Petr Zima

* [Institute page](https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=9424)

## Selected writings

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

* [[Martin Doubek]], [[Martin Markl]], [[Petr Zima]], equation (31) in: *Deformation Theory (lecture notes)*, Archivum mathematicum 43(5), 2007, 333-371 ([arXiv:0705.3719](https://arxiv.org/abs/0705.3719))

category: people


Petri net

# Contents
* table of contents
{: toc}

See also the [Petri net](http://www.azimuthproject.org/azimuth/show/Petri+net) in The Azimuth Project.

## Introduction

Petri nets are a well known model of concurrent computation, generalising [[transition systems]] by using a built in notion of resource.  Their use is widespread in modelling manufacturing systems, optimising control systems and in resource critical aspects of operational research, as well as being a useful model of computation.  Because of this they exist in many variants : colored, algebraic, probabilistic, timed, ..., and hence there are many forms of the basic definition, each thought best of that particular application.

The initial use, here, is for their links with [[transition systems]], [[event structures]], [[asynchronous automata]] etc., leading on to their comparison with [[higher dimensional automaton|higher dimensional automata]] and [[higher dimensional transition system]]s, so the first definition we will use is that given by Winskel and Nielsen (see references below), but first we will attempt to give some idea of what a Petri net is and what it does.

## Idea

The idea of a simple Petri net is based on a simple manufacturing shop.  You have various 'transitions' or manufacturing 'events' that form part of the various process occurring in the 'shop'. These typically take parts (of the final object), do something to them, such as combining two or more different parts to make a more complicated one, (for instance, putting the wheels on a car).

Parts, represented by 'tokens' are stored in 'places' and the assembly process are known, as above, as 'events' or 'transitions' (depending on the source being used for the theory).  The relationships are typically represented graphically.  For instance:

<svg width="294" height="253" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="60508">
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  <title>Layer 1</title>
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  <path fill="#ffffff" stroke="#000000" stroke-width="2" d="m222.007813,27c0,-14.36464 12.082886,-26 27,-26c14.917114,0 27,11.63536 27,26c0,14.364639 -12.082886,26 -27,26c-14.917114,0 -27,-11.635361 -27,-26z" id="svg_60508_2" transform="rotate(3.74069e-06, 320.272, -74)"/>
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  <line fill="none" stroke="#000000" stroke-width="2" x1="230.007808" y1="47.000002" x2="149.007807" y2="116.000003" id="svg_60508_6"/>
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  <text fill="#000000" stroke="#000000" stroke-width="0" x="105.008" y="65" id="svg_60508_8" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">1</text>
  <text fill="#000000" stroke="#000000" stroke-width="0" x="188.008" y="67" id="svg_60508_9" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">1</text>
  <text fill="#000000" stroke="#000000" stroke-width="0" x="157.008" y="170" id="svg_60508_10" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">1</text>
  <text fill="#000000" stroke="#000000" stroke-width="0" x="212.008" y="128" id="svg_60508_11" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">e</text>
  <text fill="#000000" stroke="#000000" stroke-width="0" x="8.00781" y="26" id="svg_60508_12" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">A</text>
  <text fill="#000000" stroke="#000000" stroke-width="0" x="286.008" y="27" id="svg_60508_13" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">B</text>
  <text fill="#000000" stroke="#000000" stroke-width="0" x="187.008" y="226" id="svg_60508_14" font-size="24" font-family="serif" text-anchor="middle" xml:space="preserve">C</text>
  <ellipse fill="#000000" stroke="#000000" stroke-width="2" cx="36.007813" cy="23" id="svg_60508_15" rx="4" ry="4"/>
  <ellipse fill="#000000" stroke="#000000" stroke-width="2" cx="60.007813" cy="24" rx="4" ry="4" id="svg_60508_16"/>
  <ellipse fill="#000000" stroke="#000000" stroke-width="2" cx="247.007813" cy="23" rx="4" ry="4" id="svg_60508_17"/>
 </g>
</svg>

represents a system with three places (labelled $A$, $B$, and $C$) and one event/ transition (labelled $e$).  Shown is a situation represented by an initial 'marking' where there are two tokens in $A$, one in $B$, and none in $C$. (The convention is that places are shown as circles and events as rectangles.)

Suppose that the 'event' takes one 'part' of type $A$, one of type $B$ and produces one of type $C$.  (This is indicated on the diagram by the labels on the edges.  Clearly with the available  resources the even $e$ is able to be performed.  (The usual Petri net jargon for this is that '$e$ is enabled'.) In this case, $e$ can be 'fired' and the result will yield a marking of one token in $A$, none in $B$ and one in $C$.  This sort of structure gets abstracted as follows:

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

A **Petri net** $N$ consists of

* a set $P$ of _places_;

* an _initial marking_ $M_0 \in \mathbb{N}^P$, so $M_0 : P \to \mathbb{N}$;

* a set $E$ of _events_; and

* two functions

  *  $pre: P\times E\to \mathbb{N}$, and

  *   $post:  P\times E\to \mathbb{N}$.
=--

## Categorical Semantics of Petri Nets {#semantics}

The functions $pre$ and $post$ can be [[curried]] to write a Petri net as a pair of functions

\begin{centre}
\begin{tikzcd}
E \ar[r,shift left=.5ex,"\mathrm{pre}"] \ar[r,shift right=.5ex,"\mathrm{post}",swap] & \mathbb{N} [P]
\end{tikzcd}
\end{centre}

where $\mathbb{N}[P]$ denotes the [[free monoid|free commutative monoid]] on the set $P$ of places. Written this way, the analogy to [[directed graph|graphs]] is clearer. The main difference is that now the source and target functions land in a [[free monoid|free commutative monoid]] rather than just a set. Thus, Petri nets can be thought of as _symmetric monoidal graphs_ because each edge is a multi-edge between a permutable sum of vertices. Just as [[directed graph|graphs]] generate [[free categories]], Petri nets should generate free [[symmetric monoidal categories]]. As Petri nets often model processes in various sciences, the free symmetric monoidal categories that they generate model the operational semantics of these processes. For a given Petri net $P$, the free symmetric monoidal $FP$ that it generates should have objects given by possible markings of $P$ and morphisms given by ways of composing the events in sequence and in parallel.

The first to explore this idea were [Meseguer and Montanari](#monoids) who constructed an adjunction between Petri nets and a subcategory of the category $CMC$ of [[commutative monoidal categories]]. This adjunction was modified in [_Petri nets based on Lawvere theories_](#gen) to get

\begin{center}
\begin{tikzcd}
\mathsf{Petri} \ar[r,bend left,"F"] & \ar[l,bend left,"U"] \mathsf{CMC}
\end{tikzcd}
\end{center}
Although morally Petri nets generate free symmetric monoidal categories, it turns out that because they have a free _commutative_ monoid of places, they more naturally generate [[commutative monoidal categories]], symmetric monoidal categories of an unusually strict type. [[Commutative monoidal categories]] don't show up often in nature because they aren't the result of a [[coherence]] theorem for symmetric monoidal categories. [[Saunders Mac Lane|Mac Lane's]] [[coherence theorem for symmetric monoidal categories]] says that every symmetric monoidal category is symmetric monoidally equivalent to a [[symmetric monoidal category|strict symmetric monoidal category]] which may have nontrivial symmetry morphisms. However, commutative monoidal categories have symmetry morphisms which are given by the identity so they aren't the [strictifications](https://ncatlab.org/nlab/show/coherence+theorem#strictification) of symmetric monoidal categories.

Computer scientists also haven't been satisfied with commutative monoidal categories as a categorical semantics for Petri nets. Commutative monoidal categories are said to exhibit the collective token philosophy (see e.g. [Glabeek and Plotkin](#philosophy)), where tokens do not have individual identities. On the other hand, symmetric monoidal categories are said to have the individual token philosophy where the individual identities of the tokens are retained. The easiest to see where an issue shows up is with a variation of the [[Eckmann-Hilton argument]]. Let $C$ be a commutative monoidal category and let $f,g \colon a \to a$ be morphisms from an object a to itself. Then,
$$\begin{aligned}
g \circ f + 1_a & = g \circ f + 1_a \circ 1_a \\
&= (g+ 1_a)\circ (f + 1_a) \\
&= (g + 1_a) \circ (1_a + f) \\
& = g \circ 1_a + 1_a \circ f \\
&= g + f
\end{aligned}$$

So in a commutative monoidal category $g \circ f + 1_a = g+ f$ even though they have very different semantic interpretations: one represents sequential composition and the other represents parallel composition.

One possible fix to this is to change the definition of Petri net. In [_Functorial Models for Petri Nets_](#functorial) the authors introduced [[Pre-net|pre-nets]]. A pre-net is a pair of functions from a set of events to the free monoid on a set of places. The key difference is that now every event is equipped with an ordering on the input and output of each transition. With pre-nets, there is a natural [[adjunction]] between the category $PreNet$ of pre-nets and $SMC$ the category of strict monoidal categories

\begin{center}
\begin{tikzcd}
\mathsf{PreNet} \ar[r,bend left,"A"] & \ar[l,bend left,"B"] \mathsf{SMC}
\end{tikzcd}
\end{center}

which can be composed with the adjunction which freely adds symmetries to a strict monoidal category to get an adjunction

\begin{center}
\begin{tikzcd}
\mathsf{PreNet} \ar[r,bend left,"L"] & \ar[l,bend left,"R"] \mathsf{SSMC}
\end{tikzcd}
\end{center}

where $SSMC$ is the category of [[symmetric monoidal category|strict symmetric monoidal categories]]. This gives a categorical operational semantics for pre-nets which obeys the individual token philosophy and avoids the pitfall of the above computation. Pre-nets can be turned into Petri nets via [[abelianization]]. For a given Petri net $P$, its abelianization forgets about the ordering on the input and output of each event. Therefore, to obtain a categorical semantics for a Petri net $P$ which obeys the individual token philosophy, a pre-net $Q$ can be chosen which abelianizes to $P$. Then, the categorical semantics for $P$ can defined as the free strict symmetric monoidal category $L(Q)$.

## References

* [[John Baez]], Fabrizio Genovese, [[Jade Master]], [[Michael Shulman]], _Categories of Nets_, ([arXiv:2101.04238](https://arxiv.org/abs/2101.04238))

* [[G. Winskel]], M. Nielsen, _Models for concurrency_, Handbook of Logic in Computer Science vol. 3, pp. 100 -- 200, Oxford Univ. Press 1994. (see also [online technical report](http://www.daimi.au.dk/PB/463/PB-463.pdf)).

* [[John C. Baez]] and [[Jacob Biamonte]], _Quantum techniques in stochastic mechanics_, World Scientific, 2018. [World Scientific](https://www.worldscientific.com/worldscibooks/10.1142/10623#t=suppl) [pdf](http://math.ucr.edu/home/baez/stoch_stable.pdf)

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

For an introduction to Petri nets (en francais, which is very clear and accessible) look at

* [[Robert Valette]], [Les Réseaux de Petri](http://homepages.laas.fr/robert/enseignement.d/cour01.pdf).

On the connection to linear logic:

* [[Uffe Engberg]] and [[Glynn Winskel]], _Petri nets as models of linear logic_. CAAP 1990. Lecture Notes in Computer Science, vol 431. Springer, Berlin, Heidelberg. [Springer](https://link.springer.com/chapter/10.1007/3-540-52590-4_46), [pdf on ResearchGate](https://www.researchgate.net/publication/318253097_Petri_Nets_as_Models_of_Linear_Logic)

On the categorical semantics of Petri nets:

* {#monoids} [[José Meseguer]] and [[Ugo Montanari]], _Petri nets are monoids_, Information and Computation, Volume 88, Issue 2, Pages 105-155, 1990. [journal](https://www.sciencedirect.com/science/article/pii/0890540190900138), [pdf](https://core.ac.uk/download/pdf/82342688.pdf)

* {#gen} [[Jade Master]], _Petri nets based on Lawvere theories_, Mathematical Structures in Computer Science 30, no. 7 (2020): 833-864. Available at [journal](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/petri-nets-based-on-lawvere-theories/9E0F011860A1285DDC498B740774FBDC)

* {#functorial} [[Roberto Bruni]], [[José Meseguer]], [[Ugo Montanari]], and [[Vladimiro Sassone]]. _Functorial models for Petri nets_ Information and Computation 170, no. 2 (2001): 207-236. [pdf](https://eprints.soton.ac.uk/264742/1/prenetsIandCOff.pdf)

* {#philosophy} [[Rob J. van Glabbeek]], and [[Gordon Plotkin]]. _Configuration structures, event structures and Petri nets_ Theoretical Computer Science 410, no. 41 (2009): 4111-4159. Available at [pdf](https://arxiv.org/abs/0912.4023)

[[!redirects Petri net]]
[[!redirects Petri nets]]
category: computer science


Petter Brändén

* [Institute page](https://www.kth.se/profile/pbranden)

* [GoogleScholar page](https://scholar.google.se/citations?user=5rRTx-YAAAAJ&hl=sv)

## Selected writings

On [[Lorentzian polynomials]]:

* [[Petter Brändén]], [[June Huh]], *Lorentzian polynomials*, Annals of Mathematics, **192** 3 (2020) [[arXiv:1902.03719](https://arxiv.org/abs/1902.03719), [doi:10.4007/annals.2020.192.3.4](https://doi.org/10.4007/annals.2020.192.3.4)]

category: people

[[!redirects Petter Branden]]


Petteri Kaski

* [Institute page](https://www.aalto.fi/en/people/petteri-kaski)

## Selected writings

On [[Cayley graph spectra]]:

* {#Kaski02} [[Petteri Kaski]], *Eigenvectors and spectra of Cayley graphs*, Lecture at *[T-79.300 Postgraduate Course in Theoretical Computer Science](http://www.tcs.hut.fi/Studies/T-79.300/2002S/)*, Helsinki University of Technology, Spring 2002 ([pdf](http://www.tcs.hut.fi/Studies/T-79.300/2002S/esitelmat/kaski_paper_020506.pdf), [[KaskiSpectraOfCayleyGraphs.pdf:file]])

category: people


Pfaffian

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#### Algebra
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* table of contents
{:toc}

## Idea

The _Pfaffian_ of a [[skew-symmetric matrix]] is a [[square root]] of its [[determinant]].

## Definition

Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-[[matrix]]  with entries in some [[field]] (or [[ring]]) $k$.

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

The _Pfaffian_ $Pf(A) \in k$ is the element

\[
  \label{PfaffianViaSignatureOfPermutations}
  Pf(A)
  \;\coloneqq\;
  \frac{1}{2^n n!}
  \sum_{\sigma \in Sym_{2n}}
   sgn(\sigma)
   \prod_{i = 1}^n
     A_{\sigma(2i -1), \sigma(2i)}
  \,,
\]

where

* $\sigma$ runs over all [[permutation]]s of $2n$ elements;

* $sgn(\sigma)$ is the [[signature of a permutation]].

Expressed equivalently in terms of the [[Levi-Civita symbol]] $\epsilon$ and using the [[Einstein summation convention]] the Pfaffian is

\[
  \label{PfaffianInTermsOfLCTensor}
  Pf(A)
  \;\coloneqq\;
  \frac{1}{2^n n!}
  A_{i_1 j_1}
  A_{i_2 j_2}
  \cdots
  A_{i_n j_n}
  \epsilon^{ i_1 j_1 i_2 j_2 \cdots  i_n j_n }
  \,.
\]

=--

## Properties

### Relation to the determining

+-- {: .num_prop #PfaffianIsSquareRootOfDeterminant}
###### Proposition
([[Pfaffian]] is [[square root]] of [[determinant]])

Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-[[matrix]]  with entries in some [[field]] (or [[ring]]) $k$.

Then the Pfaffian of $A$ (eq:PfaffianViaSignatureOfPermutations) is a [[square root]] of the [[determinant]] of $A$:

\[
  \label{DeterminantIsPfaffianSquared}
  \big(
    Pf(A)
  \big)^2
  \;=\;
  det(A)
  \,.
\]

=--

Proofs are spelled out for instance in [Haber 15, Sections 2 and 3](#Haber15)

### In terms of Berezinian integrals

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

Let $\Lambda_{2n}$ be the [[Grassmann algebra]] on $2n$ generators $\{\theta_i\}$, which we think of as a [[vector]] $\vec \theta$

Then the Pfaffian $Pf(A)$ is the [[Berezinian integral]]

$$
  Pf(A)
  =
  \int
    \exp( \langle \vec \theta, A \cdot \vec \theta \rangle )
  d \theta_1 d \theta_2 \cdots d \theta_{2n}
  \,.
$$

=--

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

Compare this to the Berezinian integral representation of the [[determinant]], which is

$$
  det(A)
  \propto
  \int
    \exp( \langle \vec \theta, A \cdot \vec \psi \rangle )
  d \theta_1 d \theta_2 \cdots d \theta_{2n}
  d \psi_1 d \psi_2 \cdots d \psi_{2n}
  \,.
$$

=--

## Pfaffian state

Pfaffians appear in the expression of certain multiparticle wave functions. Most notable is the pfaffian state of $N$ spinless electrons

$$
\Psi_{Pf}(z_1,\ldots,z_N) = pfaff\left(\frac{1}{z_k-z_l}\right)\prod_{i\lt j}(z_i-z_j)^q
exp(-\frac{1}{4}\sum |z|^2)
$$
where $pfaff(M_{k l})$ denotes the Pfaffian of the matrix whose labels are $k,l$ and $q= 1/\nu$ is the filling fraction, which is an even integer. For Pfaffian state see

* [[Gregory Moore]], N. Read, _Nonabelions in the fractional quantum hall effect_, Nucl. Phys. 360B(1991)362 [pdf](http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf)

## Related entries

* [[determinant]],

* [[Pfaffian line bundle]],

* [[hafnian]]

## References

### General

Basics:

* {#Haber15} Howard E. Haber, _Notes on antisymmetric matrices and the pfaffian_, 2015 ([pdf](http://scipp.ucsc.edu/~haber/webpage/pfaffian2.pdf), [[HaberPfaffian.pdf:file]])

See also

* J.-G. Luque, J.-Y. Thibon, _Pfaffian and hafnian identities in shuffle algebras_, [math.CO/0204026](http://arxiv.org/abs/math.CO/0204026)

* Claudiu Raicu, Jerzy Weyman, _Local cohomology with support in ideals of symmetric minors and Pfaffians_, [arxiv/1509.03954](http://arxiv.org/abs/1509.03954)

* Haber, _Notes on antisymmetric matrices and the pfaffian_, [pdf](http://scipp.ucsc.edu/~haber/webpage/pfaffian.pdf)

There is also a deformed noncommutative version of Pfaffian related to [[quantum linear group]]s:

* Naihuan Jing, Jian Zhang, _Quantum Pfaffians and hyper-Pfaffians_, Adv. Math. 265 (2014), 336--361, [arxiv/1309.5530](http://arxiv.org/abs/1309.5530)

Pfaffian variety is subject of 4.4 in

* Alexander Kuznetsov, _Semiorthogonal decompositions in algebraic geometry_, [arxiv/1404.3143](http://arxiv.org/abs/1404.3143)

Relation to $\tau$-functions is discussed in

* J. W. van de Leur, A. Yu. Orlov, _Pfaffian and determinantal tau functions I_, [arxiv/1404.6076](http://arxiv.org/abs/1404.6076)

Other articles:

* András C. Lőrincz, Claudiu Raicu, Uli Walther, Jerzy Weyman, _Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians_, [arxiv/1601.06688](http://arxiv.org/abs/1601.06688)
* M. Spera, _A C*–algebraic approach to determinants and Pfaffians, Acta Cosmologica Fasc. XXI-2, (1995) 203–208.

### Euler forms

Discussion of _[[Euler forms]]_ ([[differential form]]-representatives of [[Euler classes]] in [[de Rham cohomology]]) as [[Pfaffians]] of [[curvature forms]]:

* {#Chern44} [[Shiing-Shen Chern]], _A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds_, Annals of Mathematics Second Series __45__:4 (1944) 747-752 ([jstor:1969302](https://www.jstor.org/stable/1969302))

* {#MathaiQuillen86} [[Varghese Mathai]], [[Daniel Quillen]], below (7.3) of _Superconnections, Thom classes, and equivariant differential forms_, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

* {#Wu05} Siye Wu, Section 2.2 of _Mathai-Quillen formalism_, pages 390-399 in _Encyclopedia of Mathematical Physics_ 2006 ([arXiv:hep-th/0505003](https://arxiv.org/abs/hep-th/0505003))

* {#Walschap04} Gerard Walschap, ch. 6.3 of _Metric structures in differential deometry_, Graduate Texts in Mathematics, Springer 2004

* [[Hiro Lee Tanaka]], _Pfaffians and the Euler class_, 2014 ([pdf](http://www.hiroleetanaka.com/pdfs/2014-fall-230a-lecture-26-gauss-bonnet-chern.pdf))

* {#Nicolaescu18} [[Liviu Nicolaescu]], Section 8.3.2 of _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf), [MO comment](https://mathoverflow.net/a/117982/381))

category: algebraic geometry, representation theory
[[!redirects Pfaffians]]
[[!redirects pfaffian]]


Pfaffian equation

Exterior differential systems and, in particular, Pfaffian equations

* J.F. Pfaff, Berl. Abh. (1814–1815) pp. 76–135
* Springer [[eom]], [Pfaffian equation](http://www.encyclopediaofmath.org/index.php/Pfaffian_equation) article
* E. Berger, R. L. Bryant, P. A. Griffiths, _The Gauss equations and rigidity of isometric embeddings_, Duke Math. J. __50__ (1983), 803–892 [pdf](http://publications.ias.edu/sites/default/files/gausseq.pdf)
* Robert Bryant, Phillip Griffiths, Daniel Grossman, _Exterior differential systems and Euler-Lagrange partial differential equations_, Chicago Lectures in Mathematics 2003, 205+xiv pp. [math.DG/0207039](http://arxiv.org/abs/math/0207039) [bookpage](http://www.press.uchicago.edu/ucp/books/book/chicago/E/bo3637779.html)
* R. L. Bryant, S. S. Chern, R. B. Gardner, H. Goldschmidt, P. A. Griffiths, _Exterior differential systems_, MSRI Publications, Springer, [pdf](http://library.msri.org/books/Book18/MSRI-v18-Bryant-Chern-et-al.pdf)
* P. A. Griffiths, _Exterior differential systems and the calculus of variations_, Progress in Math. __25__, Birkhauser 1983
* R. B. Gardner, _Invariants of Pfaffian systems_, Trans. Amer. Math. Soc. __126__ (1967) 514–533.
* R. L. Bryant, P. A. Griffiths, D. Yang, _Characteristics and existence of isometric embeddings_, Duke Math. J. __50__ (1983), 893–994
* Phillip A. Griffiths, Gary R. Jensen, _Differential systems and isometric embeddings_, Annals of Math. Studies __114__, 1987, 238 pages [gBooks](http://books.google.fr/books/p/princeton?id=EwiTAuWIowYC&lpg=PP1&pg=PP1#v=onepage&q&f=false)
* T. Ivey, J. M. Landsberg, _Cartan for beginners: differential geometry via [[moving frame]]s and exterior differential systems_, American Mathematical Society
* Paul Popescu, _A Lagrangian form of Pfaff forms_, [arxiv/1212.4873](http://arxiv.org/abs/1212.4873)
* Pfaffian equations, [Bobby's notes](http://bobbyness.net/NerdyStuff/Differential%20Equations/1st-order-PDEs/Pfaff.html)
* Lev Elsgolts, _Differential equations and the calculus of variations_, [pdf](http://bobbyness.net/NerdyStuff/Differential%20Equations/1st-order-PDEs/Pfaff_files/Elsgolts%20Pfaffian%20Equations.pdf)
[[!redirects Pfaffian equations]]
[[!redirects Pfaffian differential system]]
[[!redirects exterior differential system]]


Pfaffian line bundle

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## Idea

A $B$-parameterized family of [[Dirac operator]]s on $d$-dimensional space gives rise to their [[determinant line bundle]] on $B$. If $d = 8k+2$ for $k \in \mathbb{N}$ this has a canonical square root line bundle, the **Pfaffian line bundle**.

## As quantum anomaly

In the [[path integral]] [[quantization]] of [[quantum field theory]], the partial [[path integral]] of an [[action functional]] over just the [[fermionic field]]s yields in general not a function on the remaining space of bosonic fields, but a section of a [[line bundle]] on this bosonic configuration space, the [[determinant line bundle]] of the family of [[Dirac operator]]s. The [[Lorentzian metric]] assumed in relativistic quantum field theory leads in the Wick-rotated theory to the passage to the square root of this line bundle.

Therefore the nontriviality of the Pfaffian line bundle is in these dimensions the fermionic [[quantum anomaly]].

## Related entries

* [[Pfaffian]]

* **Pfaffian line bundle**

* [[fermionic path integral]]

[[!include square roots of line bundles - table]]

## References

The general notion of Pfaffian line bundle is described in section 3 of

* [[Dan Freed]], _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987,  (revised [pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf), [dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002))

* D. Borthwick, _The Pfaffian line bundle_, Commun. Math. Phys. __149__ (1992) (3) 463–493 [doi](https://doi.org/10.1007/BF02096939) [euclid](https://projecteuclid.org/euclid.cmp/1104251304)

* Section 1.3 in: [[Emma Previato]], [[Mauro Spera]], _Isometric embeddings of infinite-dimensional Grassmannians_, Regul. Chaot. Dyn. (2011) 16: 356 [doi](https://doi.org/10.1134/S1560354711030099)

* F. Balogh, J. Harnad, J. Hurtubise, _Isotropic Grassmannians, Plücker and Cartan maps_, [arxiv/2007.03586](https://arxiv.org/abs/2007.03586)

> This work is motivated by the relation between the KP and BKP integrable hierarchies, whose τ-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space $V$ of dimension $N$, embeds the Grassmannian $Gr^0_V(V+V^*)$ of maximal isotropic subspaces of $V+V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $\Lambda(V)$, and the Plücker map, which embeds the Grassmannian $Gr_V(V+V^*)$ of all N-planes in $V+V^*$ into the projectivization of $\Lambda^N(V+V^*)$. The Plücker coordinates on $Gr^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle $Pf^*\to Gr^0_V(V+V^*,Q)$. In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric N×N matrix as bilinear sums over the Pfaffians of their principal minors

The [[nLab:string theory|string]] worldsheet [[Green-Schwarz mechanism]] which trivializes the worldsheet Pfaffian line bundle, and its relation to [[string structure]]s, that goes back to Killingback and [[Edward Witten]], has been formalized in

* [[Ulrich Bunke]], _String structures and trivialisations of a Pfaffian line bundle_ ([arXiv](http://arxiv.org/abs/0909.0846))

[[!redirects Pfaffian line bundles]]

[[!redirects Pfaffian bundle]]
[[!redirects Pfaffian bundles]]

[[!redirects pfaffian line bundle]]
[[!redirects pfaffian line bundles]]


phantom category

A **phantom category** is a [[localization of an (infinity,1)-category|localization]] $C \subset D(X)$ of the [[stable (infinity,1)-category]] of [[perfect complexes]] on some [[scheme]] $X$, with trivial [[algebraic K-theory]] $K(C)$.

## Related

* [[derived noncommutative geometry]]

* [[noncommutative motives]]

## References

* [[Sergey Gorchinskiy]], [[Dmitri Orlov]], _Geometric Phantom Categories_, [arXiv:1209.6183](http://arxiv.org/abs/1209.6183).

* [[Pawel Sosna]], _Some remarks on phantom categories and motives_, [arXiv:1511.07711](http://arxiv.org/abs/1511.07711).


phantom dark energy

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{:toc}

## Idea

What is called _phantom dark energy_ ([Caldwell 02](#Caldwell02)) is [[dark energy]] with an [[equation of state]] with $w \lt 1$, hence with [[stress-energy tensor]] of the form

$$
  T_{\mu \nu}
  \;=\;
  (\epsilon - P) u_\mu u_\nu - g_{\mu \nu} P
$$

(for $u$ its density/flow [[vector field]] and $P \in \mathbb{R}$ the [[pressure]]) with

\[
  \label{wSmallerThanMinusOne}
  w \;\coloneqq\; \frac{P}{\epsilon} \;\lt\;
  1
  \,.
\]

It has been argued that data of observational [[cosmology]] actually prefers [[phantom dark energy]] over a plain [[cosmological constant]] (which correspondes to $w = 1$), see e.g. [Caldwell 02](#Caldwell02), [Di Valentino-Melchiorri-Silk 20](#ValentinoMelchiorriSilk20).

## As an effective phenomenon
 {#AsAnEffectivePhenomenon}

Taken at face value a fundamental [[field (physics)]] with such an equation of state with $w \lt 1$ (eq:wSmallerThanMinusOne) appears to be inconsistent, as it violates the [[null energy condition]]. (Though this is subtle in itself, even fields with $w \gt -1$ may violate the [[null energy condition]] [Sawicku-Vikman 12](#SawickuVikman12)).

But apparent phantom dark energy may appear as an effective description of more fundamental effects, for instance of an [[inhomogeneous cosmology]] ([Ludwick 15](#Ludwick15), [Ludwick 18](#Ludwick18), review in [Ludwick 17, Section 6](#Ludwick17))

## Related concepts

* [[dark energy]]

* [[quintessence]]

* [[Witten's Dark Fantasy]]

## References

Review:

* {#Ludwick17} Kevin J. Ludwick, _The Viability of Phantom Dark Energy: A Brief Review_, Modern Physics Letters A 32, 28 (2017) ([arXiv:1708.06981](https://arxiv.org/abs/1708.06981))

Original articles:

* {#Caldwell02} R.R. Caldwell, _A Phantom Menace? Cosmological consequences of a dark energy component with super-negative equation of state_, Phys. Lett. B545:23-29, 2002 ([arXiv:astro-ph/9908168](https://arxiv.org/abs/astro-ph/9908168))

* {#SawickuVikman12} Ignacy Sawicki, Alexander Vikman, _Hidden Negative Energies in Strongly Accelerated Universes_, Phys. Rev. D 87, 067301 (2013) ([arXiv:1209.2961](https://arxiv.org/abs/1209.2961))

* {#Ludwick15} Kevin J. Ludwick, _Examining the Viability of Phantom Dark Energy_, Phys. Rev. D 92, 063019 (2015) ([arXiv:1507.06492](https://arxiv.org/abs/1507.06492))

* {#Ludwick18} Kevin J. Ludwick, _The Viability of Phantom Dark Energy as a Quantum Field in 1st-Order FLRW Space_, Phys. Rev. D 98, 043519 (2018) ([arXiv:1804.02987](https://arxiv.org/abs/1804.02987))

Observation:

* {#ValentinoMelchiorriSilk20} Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Cosmic Discordance: Planck and luminosity distance data exclude LCDM_ ([arXiv:2003.04935](https://arxiv.org/abs/2003.04935))


phantom map

[[!redirects phantom maps]]

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* table of contents
{:toc}

## Definition

In [[stable homotopy theory]]/[[cohomology theory]] a [[homomorphism]] of [[spectra]] $E \longrightarrow F$ is called a _phantom map_ if it induces (up to [[homotopy]]) the [[zero morphism]] on [[generalized homology theories]]
$E_\bullet \longrightarrow F_\bullet$, hence if for every [[topological space]]/[[homotopy type]] $X$ the induced map of [[homology groups]]

$$
  E_\bullet(X) \longrightarrow F_\bullet(X)
$$

is the [[zero morphism]].

The existence of phantom maps implies that despite the [[Brown representability theorem]], there is a subtle difference between generalized (Eilenberg-Steenrod) homology theories and the [[spectra]] which represent them: the latter contain in general more information. (Note that this issue is not present when working with cohomology theories instead of homology theories).

In even more classical context, a continuous map $f$ from a [[CW-complex]] $X$ to a topological space $Y$ is a phantom map if it is not homotopic to a constant but for every finite CW-subcomplex $Z\subset X$ the restriction $f|_Z:Z\to Y$ is homotopic to a constant.

## Properties

### Between Landweber-exact spectra

Between [[Landweber exact spectra]], every phantom map is already null-homotopic. (e.g. [Lurie, 10, corollary 7](#Lurie)).

## References

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 17 _Phantom maps_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf))
 {#Lurie}

[[!redirects phantom morphism]]
[[!redirects phantom morphisms]]

[[!redirects phantom homomorphism]]
[[!redirects phantom homomorphisms]]


phase

The term "phase" appears in different contexts:

* the [[complex phase]] of a [[complex number]] $c$ is the real number $\phi$ in the decomposition $c = {\vert c \vert} e^{i \phi}$;

* the phase of a [[wave function]] in [[physics]] is just this complex phase, the wave function being (locally) a function with values in the complex numbers;

* correspondingly then the _phase difference_ between two linearly dependent vectors of a complex [[Hilbert space]] is the phase of the complex number that relates them;

* in [[thermodynamics]] and [[statistical physics]] different _[[phase of matter|phases]]_ of a [[physical system]] label different order parameters that distinguish macroscopically different regions of the configuration space of the system;

* the [[phase space]] of a [[physical system]] is its space of classical solutions, typically given coordinates involving [[generalised momenta]] (for a relation of this to the phase of a [[wave function]] above see at _[[phase and phase space in physics]]_).

[[!redirects phase]]
[[!redirects phases]]


phase and phase space in physics

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## Idea

In [[physics]] the term [[phase]] appears prominently in two superficially different uses:

Inside the term "[[phase space]]", introduced by [[Boltzmann]] in 1872 in the context of [[classical mechanics]], it refers to the instantaneous position [[coordinates]] and [[momenta]] of a [[particle]]; its _Bewegungsphase_, it _phase of motion_. ([Nolte, p. 4](#Nolte10)).

But the term "[[phase]]" re-appeared, independently, with the advent of [[quantum physics]]: it refers to the [[complex phase]] of the [[complex numbers]] that [[wave functions]] take values in.

But it turns out that in the [[semiclassical approximation]] to [[quantum physics]] the phase of wave functions (or rather its [[derivative]]) precisely parametrizes points in [[phase space]]! Even though this is historically a pure coincidence of terminology, fortunately it matches to some extent.

This relation is discussed in some detail at _[[semiclassical state]]_. It is an old observation that drove the pre-history of [[quantum mechanics]] at a time when this was discovered by drawing analogies with [[wave mechanics|wave]] [[optics]]:

the [[derivative]] of the [[phase]] (as in: [[complex numbers]]) of a [[wave function]] is (locally, or else in semiclassical approximation) the [[momentum]] of the corresponding [[particle]] (of which the wave function is the [[quantum state]]).  This momentum is, in addition to the spatial dependence of the wave function, what parameterizes [[phase space]].

More precisely: a stationary [[semiclassical state]] of a [[particle]] propagating on some [[Riemannian manifold]] $(X,g)$ (and subject to some [[force]] given by some scalar potential but *not* charged under a [[gauge field]]) is a [[wave function]] of the form

$$
  \psi \colon (x,t) \mapsto \exp(i S(x)/\hbar) a(x) \exp(-i \omega t)
  \,,
$$

where $S$ and $A$ are real-valued functions. The [[de Rham differential]] of the **[[complex phase]]-function**

$$
  S\;\colon\; X \to \mathbb{R}
$$

appearing here defines a map from $X$ into its [[phase space]], which here is the [[cotangent bundle]] over $X$:

$$
  \mathbf{d}S \;\colon\; X \to T^\ast X
  \,.
$$

The phase-space image of $X$ defined this way is the level set of the given [[Hamiltonian]] $H \colon T^* X \to \mathbb{R}$ at [[energy]] $E = \hbar \omega$.

So the de Rham differential $\mathbf{d}S$ of the complex phase function $S$ of a semiclassical wave function precisely parametrizes this semiclassical state as a submanifold in phase space.

## References

The origin of the term "phase" in "phase space" is discussed in

* David Nolte, _The tangled tale of phase space_, Physics Today (April 2010) ([pdf](http://www.physics.purdue.edu/nlo/NoltePT10.pdf))
 {#Nolte10}

The close relation between complex phases of semiclassical wave functions and phase space points is discussed for instance in the introduction (around p. 9) of

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_ ([pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf))

[[!redirects phase and phase space in physics]]

[[!redirects phases and phase spaces in physics]]

[[!redirects phase (physics)]]


phase of matter

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* table of contents
{:toc}

## Idea

(...) concept in [[thermodynamics]]/[[condensed matter physics]] (...)

## Examples

* [[solid state physics|solid]]

* [[fluid dynamics|fluid]]

* gas

* [[plasma]]

* [[quark-gluon plasma]]

* [[topological phase of matter]], [[topological state of matter]]

  [[topological order]]

* [[quasicrystal]]

## Related concepts

* [[phase transition]]

* [[thermodynamic limit]]

* [[quantum material]]

## References

### General

See also

* Wikipedia, _Phase_(matter)_

* Wikipedia, _[State of matter](https://en.wikipedia.org/wiki/State_of_matter)_

* Wikipedia, _[Phase diagram](https://en.wikipedia.org/wiki/Phase_diagram)_

### In QCD

On the phase diagram of [[quantum chromodynamics]]:

* [[Edward Shuryak]], _The Phases of QCD_ ([arXiv:hep-ph/9609249](https://arxiv.org/abs/hep-ph/9609249))

* {#Hands01} Simon Hands, _The Phase Diagram of QCD_, Contemp. Phys. 42:209-225, 2001 ([arXiv:physics/0105022](https://arxiv.org/abs/physics/0105022))

* {#Schaefer05} [[Thomas Schaefer]], _Phases of QCD_ ([arXiv:hep-ph/0509068](https://arxiv.org/abs/hep-ph/0509068))

[[!redirects phases of matter]]

[[!redirects phase (matter)]]

[[!redirects state of matter]]
[[!redirects states of matter]]

[[!redirects phase diagram]]
[[!redirects phase diagrams]]