Patrick Schultz

Graduate Student at University of Oregon.

## Selected writings

On [[categorical systems theory]]:

* [[Patrick Schultz]], [[David I. Spivak]], [[Christina Vasilakopoulou]], *Dynamical Systems and Sheaves*, Appl Categor Struct **28** (2020) 1–57 [[arXiv:1609.08086](https://arxiv.org/abs/1609.08086), [doi:10.1007/s10485-019-09565-x](https://doi.org/10.1007/s10485-019-09565-x)]

category: people


Patrick Walls

* [webpage](http://ms.mcmaster.ca/~pwalls/)

category: people


Paul Aspinwall

**Paul S Aspinwall** is a theoretical physicist at Duke University.

* [publication list](http://fds.duke.edu/db/aas/math/faculty/psa/publications)

## Selected publications

On [[small instantons]] under [[duality between F-theory and heterotic string theory]]:

* {#AspinwallGross96} [[Paul Aspinwall]], [[Mark Gross]], _The $SO(32)$ Heterotic String on a K3 Surface_, Phys. Lett. B387 (1996) 735-742 ([arXiv:hep-th/9605131](https://arxiv.org/abs/hep-th/9605131))

* {#AspinwallMorrison97} [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_, Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))

On [[ADE-singularities]] in [[string theory]]:

* [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_,  Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))

On [[moduli stabilization]] in [[M-theory on 8-manifolds]] for the [[product manifold]] of [[K3]] with itself:

* [[Paul Aspinwall]], [[Renata Kallosh]], _Fixing All Moduli for M-Theory on $K3 \times K3$_, JHEP 0510:001, 2005 ([arXiv:hep-th/0506014](https://arxiv.org/abs/hep-th/0506014))

On [[mirror symmetry]]:

* [[Paul Aspinwall]], [[Tom Bridgeland]], [[Alastair Craw]], [[Michael Douglas]], Mark Gross, _Dirichlet branes and mirror symmetry_, Amer. Math. Soc. Clay Math. Institute 2009.

category: people


Paul Balmer

__Paul Balmer__ is a mathematician working in algebraic geometry, algebraic K-theory, homotopy theory and category theory.

* [web](http://www.math.ucla.edu/~balmer);

* [publication list](http://www.math.ucla.edu/~balmer/research/publications.html)

## Selected writings

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

* {#Balmer04} [[Paul Balmer]], _The spectrum of prime ideals in tensor triangulated categories_. J. Reine Angew. Math., 588:149–168, 2005 ([arXiv:math/0409360](http://arxiv.org/abs/math/0409360))

* {#Balmer10} [[Paul Balmer]], _Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings_, Algebr. Geom. Topol., 10(3):1521–1563, 2010 ([pdf](http://www.math.ucla.edu/~balmer/research/Pubfile/SSS.pdf))

On [[stacks]] of [[group representations]]:

* [[Paul Balmer]], *Stacks of group representations*, J. European Math. Soc. **17** 1 (2015) 189-228 [[arXiv:1302.6290](https://arxiv.org/abs/1302.6290), [doi:10.4171/jems/501](https://doi.org/10.4171/jems/501)]

On [[tensor triangulated categories]]:

* [[Paul Balmer]], _A guide to tensor-triangular classification_, in: [[Haynes Miller]] (ed.) _[[Handbook of Homotopy Theory]]_ ([arXiv:1912.08963](https://arxiv.org/abs/1912.08963))

category: people

[[!redirects P. Balmer]]


Paul Baum

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

## Selected writings

On the [[equivariant Chern character]]:

* [[Alain Connes]], [[Paul Baum]], _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))

## Related $n$Lab entries

* [[Baum-Connes conjecture]]

category: people


Paul Bendich

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

## Selected writings

On [[well groups]]:

* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *The Robustness of Level Sets*, In: M. de Berg, U. Meyer (eds.) _Algorithms – ESA 2010_. ESA 2010. Lecture Notes in Computer Science **6346** Springer (2010)  $[$[doi:10.1007/978-3-642-15775-2_1](https://doi.org/10.1007/978-3-642-15775-2_1)$]$

* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Homology and Robustness of Level and Interlevel Sets*, Homology, Homotopy and Applications, **15** (2013) 51-72 $[$[euclid:1383943667](https://projecteuclid.org/euclid.hha/1383943667)$]$

category: people


Paul Benioff

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

* Argonne National Laboratory, *[Remembering Paul Benioff, renowned scientist and quantum computing pioneer](https://www.anl.gov/article/remembering-paul-benioff-renowned-scientist-and-quantum-computing-pioneer)*

## Selected writings

Early proposal of [[quantum computation]]:

* [[Paul Benioff]], *The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines*, J Stat Phys 22, 563–591 (1980) [[doi:10.1007/BF01011339](https://doi.org/10.1007/BF01011339)]

* [[Paul Benioff]], *Quantum Mechanical Models of Turing Machines That Dissipate No Energy*, Phys. Rev. Lett. **48** 1581  (1982) [[doi:10.1103/PhysRevLett.48.1581](https://doi.org/10.1103/PhysRevLett.48.1581)]

category: people


Paul Bernays

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

## related $n$Lab entries

* [[ZFC]]

category: people


Paul Blain Levy

[[!redirects Paul-Blain Levy]]
Paul Blain Levy is a theoretical [[computer science|computer scientist]] at the University of Birmingham.

* [website](http://www.cs.bham.ac.uk/~pbl)

## Selected writings

On [[locally graded categories]]:

* {#Levy19} [[Paul Blain Levy]], *Locally graded categories*, talk (2019) [slides:[pdf](https://www.cs.bham.ac.uk/~pbl/papers/locgrade.pdf), [[Levy-LocallyGradedCategories.pdf:file]]]

# Related Pages

* Levy developed the [[call-by-push-value]] calculus in his PhD thesis.

* [[premulticategory]]

* [[Freyd multicategory]]

category: people

[[!redirects Paul Levy]]


Paul Bressler

**Paul Bressler** works mainly in [[geometry]] (in the wide and modern sense of the word) including [[deformation theory]], index theorems, [[algebraic geometry|algebraic]] and [[differential geometry|differential]] geometry, [[algebraic topology]], [[homological algebra]], [[K-theory]]; some of his motivation lies in mathematical [[physics]] (quantization, mirror symmetry, conformal field theory, ...). Bressler's advisor was [[Raoul Bott]].

* [homepage](http://paulbressler.com)

category: people


Paul Busch

* Wikipedia entry

## Selected writings

On [[quantum physics]] with emphasis on [[quantum probability]] via [[POVMs]]:

* [[Paul Busch]], Marian Grabowski, [[Pekka J. Lahti]], *Operational Quantum Physics*, Lecture Notes in Physics Monographs **31**, Springer (1995) [[doi:10.1007/978-3-540-49239-9](https://doi.org/10.1007/978-3-540-49239-9)]

* [[Paul Busch]], [[Pekka J. Lahti]], Juha-Pekka Pellonpää, Kari Ylinen, *Quantum Measurement*, Springer (2016) [[doi:10.1007/978-3-319-43389-9](https://doi.org/10.1007/978-3-319-43389-9)]

category: people


Paul Cook

Paul Cook is lecturer of [[mathematics]] at King's College London.

## Seelected writings

On [[D1-D7 brane intersections]] as fuzzy funnels on [[fuzzy 6-spheres]]:

* [[Paul Cook]], Robert de Mello Koch, Jeff Murugan, _Non-Abelian BIonic Brane Intersections_,  Phys. Rev. D68:126007, 2003 ([arXiv:hep-th/0306250](https://arxiv.org/abs/hep-th/0306250))

On [[Kac-Moody algebras]] in [[M-theory]] ([[U-duality]]):

* [[Paul Cook]], _Connections between Kac-Moody algebras and M-theory_ PhD thesis ([arXiv:0711.3498](http://arxiv.org/abs/0711.3498))

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

* 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

## related $n$Lab entries

* [[Kac-Moody algebra]], [[E11]]

* [[M-theory]]

category: people

[[!redirects Paul P. Cook]]


Paul de Medeiros

## Selected writings

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

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

On [[M2-brane 3-algebras]] in the [[BLG model]] for coincident [[M2-branes]]:

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], _Metric Lie 3-algebras in Bagger-Lambert theory_, JHEP 0808 : 045, 2008 ([arXiv:0806.3242](https://arxiv.org/abs/0806.3242))

On the equivalence of [[M2-brane 3-algebras]] to [[Lie algebra representations]] over [[metric Lie algebras]]:

* {#MFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

On the [[ADE classification]] of [[black brane|black]] [[BPS state|BPS]]-[[M2-brane]] solutions to [[D=11 supergravity]] (via [[spherical space forms]] of the [[7-sphere]]):

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* {#MedeirosFigueroa10} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

category: people


Paul Dedecker

Paul Dedecker was a Belgian mathematician, one of the pioneers of the subject of [[nonabelian cohomology]].

He also worked on [[category theory]], [[variational calculus]] and its relations to [[homological algebra]], [[exterior calculus]] on [[manifolds]] and on [[mathematical physics]].

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

## Selected writings

On [[nonabelian cohomology]]:

* [[Paul Dedecker]], *Cohomologie à coefficients non abéliens et espaces fibrés*,  Bulletins de l'Académie Royale de Belgique **41** (1955) 1132-1146 [[persee:barb_0001-4141_1955_num_41_1_69497](https://www.persee.fr/doc/barb_0001-4141_1955_num_41_1_69497)]

* [[Paul Dedecker]], *Cohomologie de dimension 2 à coefficients non abéliens*, C. R. Acad. Sci. Paris **247** (1958) 1160-1163 [[BnF](https://gallica.bnf.fr/ark:/12148/bpt6k7258/f192.item.r=dedecker)]

* [[Paul Dedecker]], *Sur La Cohomologie Non Abelienne I (Dimension Deux)*, Canadian Journal of Mathematics **12** (1960) 231-251 [[doi:10.4153/CJM-1960-019-7](https://doi.org/10.4153/CJM-1960-019-7)]

* Groupoïdes de cohomologie à coefficients non abéliens et espaces fibrés. Colloque de topologie algébrique, Louvain, 1956,  pp. 135--149. Georges Thone, Liège; Masson & Cie, Paris, 1957.

* On the exact cohomology sequence of a space with coefficients in a nonabelian sheaf.  1958  Symposium internacional de topología algebraica International symposium on algebraic topology  pp. 309--322 Universidad Nacional Autónoma de México and UNESCO, Mexico City

* Sur la cohomologie non abélienne. I. Canad. J. Math.  12  1960 231--251; Sur la cohomologie non abélienne. II.  Canad. J. Math.  15  1963 84--93.

* Le foncteur $Hom$ non abélien. Notion de poulpe. C. R. Acad. Sci. Paris  257  1963 2384--2387.

* Sur la $n$-cohomologie non abélienne. C. R. Acad. Sci. Paris  260  1965 4137--4139.

* (with Armin Frei), Généralisation de la suite exacte de cohomologie non abélienne. C. R. Acad. Sci. Paris Sér. A-B  263  1966 A203--A206.

* (with A. S-T. Lue) A nonabelian two-dimensional cohomology for associative algebras.  Bull. Amer. Math. Soc.  72  1966 1044--1050.

* Algèbre homologique non abélienne. 1966  Colloque de Topologie (Brussels, 1964)  pp. 209--234 Librairie Universitaire, Louvain

* (with René Lavendhomme), Catégories fibrées et cohomologie non abélienne de dimension deux, C. R. Acad. Sci. Paris Sér. A-B  266  1968 A453--A456.

* (with René Lavendhomme), Sur les catégories fibrées. C. R. Acad. Sci. Paris Sér. A-B  266  1968 A409--A412.

* Three dimensional non-abelian cohomology for groups.  1969  Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two)  pp. 32--64 Springer, Berlin

* (with E. Luks), Sur la non fonctorialité du $H^{2}$ en cohomologie non abélienne.  C. R. Acad. Sci. Paris Sér. A-B  282  (1976), no. 4, Ai, A183--A185.

[[!redirects Dedecker]]

category: people


Paul Dirac

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

## Selected writings

Early discussion of the foundations of [[quantum mechanics]] (including the possibly first consideration of what came to be known as the [[Slater determinant]]):

* [[Paul A. M. Dirac]], *On the theory of quantum mechanics*, Proceedings of the Royal Society **112** 762 (1926) [[doi:10.1098/rspa.1926.0133](https://doi.org/10.1098/rspa.1926.0133)]

and with early emphasis of the [[Born-Pauli rule]]:

*  [[Paul A. M. Dirac]], *The physical interpretation of the quantum dynamics*, Proceedings of the Royal Society of London **113** 765 (1927) [[doi:10.1098/rspa.1927.0012](https://doi.org/10.1098/rspa.1927.0012)]

Precursor discussion to the [[path integral]]-formulation of [[quantum mechanics]]:

* [[Paul A. M. Dirac]], *The Lagrangian in Quantum Mechanics*, Physikalische Zeitschrift der Sowjetunion, Band **3**, Heft 1 (1933) 64–72 [[pdf](https://www.informationphilosopher.com/solutions/scientists/dirac/Lagrangian_1933.pdf)], reprinted in [Brown 2005](path+integral#Brown05) [[doi:10.1142/9789812567635_0003](https://doi.org/10.1142/9789812567635_0003)]

Introducing [[bra-ket]]-notation to [[quantum mechanics]]:

* {#Dirac39} [[Paul A. M. Dirac]], *A new notation for quantum mechanics*, Mathematical Proceedings of the Cambridge Philosophical Society **35** 3 (1939) 416-418 [[doi:10.1017/S0305004100021162](https://doi.org/10.1017/S0305004100021162), [pdf](https://www.ifsc.usp.br/~lattice/wp-content/uploads/2014/02/Dirac_notation.pdf)]

Introducing the modern theory of [[constrained mechanics|constrained]] [[Hamiltonian mechanics]]:

* [[Paul A. M. Dirac]], *Generalized Hamiltonian Dynamics*, Canadian Journal of Mathematics **2** (1950) 129-148 [[doi:10.4153/CJM-1950-012-1](https://doi.org/10.4153/CJM-1950-012-1)]

  reprinted in:

  Alexander S. Blum, Dean Rickles (eds.), Ch.34 in: *Quantum Gravity in the First Half of the Twentieth Century: A Sourcebook*, Edition Open Sources **10** (2018) 484-503 [[doi:10.34663/9783945561317-00](https://edition-open-sources.org/sources/10/), [pdf](https://edition-open-sources.org/media/sources/10/36/sources10chap34.pdf)]

On a [[relativistic membrane]] [[model (in theoretical physics)|model]] for the [[electron]]:

* {#Dirac62} [[Paul Dirac]], *An Extensible Model of the Electron*, Proc. Roy. Soc. A **268** (1962) 57-67 [[jstor:2414316](https://www.jstor.org/stable/2414316)]

  > (also proposing the [[Dirac-Born-Infeld action]])

* [[Paul Dirac]], _The motion of an Extended Particle in the Gravitational Field_, in: [[Leopold Infeld|L. Infeld]] (ed.), *Relativistic Theories of Gravitation*, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175 [[spire:1623740](http://inspirehep.net/record/1623740), article:[[Dirac-MotionOfAnExtendedParticle.pdf:file]], full proceedings:[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)]

* [[Paul Dirac]], _Particles of Finite Size in the Gravitational Field_, Proc. Roy. Soc. A **270** (1962) 354-356 [[doi:10.1098/rspa.1962.0228](https://doi.org/10.1098/rspa.1962.0228)]

On [[quantum mechanics]] (and some [[quantum field theory]]):

* {#Dirac30} [[Paul Dirac]], *[[The Principles of Quantum Mechanics]]*, International series of monographs on physics, Oxford University Press (1930) [[ISBN:9780198520115](https://global.oup.com/academic/product/the-principles-of-quantum-mechanics-9780198520115)]

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

Early discussion of the [[positron]]:

* {#Dirac33} [[Paul A. M. Dirac]], *Théorie du Positron*, Proceedings of *Structure et propriétés des noyaux atomiques*, [Solvay](http://www.solvayinstitutes.be/) Conference (1933) 203-212 [full:[pdf](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/1933.pdf)]

## Quotes

{#OnNotation} On [[syntax]]:

> [[Dirac 1939](#Dirac39)] In mathematical theories the question of notation, while not of primary importance, is yet worthy of careful consideration, since a good notation can be of great value in helping the development of a theory, by making it easy to write down those quantities or combinations of quantities that are important, and difficult or impossible to write down those that are unimportant.

On [[mathematical physics]]:

> [[Dirac 1978](#Dirac78):] One should keep the need for a sound mathematical basis dominating one’s search for a new theory. Any physical or philosophical ideas that one has must be adjusted to  fit the mathematics. Not the way round.

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

## Related entries

* [[Planck's constant]]

* [[The Principles of Quantum Mechanics]]

* [[quantum electrodynamics]]

* [[Dirac operator]]

* [[Dirac field]]

* [[Dirac spinor]]

category: people

[[!redirects Dirac]]
[[!redirects Paul Dirac]]
[[!redirects Paul A. M. Dirac]]
[[!redirects PAM Dirac]]
[[!redirects P.A.M. Dirac]]
[[!redirects P. A. M. Dirac]]


Paul Downen

* [personal page](https://www.pauldownen.com/)

* [institute page](https://www.uml.edu/sciences/computer-science/faculty/downen-paul.aspx)

## Selected writings

Introduction to [[proof theory]] via [[natural deduction]], [[sequent calculus]]:

* [[Paul Downen]], [[Zena M. Ariola]], *A tutorial on computational classical logic and the sequent calculus*, Journal of Functional Programming **28** (2018) e3 [[doi:10.1017/S0956796818000023](https://doi.org/10.1017/S0956796818000023), [pdf](https://www.pauldownen.com/publications/sequent-intro.pdf)]

category: people


Paul E. Schupp

* [homepage](http://www.math.uiuc.edu/People/schupp.html)

[[!redirects P. E. Schupp]]

[[!redirects P. Schupp]]


Paul Edwin Bland

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=101004)

## Selected writings

On [[rings]] and [[modules]] ([[algebra]]):

* [[Paul Edwin Bland]], *Rings and Their Modules*, De Gruyter (2011) [[doi:10.1515/9783110250237](https://doi.org/10.1515/9783110250237), [pdf](http://site.iugaza.edu.ps/mashker/files/%D9%85%D8%AD%D8%A7%D8%B6%D8%B1%D8%A7%D8%AA-%D8%AC%D8%A8%D8%B1-%D8%AF%D9%83%D8%AA%D9%88%D8%B1%D8%A7%D8%A9.pdf)]

category: people

[[!redirects Paul E. Bland]]


Paul Ehrenfest

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

## related $n$Lab entries

* [[spinor]]

category: people


Paul Fendley

Paul Fendley is a Senior Research Fellow at All Souls College Oxford, and also a professor in the Rudolf Peierls Centre for Theoretical Physics, Oxford.

His research interests include [[statistical mechanics]] and [[condensed matter physics]].

* [Department of Physics website](https://www2.physics.ox.ac.uk/contacts/people/fendley)

* [All Souls Website](https://www.asc.ox.ac.uk/person/18)

## Selected writings

* [Selected papers](https://arxiv.org/find/all/1/au:+fendley_p/0/1/0/all/0/1)

On [[parafermions]]:

* [[Paul Fendley]], *Free parafermions*, J. Phys. A **47** (2014) 075001 $[$[arXiv:1310.6049](https://arxiv.org/abs/1310.6049), [doi:10.1088/1751-8113/47/7/075001](https://doi.org/10.1088/1751-8113/47/7/075001)$]$

* [[Paul Fendley]], *Parafermionic edge zero modes in $\mathbb{Z}_n$-invariant spin chains*,  J. Stat. Mech. (2012) P11020 $[$[doi:10.1088/1742-5468/2012/11/P11020](https://iopscience.iop.org/article/10.1088/1742-5468/2012/11/P11020)$]$

and in relation to [[topological phases of matter]] with [[topological order]]:

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

## Recorded talks

* [MPPM](https://www.youtube.com/watch?v=oehDq-cvFyM) on-line seminar from 28 April 2020, Title: _Statistical Mechanics and Fusion Categories_.  The talk includes discussion of relation to knot invariants and lattice topological defects.

category: people


Paul Francis Mendler

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=77001)

## Selected writing

Early definition of [[inductive types]] in [[type theory]]:

* [[Robert L. Constable]], [[N. Paul Francis Mendler]], *Recursive definitions in type theory*, in *Logic of Programs 1985*, Lecture Notes in Computer Science **193** Springer (1985) $[$[doi:10.1007/3-540-15648-8_5](https://doi.org/10.1007/3-540-15648-8_5)$]$

* [[Paul Francis Mendler]], *Inductive Definition in Type Theory*, Cornell (1987) [[hdl:1813/6710](https://hdl.handle.net/1813/6710)]

category: people

[[!redirects Paul Mendler]]
[[!redirects Paul F. Mendler]]
[[!redirects Nax Paul Francis Mendler]]
[[!redirects N. Paul Francis Mendler]]


Paul Garrett

* [webpage](http://www.math.umn.edu/~garrett/)

## Selected writings

On [[exceptional isomorphisms]] of [[classical Lie groups]]:

* {#Garrett13} [[Paul Garrett]], _Sporadic isogenies to orthogonal groups_ (July 2013) [[pdf](http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf), [[Garrett-SporadicIsogenies.pdf:file]] ]

category: people


Paul Ginsparg

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

## Selected writings

On [[quantum anomalies]]:

* [[Luis Alvarez-Gaumé]], [[Edward Witten]], _Gravitational Anomalies_ Nucl. Phys. B234 (1984) 269 (doi:10.1016/0550-3213(84)90066-X, [spire:192309](http://inspirehep.net/record/192309))

* [[Luis Alvarez-Gaumé]], [[Paul Ginsparg]], _The structure of gauge and gravitational anomalies_, Ann. Phys. 161 (1985) 423. (doi:10.1016/0003-4916(85)90087-9, [spire:202565](http://inspirehep.net/record/202565))

On [[conformal field theory]]:

* [[Paul Ginsparg]], *Applied Conformal Field Theory*, lectures at: *[Fields, strings, critical phenomena, Les Houche Summer School 1988](https://inis.iaea.org/search/search.aspx?orig_q=RN:21047524)([arXiv:hep-th/9108028](https://arxiv.org/abs/hep-th/9108028))

On [[superstring theory]]:

* [[Paul Ginsparg]], [[Sheldon Glashow]], _Desperately Seeking Superstrings_, Phys.Today 86N5 (1986) 7-9 ([arXiv:physics/9403001](https://arxiv.org/abs/physics/9403001), [doi:10.1063/1.2814991](https://physicstoday.scitation.org/doi/10.1063/1.2814991))

category: people


Paul Glenn

Paul Glenn is an American mathematician, whose thesis work (with [[Jack Duskin]] as his thesis advisor) involved the development of the use of [[hypergroupoid]]s in the representation of cohomology classes.

* [homepage](http://faculty.cua.edu/glenn/welcome.html)

* [Math Genealogy page](http://genealogy.mathematik.uni-bielefeld.de/id.php?id=30964)

## Selected writings

On [[principal infinity-bundles|principal $\infty$-bundles]],  [[nonabelian cohomology]] and [[hypergroupoids]]:

* [[Paul G. Glenn]], *Realization of cohomology classes in arbitrary exact categories*, J. Pure Appl. Algebra **25** 1 (1982) 33-105 [doi:10.1016/0022-4049(82)90094-9]

category:people

[[!redirects Paul G. Glenn]]


Paul Goerss

* [website](http://www.math.northwestern.edu/~pgoerss/)

## Selected writings

On [[simplicial homotopy theory]]:

* [[Paul Goerss]], [[John Frederick Jardine]], *[[Simplicial homotopy theory]]*, Progress in Mathematics, Birkhäuser (1999), Modern Birkhäuser Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4)]

On [[model categories]] and [[simplicial homotopy theory]] ([[simplicial model categories]]):

* [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago in August 2004, published as: _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics **436** AMS (2007) [[arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436)]

On [[elliptic cohomology]] and [[topological modular forms]]:

* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller, and Lurie)_, Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 ([arXiv:0910.5130](https://arxiv.org/abs/0910.5130), [numdam:AST_2010__332__221_0](http://www.numdam.org/item/AST_2010__332__221_0))

## Related enties

* [[Topological Algebraic Geometry - A Workshop]]

category: people


Paul H. Frampton

* [personal page](http://www.paulframpton.org/about/)

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

## Selected writings

On the origin of [[string theory]] as a model for [[hadrodynamics]] (cf. *[[Polyakov  gauge-string duality]]*):

* [[Paul H. Frampton]], *Dual Resonance Models and Superstrings*, World Scientific (1986) [[doi:10.1142/0249](https://doi.org/10.1142/0249)]

category: people

[[!redirects Paul Frampton]]


Paul Halmos

Paul Halmos was a Hungarian-born American-based mathematician.

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

## Selected writings

On [[finite-dimensional Hilbert spaces]]:

* [[Paul R. Halmos]], *Finite-Dimensional Hilbert Spaces*, The American Mathematical Monthly, **77** 5 (1970) 457-464 [[doi:10.2307/2317378](https://doi.org/10.2307/2317378), [jstor:2317378](https://www.jstor.org/stable/2317378)]

See also:

* Measure Theory. D. Van Nostrand Company, 1950.

* A Hilbert space problem book. D. Van Nostrand Co., 1967.  Second edition, Graduate Texts in Mathematics, 19, Springer, 1982. ISBN: 0-387-90685-1.

* Lectures on ergodic theory. Chelsea Publishing Co., 1960.

* Introduction to Hilbert Space and the theory of Spectral Multiplicity. Chelsea Publishing Company, 1951.

* Lectures on Boolean algebras. Van Nostrand Mathematical Studies, No. 1.  D. Van Nostrand Co., 1963.

* Algebraic logic. Chelsea Publishing Co., 1962.

* Finite Dimensional Vector Spaces.  Annals of Mathematics Studies, no. 7.  Princeton University Press, Princeton, N.J., 1942.  Second edition, D. Van Nostrand Co., 1958.

* Naive set theory.  D. Van Nostrand Co., 1960.

* Linear algebra problem book. The Dolciani Mathematical Expositions, 16. Mathematical Association of America, 1995.  ISBN: 0-88385-322-1.

[[!redirects Paul R. Halmos]]


Paul Hertz

[[!redirects empty 156]]
[[!redirects empty 156]]

**Paul Hertz** (1881-1940) was a German theoretical physicist and philosopher who made important pioneering contributions to structural proof theory. In particular, his explorations of axiomatic "Satzsysteme" (systems of sentences) in the 1920s was a stepping stone to [[Gerhard Gentzen|Gerhard Gentzen's]] [[sequent calculus]].

## Related entries

* [[Gerhard Gentzen]]

* [[sequent calculus]]

## References

 * [[Paul Bernays]], *Hertz, Paul*, Neue Deutsche Biographie **8** (1969) p.711. ([link](http://www.deutsche-biographie.de/sfz30354.html))

* [[Gerhard Gentzen]], *Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen*,
Math. Ann.
**107** (1933) pp. 329-350. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002275872))

* Paul Hertz, *Über die Axiomensysteme kleinster Satzzahl für ein System von Sätzen und den Begriff des idealen Elementes*, Jahresbericht DMV **31** (1922) pp.154-157. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002126826))

* Paul Hertz, *Über Axiomensysteme für beliebige Satzsysteme I*, Math. Ann. **87** (1922) pp.246-269. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002269023))

* Paul Hertz, *Über Axiomensysteme für beliebige Satzsysteme II*, Math. Ann. **89** (1923) pp.76-102. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002269260))

* Paul Hertz, *Über Axiomensysteme für beliebige Satzsysteme*, Math. Ann. **101** (1929) pp.457-514. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002273322))

* Javier Legris, *Paul Hertz and the Origin of Structural Reasoning*, introduction to an English translation of Hertz (1922) pp.3-10 in Béziau (ed.), *Universal logic from Paul Hertz to Dov Gabbay*, Birkhäuser Basel 2012.

* Peter Schroeder-Heister, *Resolutions and the Origins of Structural Reasoning: Early Proof-Theoretic Ideas of Hertz and Gentzen*, Bulletin Symbolic Logic **8** no.2 (2002) pp.246-265. ([extended draft](http://www.uni-tuebingen.de/index.php?eID=tx_nawsecuredl&u=0&g=0&t=1465418144&hash=fa96b52e07f5fdecd06840e4f940393fa676cd05&file=fileadmin/Uni_Tuebingen/Fakultaeten/InfoKogni/WSI/LS/psh/forschung/publikationen/logic/c-00-09/HertzResolution00long.pdf))

category:people


Paul Horja

__Richard Paul Horja__ is an American [[algebraic geometer]] of Romanian origin, currently at the Oklahoma State University. His work is partly motivated by [[string theory]], especially [[mirror symmetry]].

* [homepage](http://cauchy.math.okstate.edu/~horja), [publications](http://cauchy.math.okstate.edu/~horja/publications.html)

category: people

[[!redirects Richard Paul Horja]]


Paul Howe

Paul Howe is emeritus professor in theoretical [[physics]] at King's College London-

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

* [inspire page](https://inspirehep.net/authors/1005477)

## Selected writings

Making manifest the local [[worldsheet]] [[supersymmetry]] of the [[spinning string]] (via the supersymmetric [[Polyakov action]]):

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], _A locally supersymmetric and reparametrization invariant action for the spinning string_ , Physics Letters B **65** Issue 5, 20 (1976) 471-474 [doi:10.1016/0370-2693(76)90445-7]

On [[spinning particles]] and their [[worldline]] [[supersymmetry]]:

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], *A Lagrangian formulation of the classical and quantum dynamics of spinning particles*, Nucl. Phys. B **118** (1977) 76 [doi:10.1016/0550-3213(77)90364-9]

Formulation of [[D=11 supergravity]] in [[superspace]] (via [[supergeometry]]):

* [[Lars Brink]], [[Paul Howe]], *Eleven-Dimensional Supergravity on the Mass-Shell in Superspace*, Phys.Lett. B **91** (1980) 384 [doi:10.1016/0370-2693(80)91002-3]

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 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))

* {#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))

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

* {#HoweSezgin97} [[Paul Howe]], [[Ergin Sezgin]], _$D=11$, $p=5$_, Phys. Lett. B **394** (1997) 62-66 [[arXiv:hep-th/9611008](https://arxiv.org/abs/hep-th/9611008), doi:10.1016/S0370-2693(96)01672-3]

* {#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 how the [[torsion constraints in supergravity]] for [[11-dimensional supergravity]] already imply the [[equations of motion]]:

* {#Howe97} [[Paul Howe]], _Weyl Superspace_, Physics Letters B, Volume 415, Issue 2, 11 December 1997, Pages 149–155 ([arXiv:hep-th/9707184](http://arxiv.org/abs/hep-th/9707184))

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

* [[Paul S. Howe]], [[Neil Lambert]], [[Pete West]], _The Self-Dual String Soliton_, Nucl. Phys. B **515** (1998) 203-216 [[arXiv:hep-th/9709014](http://arxiv.org/abs/hep-th/9709014), doi:10.1016/S0550-3213(97)00750-5]

Discussion of [[D=4 N=4 SYM]] with emphasis on [[multi-trace operators]]:

* P. J. Heslop, [[Paul Howe]], _Aspects of $\mathcal{N}$=4 SYM_, JHEP 0401 (2004) 058 ([arXiv:hep-th/0307210](https://arxiv.org/abs/hep-th/0307210))

On the [[M2-brane]] via the [[superembedding approach]]:

* {#HoweSezgin05} [[Paul Howe]], [[Ergin Sezgin]], _The supermembrane revisited_, Class.Quant.Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](https://arxiv.org/abs/hep-th/0412245))

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

* {#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))

and specifically via the [[superembedding approach]]:

* {#HoweSezgin97} [[Paul Howe]], [[Ergin Sezgin]], _$D=11$, $p=5$_, Phys.Lett. B394 (1997) 62-66 ([arXiv:hep-th/9611008](https://arxiv.org/abs/hep-th/9611008))

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

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

## Related entries

* [[spinning string]]

* [[supergravity torsion constraints]]

* [[3-brane in 6d]]

category: people

[[!redirects Paul S. Howe]]
[[!redirects P. S. Howe]]


Paul Hoyer

* [webpage](https://www.mv.helsinki.fi/home/hoyer/)

## Selected writings

On [[bound states]] in [[QED]] and [[QCD]] (notably [[positronium]] and [[hadrons]]):

* [[Paul Hoyer]], _Bound states -- from QED to QCD_, Lectures at "Mini-school on theoretical methods in particle physics" at the Higgs Centre for Theoretical Physics, University of Edinburgh on 30 September to 4 October 2013 ([arXiv:1402.5005](https://arxiv.org/abs/1402.5005))

* [[Paul Hoyer]], _Journey to the Bound States_ ([arXiv:2101.06721](https://arxiv.org/abs/2101.06721))

category: people


Paul Hudak

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

* [obituary](https://news.yale.edu/2015/04/30/memoriam-paul-hudak-computer-scientist-and-saybrook-college-master)

## Selected writings

On [[monad transformers]] in [[functional programming]] with [[monadic effects]]:

* {#LiangHudakJones95} Sheng Liang, [[Paul Hudak]], [[Mark Jones]], *Monad transformers and modular interpreters*, POPL '95 (1995) 333–343 [[doi:10.1145/199448.199528](https://doi.org/10.1145/199448.199528)]

On [[domain specific programming languages]]:

* [[Paul Hudak]], *Building Domain-Specific Embedded Languages*, ACM Computing Surveys **28** 4e (1996) 196–es [[doi:10.1145/242224.242477](https://doi.org/10.1145/242224.242477), [web](https://dl.acm.org/doi/fullHtml/10.1145/242224.242477)]

* [[Paul Hudak]], *Domain Specific Languages*, Chapter 3 (pp. 39–60) in:  Peter H. Salas (ed), *Handbook of Programming Languages*, Vol. III: *Little Languages and Tools*, MacMillan  (1998) [[pdf](http://cs448h.stanford.edu/DSEL-Little.pdf), [[Hudak-DSLs.pdf:file]]]

More specifically on [[domain specific embedded programming languages]]:

* [[Paul Hudak]], _Modular Domain Specific Languages and Tools_, in: _Proceedings of Fifth International Conference on Software Reuse_, IEEE Computer Society Press  (1998)  [[pdf](https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=685738), [doi:10.5555/551789.853532](https://dl.acm.org/doi/10.5555/551789.853532)]

On the history of the development of the [[Haskell]] [[programming language]]:

* [[Paul Hudak]], [[John Hughes]], [[Simon Peyton Jones]], [[Philip Wadler]], *A history of Haskell: being lazy with class*, History of Programming Languages (2007) 12-1 - 12-55 [[doi:10.1145/1238844.1238856](https://doi.org/10.1145/1238844.1238856)]

## Related entries

* [[Haskell]]


Paul J. Cohen

**Paul J. Cohen** (1934-2007) was an American mathematician who received the [[Fields medal]] in 1966 for his groundbreaking work in [[set theory]].

In his proof of the independence of the [[continuum hypothesis]] from the [[ZFC]] axioms of set theory he introduced [[forcing]] into the arsenal of [[mathematical logic]] and [[model theory]]:

>It was in 1963 that we were hit by a real bomb, however, when Paul J. Cohen discovered his method of 'forcing'... Set theory could never be the same after Cohen, and there is simply no comparison whatsoever in the sophistication of our knowledge about models for set theory today as contrasted to the pre-Cohen area. One of the most striking consequences of his work is the realization of the extreme *relativity* of the notion of cardinal number. [[Dana Scott]] in (Bell 2005, p.xiv)

From a broader perspective Cohen's results fit nicely into the landscape of relativising set theory and the rise of 'variable sets'in the work of [[Grothendieck]], [[Lawvere]] and [[Miles Tierney|Tierney]] on [[topos theory]] in the 60s and afterwards.

## Related pages

* [[forcing]]

* [[continuum hypothesis]]

## Links

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

## References

* [[John Bell|J. L. Bell]], _Set Theory - Boolean-Valued Models and Independence Proofs_ , Oxford Logic Guides **47** 3rd ed. Oxford UP 2005.

* [[Alonzo Church|A. Church]], _Paul J. Cohen and the Continuum Problem_, pp.15-20 in Proceedings ICM Moscow 1966. ([pdf](http://www.mathunion.org/ICM/ICM1966.1/Main/icm1966.1.0015.0020.ocr.pdf))

* P. J. Cohen, _The independence of the continuum hypothesis I_ , Proc.Nat.Acad.Sci. **50** (1963) pp.1143-1148. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287/pdf/pnas00240-0135.pdf))

* P. J. Cohen, _The independence of the continuum hypothesis II_ , Proc.Nat.Acad.Sci. **51** (1963) pp.105-110. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611/pdf/pnas00175-0117.pdf))

* P. J. Cohen, _Set Theory and the Continuum Hypothesis_ , Benjamin New York 1966. (Dover reprint 2008)

category:people

[[!redirects Paul Joseph Cohen]]
[[!redirects Paul Cohen]]
[[!redirects P. J. Cohen]]


Paul Johnson

[Paul Johnson](http://www2.imperial.ac.uk/~pjohnso1/)

Testing this out...


Paul Kainen

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

## Selected writings

On [[weak adjoint functors]]:

* [[Paul Kainen]], *Weak adjoint functors*, Mathematische Zeitschrift **122** 1 (1971) 1-9 [[dml:171575](https://eudml.org/doc/171575)]

## Related entries

* [[weak colimit]], [[weakly initial object]]

* [[four-colour theorem]]

category: people

[[!redirects Paul C. Kainen]]


Paul Kirk

Paul Kirk is an algebraic topologist working at Indiana University.

* [webpage](http://php.indiana.edu/~pkirk/home.html).

##Publications include

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

category:people


Paul Koerber

* [webpage](http://itf.fys.kuleuven.be/~koerber/research.html)

## Selected writings

On [[generalized complex geometry]] from the point of view of [[G-structures]]:

* [[Paul Koerber]], _Lectures on Generalized Complex Geometry for Physicists_, Fortsch. Phys. 59: 169-242, 2011 ([arXiv:1006.1536](https://arxiv.org/abs/1006.1536))

category: people


Paul Langacker

* [webpage](http://www.sns.ias.edu/~pgl/)

## Selected writings

On [[intersecting D-brane models]]:

* [[Ralph Blumenhagen]], [[Mirjam Cvetic]], [[Paul Langacker]], [[Gary Shiu]], *Towards Realistic Intersecting D-Brane Models*, Ann. Rev. Nucl. Part. Sci. **55** (2005) 71-139 [[arXiv:hep-th/0502005](http://arxiv.org/abs/hep-th/0502005), [doi:10.1146/annurev.nucl.55.090704.151541](https://doi.org/10.1146/annurev.nucl.55.090704.151541)]

## Related entries

* [[standard model of particle physics]]

category: people


Paul Lorenzen

[[!redirects Paul Lorentzen]]
* [Wikipedia entry](https://en.wikipedia.org/wiki/Paul_Lorenzen)

## Related entries

* [[game semantics]]

category: people


Paul M. Cohn

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

## Selected writings

On [[localization of rings]] ([[Cohn localization]]):

* [[Paul M. Cohn]], _Free rings and their relations_, Academic Press (1971) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/cohnfree.pdf)]

* [[Paul M. Cohn]], _Inversive localization in noetherian rings_, Communications on Pure and Applied Mathematics __26__ 5-6 (1973 ) 679-691 [[doi:10.1002/cpa.3160260510](http://dx.doi.org/10.1002/cpa.3160260510)]

category: people
[[!redirects P. M. Cohn]]


Paul Melvin

* [webpage](http://www.brynmawr.edu/math/people/melvin/)

category: people


Paul Mitchener

* [webpage](http://www.mitchener.staff.shef.ac.uk/)

category: people


Paul Mostert

* [institute page](https://mathematics.ku.edu/paul-mostert)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=16668&fChrono=1)

## Selected writings

On existence of [[local sections]] of [[coset space]] [[coprojections]] (for [[closed subgroups]]):

* {#Mostert53} [[Paul Mostert]], _Local Cross Sections in Locally Compact Groups_, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 ([jstor:2032540](https://www.jstor.org/stable/2032540), [doi:10.2307/2032540](https://doi.org/10.2307/2032540))

category: people


Paul Muhly

* [webpage](http://www.math.uiowa.edu/~pmuhly/)

category: people


Paul Norbury

* [personal page](https://researchers.ms.unimelb.edu.au/~norbury@unimelb/)

* [GoogleScholar page](https://scholar.google.com/citations?user=wpSS3d4AAAAJ&hl=en)

## Selected writings

Proof of the [[Atiyah-Sutcliffe conjecture]] for the case of $n = 4$ points:

* {#EastwoodNorbury01} [[Michael Eastwood]], [[Paul Norbury]], _A proof of Atiyah’s conjecture of four points in Euclidean three-space_, Geom. Topol. 5 (2001) 885–893 ([arXiv:math/0109161](https://arxiv.org/abs/math/0109161), [doi:10.2140/gt.2001.5.88510.2140/gt.2001.5.885](https://msp.org/gt/2001/5-2/p12.xhtml))

category: people


Paul Olum

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

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=42981)

## Selected writings

On [[nonabelian algebraic topology]]:

* [[Paul Olum]], _Non-abelian cohomology and van Kampen's theorem_, Ann. Math. 68 (1958) 658--667.

On classification of [[lens spaces]]:

* {#Olum53} [[Paul Olum]], _Mappings of manifolds and the notion of degree_, Ann. of Math. (2) *58* (1953), p. 458–480. [JSTOR:1969748](https://www.jstor.org/stable/1969748)

category: people


Paul Purdon Martin

* [personal page](https://www1.maths.leeds.ac.uk/~ppmartin/)

* [institute page](https://eps.leeds.ac.uk/maths/staff/4061/professor-paul-martin)

## Selected writings

On [[motion groupoids]] and [[mapping class groups]]:

* [[Fiona Torzewska]], [[João Faria Martins]], [[Paul Purdon Martin]], *Motion groupoids and mapping class groupoids*, Comm. Math. Phys. **402** (2023) 1621-1705 [[arXiv:2103.10377](https://arxiv.org/abs/2103.10377), [doi:10.1007/s00220-023-04755-0](https://doi.org/10.1007/s00220-023-04755-0)]

category: people


Paul Redding

Paul Redding is a philosopher who studies Continental idealist philosophy.

* [Website](http://paulredding.net)

##Related entries

* [[Georg Hegel]]
* [[Hegel's "Logic" as Modal Type Theory]]

##Works

* _Analytic Philosophy and the Return of Hegelian Thought_, Cambridge University Press, 2007 ([review](http://ndpr.nd.edu/news/23465-analytic-philosophy-and-the-return-of-hegelian-thought/) by Willem A. deVries)

category: people


Paul Romatschke

* [institute page](https://www.colorado.edu/physics/paul-romatschke)

* [GoogleScholar page](https://scholar.google.de/citations?user=ZM9wLcsAAAAJ&hl=en)

## Selected writings

On the [[Landau pole]]:

* [[Paul Romatschke]], *Life at the Landau pole* [[arXiv:2212.03254](https://arxiv.org/abs/2212.03254)]

On the [[large-N limit]] in [[quantum mechanics]] and [[quantum field theory]]:

* [[Paul Romatschke]], *Quantum Field Theory in Large $N$ Wonderland: Three Lectures* [[arXiv:2310.00048](https://arxiv.org/abs/2310.00048)]

On the [[confinement]]-problem in [[QCD]]:

* Seth Grable, [[Paul Romatschke]], *Elements of Confinement for QCD with Twelve Massless Quarks* $[$[arXiv:2310.12203](https://arxiv.org/abs/2310.12203)$]$

category: people


Paul Seidel

[[!redirects Blank]]
Paul Seidel is a professor in the Department of Mathematics at MIT.

* [webpage](http://www-math.mit.edu/~seidel/)


Paul Steinhardt

* [webpage](http://wwwphy.princeton.edu/~steinh/)

## Selected writings

On [[non-perturbative effects]] in [[cosmology]]:

* Anna Ijjas, Frans Pretorius, [[Paul Steinhardt]], _Stability and the gauge problem in non-perturbative cosmology_,  Journal of Cosmology and Astroparticle Physics, Volume 2019, January 2019 ([arXiv:1809.07010](https://arxiv.org/abs/1809.07010), [doi:10.1088/1475-7516/2019/01/015](https://iopscience.iop.org/article/10.1088/1475-7516/2019/01/015))

On [[string phenomenology]] via [[heterotic M-theory]]:

* Cédric Deffayet, [[Burt Ovrut]], [[Paul Steinhardt]], *Stable Vacua with Realistic Phenomenology and Cosmology in Heterotic M-theory Satisfying Swampland Conjectures* [[arXiv:2401.04828](https://arxiv.org/abs/2401.04828)]

## Related entries

* [[cosmology]]

* [[cosmic inflation]]

* [[ekpyrotic cosmology]]

category: people


Paul Sutcliffe

* [webpage](http://maths.dur.ac.uk/~dma0pms/)

## Selected writings

On [[Yang-Mills monopoles]]:

* [[Paul Sutcliffe]], *BPS Monopoles*, Int. J. Mod. Phys. A12:4663-4706 (1997) ([arXiv:hep-th/9707009](https://arxiv.org/abs/hep-th/9707009))

On classification of [[Yang-Mills monopoles]] via [[rational maps]] ([[holomorphic maps]], really) from the [[Riemann sphere]] to a [[flag variety]]:

* {#IonnadouSutcliffe99a} [[Theodora Ioannidou]], [[Paul Sutcliffe]], *Monopoles and Harmonic Maps*, J. Math. Phys. 40:5440-5455 (1999) ([arXiv:hep-th/9903183](https://arxiv.org/abs/hep-th/9903183))

* [[Theodora Ioannidou]], [[Paul Sutcliffe]], *Monopoles from Rational Maps*, Phys. Lett. B457 (1999) 133-138 ([arXiv:hep-th/9905066](https://arxiv.org/abs/hep-th/9905066))

Discussion relating [[skyrmions]] with [[instantons]], [[calorons]] and [[solitons]] and [[monopoles]] in the [[Sakai-Sugimoto model]] ([[AdS/QCD correspondence]] via [[D=5 Yang-Mills theory]], [[hadron Kaluza-Klein theory]]):

* [[Michael Atiyah]], [[Paul Sutcliffe]], _Skyrmions, instantons, mass and curvature_, Phys. Lett. B605 (2005) 106-114 ([arXiv:hep-th/0411052](https://arxiv.org/abs/hep-th/0411052))

* {#Sutcliffe10} [[Paul Sutcliffe]], _Skyrmions, instantons and holography_, JHEP 1008:019, 2010 ([arXiv:1003.0023](https://arxiv.org/abs/1003.0023))

* {#BolognesiSutcliffe13} [[Stefano Bolognesi]], [[Paul Sutcliffe]], _The Sakai-Sugimoto soliton_, JHEP 1401:078, 2014 ([arXiv:1309.1396](https://arxiv.org/abs/1309.1396))

* {#Sutcliffe15} [[Paul Sutcliffe]], _Holographic Skyrmions_, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 ([spire:1383608](http://inspirehep.net/record/1383608))

On [[Skyrmions]] via [[rational maps]] from the [[complex plane]], hence [[holomorphic maps]] from the [[Riemann sphere]], to itself:

* [[Conor J. Houghton]], [[Nicholas Manton]], [[Paul Sutcliffe]], *Rational maps, monopoles and skyrmions*, Nucl. Phys. B510 (1998) 507-537 ([arXiv:hep-th/9705151](https://arxiv.org/abs/hep-th/9705151), doi:10.1016/S0550-3213(97)00619-6)

* [[Richard Battye]], [[Paul Sutcliffe]], *Skyrmions, Fullerenes and Rational Maps*, Rev. Math. Phys. 14 (2002) 29-86 ([arXiv:hep-th/0103026](https://arxiv.org/abs/hep-th/0103026))

* [[Derek Harland]], [[Paul Sutcliffe]], *Rational Skyrmions* [[arXiv:2307.09355](https://arxiv.org/abs/2307.09355)]

Introducing the [[Atiyah-Sutcliffe conjecture]]:

* [[Michael F. Atiyah]], [[Paul M. Sutcliffe]], _The geometry of point particles_, Proc. Roy. Soc. London Ser. A 458 (2002), 1089–1115 ([hep-th/0105179](https://arxiv.org/abs/hep-th/0105179), [doi:10.1098/rspa.2001.0913](https://doi.org/10.1098/rspa.2001.0913))

On [[skyrmions]] as [[model (in theoretical physics)|models]] of [[atomic nuclei]]:

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001)) in:  [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

and including beyond the [[pion]] also the [[rho-meson]]:

* {#NayaSutcliffe18a} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions and clustering in light nuclei_, Phys. Rev. Lett. 121, 232002 (2018) ([arXiv:1811.02064](https://arxiv.org/abs/1811.02064))

* {#NayaSutcliffe18b} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions in models with pions and rho_, JHEP 05 (2018) 174 ([arXiv:1803.06098](https://arxiv.org/abs/1803.06098))

  APS Synopsis: _[Revamping the Skyrmion Model](https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002)_, 2018

## Related $n$Lab entries

* [[Yang-Mills monopoles]]

* [[skyrmion]]

* [[atomic nucleus]]

category: people

[[!redirects Paul M. Sutcliffe]]
[[!redirects P. M. Sutcliffe]]


Paul Taylor

Paul Taylor works on the [[foundations]] of mathematics and [[computer science]].

*  [home page](http://www.paultaylor.eu/)

## Selected writings

Introducing (among other things) the notion of *categories with [[display maps]]* (later also "[[clans]]") for [[categorical semantics of dependent type theory]]:

* [[Paul Taylor]], §4.3.2 in: *Recursive Domains, Indexed Category Theory and Polymorphism*, Cambridge (1983-7) [[pdf](https://ncatlab.org/nlab/files/Taylor-IndexedCategoryTheory.pdf)]

On [[formal logic|form]] [[logic]] and [[formal proof|formal]] [[proof]] [[proof theory|theory]] ([[type theory|typed]] [[lambda-calculus|$\lambda$-calculus]], [[linear logic]],  [[coherence spaces]], ...)

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

On [[mathematical foundations|foundations]] in [[constructive mathematics]]:

*  _[[Practical Foundations of Mathematics]]_, 1999, [web](http://www.paultaylor.eu/~pt/prafm/).

On [[abstract Stone duality]]:

*  15 papers: [web](http://www.paultaylor.eu/ASD/)

On [[arithmetic pretoposes]]:

* [[Paul Taylor]], _Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe_, Electronic Notes in Theoretical Computer Science **122** (2005) 247-296 [[doi:10.1016/j.entcs.2004.06.059](https://doi.org/10.1016/j.entcs.2004.06.059)]

On the [[Dedekind real numbers]] via [[abstract Stone duality]]:

*  [[Paul Taylor]], *[Dedekind cuts](http://www.paultaylor.eu/ASD/dedras/classical)* (2007-2009?)

* [[Andrej Bauer]], [[Paul Taylor]], *The Dedekind reals in abstract Stone duality*, Mathematical Structures in Computer Science **19** 4 (2009) 757-838 [[doi:10.1017/S0960129509007695](https://doi.org/10.1017/S0960129509007695)]

category: people


Paul Townsend

Paul Townsend is professor for theoretical [[physics]] at Cambridge.

* [webpage](http://www.damtp.cam.ac.uk/people/p.k.townsend/)

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

## Selected writings

Early discussion of the [[D'Auria-Fré formulation of supergravity]]:

* [[Riccardo D'Auria]], [[Pietro Fré]], [[Paul Townsend]],  *Invariance of actions, rheonomy and the new minimal $N=1$ supergravity in the group manifold approach*, Ann. Phys. 155 (1984) 423-446 [[cds:143990](https://cds.cern.ch/record/143990), doi:10.1016/0003-4916(84)90007-1, [pdf](https://cds.cern.ch/record/143990/files/198305027.pdf)]

On [[3d gravity]] as a [[Chern-Simons theory]] and its [[holography|holographic]] relation to a [[2d CFT]] [[boundary field theory]] (well before [[AdS/CFT]] was conceived from [[string theory]]):

* A. Achucarro, [[Paul Townsend]], _A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories_, Phys. Lett. B180 (1986) 89 (doi:10.1016/0370-2693(86)90140-1, [spire:21208](http://inspirehep.net/record/21208))

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

* [[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)

Early discussion of the [[duality between type IIA string theory and M-theory]] in view of [[double dimensional reduction]] of the [[M2-brane]]:

* [[Paul K. Townsend]], *The eleven-dimensional supermembrane revisited*, Phys. Lett. B **350** (1995) 184-187 [[arXiv:hep-th/9501068](https://arxiv.org/abs/hep-th/9501068), doi:10.1016/0370-2693(95)00397-4]

On [[black brane|black]]$\,$[[D8-brane]]-solutions in [[massive type IIA supergravity]]/[[massive type IIA string theory]]:

* {#BergshoessdeRooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B470:113-135, 1996 ([arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150))

On [[M2-M5-brane bound states]]:

* {#ILPT95} [[José M. Izquierdo]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], *Dyonic Membranes*, Nucl. Phys. B **460** (1996) 560-578 [[arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177), doi:10.1016/0550-3213(95)00606-0]

* [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic $p$-branes from self-dual $(p+1)$-branes_, Phys.Lett.B384:86-92, 1996 ([arXiv:hep-th/9605146](https://arxiv.org/abs/hep-th/9605146))

On the [[3-brane in 6d]]:

* [[Joaquim Gomis]], [[David Mateos]], [[Joan Simón]], [[Paul Townsend]], _Brane-Intersection Dynamics from Branes in Brane Backgrounds_, Phys. Lett. B430 (1998) 231-236 ([arXiv:hep-th/9803040](http://arxiv.org/abs/hep-th/9803040))

On [[black holes]] in [[D=5 supergravity]]:

* {#GauntlettMyersTownsend98} [[Jerome Gauntlett]], [[Robert C. Myers]], [[Paul Townsend]], *Black Holes of $D=5$ Supergravity*, Class.Quant.Grav. **16** (1999) 1-21 [[arXiv:hep-th/9810204](http://arxiv.org/abs/hep-th/9810204), [doi:10.1088/0264-9381/16/1/001](https://doi.org/10.1088/0264-9381/16/1/001)]

On [[D-brane polarization]] into [[supertubes]]:

* [[David Mateos]], [[Paul Townsend]], _Supertubes_, Phys. Rev. Lett. 87 (2001) 011602 ([arXiv:hep-th/0103030](https://arxiv.org/abs/hep-th/0103030))

On [[higher curvature corrections]] to [[D=11 supergravity]] and [[M-theory on G2-manifolds]]:

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _Supersymmetric Deformations of $G_2$ Manifolds from Higher-Order Corrections to String and M-Theory_, JHEP 0410:019, 2004 ([arXiv:hep-th/0312002](https://arxiv.org/abs/hep-th/0312002))

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _String and M-theory Deformations of Manifolds with Special Holonomy_, JHEP 0507:075, 2005 ([arXiv:hep-th/0410176](https://arxiv.org/abs/hep-th/0410176))

On open [[M5-branes]] [[intersecting brane|intersecting]] [[M9-branes]] in a [[Yang monopole]]:

* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys. Rev. Lett. 97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193), [doi:10.1103/PhysRevLett.97.231601](https://doi.org/10.1103/PhysRevLett.97.231601))

On [[supersymmetry and division algebras]], the corresponding [twistor space](twistor+space#TwistorSpace) and its [[anti de Sitter spacetime|AdS]] version:

* [[Alex Arvanitakis]], Alec E. Barns-Graham, [[Paul Townsend]], _Twistor description of spinning particles in AdS_, JHEP 01 (2018) 059 ([arXiv:1710.09557](https://arxiv.org/abs/1710.09557))

## Related $n$Lab entries

* [[string theory]], [[M-theory]]

* [[Green-Schwarz super p-brane]]

* [[super 1-brane in 3d]], [[super 2-brane in 4d]]

* [[M-theory super Lie algebra]]

category: people

[[!redirects Paul K. Townsend]]


Paul Turner

* [webpage](http://www.unige.ch/~turnerpa/)

## Selected writings

On the [[quaternionic cobordism ring]]:

* [[Vassily Gorbunov]], [[Nigel Ray]], [[Paul Turner]], _On the Hopf Ring for a Symplectic Oriented Spectrum_, American Journal of Mathematics
Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 ([doi:10.2307/2374960](https://doi.org/10.2307/2374960))

## Related $n$Lab entries

* [[Hopf ring spectrum]]

category: people


Paul VanKoughnett

* [webpage](https://www.math.northwestern.edu/people/gradStudentProfiles/paul.vankoughnett.html)

category: people


Paul Vojta

* [webpage](http://math.berkeley.edu/~vojta/)

category: people


Paul Wesson

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

## Selected writings

On [[Kaluza-Klein theory]] (including [[5d electromagnetism]], [[5d Einstein-Maxwell theory]], ...):

* [[Paul Wesson]], _Space-Time-Matter: Modern Kaluza-Klein Theory_, World Scientific 1989 ([doi:10.1142/3889](https://doi.org/10.1142/3889))

* [[Paul Wesson]], James M. Overduin, _Principles of Space-Time-Matter: Cosmology, Particles and Waves in Five Dimensions_, World Scientific 2018 ([doi:10.1142/10871](https://doi.org/10.1142/10871))

category: people


Paul Windey

* [webpage](http://string.lpthe.jussieu.fr/members.pl?key=18)

## related $n$Lab entries

* [[Dirac-Ramond operator]]

* [[Witten genus]]

category: people


Paul Wollan

* [homepage](http://wwwusers.di.uniroma1.it/~wollan/)

## Selected writings

On the [[axiomatization]] of [[infinite set|infinite]] [[matroids]]:

* [[Henning Bruhn]], [[Reinhard Diestel]], [[Matthias Kriesell]], [[Rudi Pendavingh]], [[Paul Wollan]], *Axioms for infinite matroids*, Advances in Mathematics, **239** (2013) 18–46 [[arXiv:1003.3919](https://arxiv.org/abs/1003.3919), [doi:10.1016/j.aim.2013.01.011](https://doi.org/10.1016/j.aim.2013.01.011)]

category: people


Paul-André Melliès

Paul-André Melliès is a CNRS Researcher,  member of the Laboratoire [[PPS]] at the Department of Mathematical Sciences and Department of Computer Science of the Université Paris 7 - Denis Diderot. He is interested in Mathematical logic, proof theory, [[game semantics]], theory and practice of programming languages, formalized mathematics, proof assistants, mathematical physics, knot theory, quantum groups, n-dimensional algebra, and operads (and in practice a lot more!)

His webpage is [here](http://www.pps.univ-paris-diderot.fr/~mellies/).

### Selected writings

On [[categorical semantics]] of [[linear logic]] with its [[exponential modality]]:

* {#Melliès02} [[Paul-André Melliès]], *Categorical models of linear logic revisited* (2002) [[hal:00154229](https://hal.science/hal-00154229)]

* {#Melliès09} [[Paul-André Melliès]], *Categorical semantics of linear logic*, in *[Interactive models of computation and program behaviour](https://smf.emath.fr/publications/modeles-interactifs-de-calcul-et-de-comportement-de-programme)*, Panoramas et synthèses **27** (2009) 1-196 [[web](https://smf.emath.fr/publications/semantique-categorielle-de-la-logique-lineaire), [pdf](https://www.irif.fr/~mellies/papers/panorama.pdf), [[Mellies-CategoricalSemanticsLinear.pdf:file]]]

On [[string diagrams]]

* {#Mellies06} [[Paul-André Melliès]], _Functorial boxes in string diagrams_, Procceding of _Computer Science Logic 2006_ in Szeged, Hungary. 2006 ([hal:00154243](https://dumas.ccsd.cnrs.fr/PPS/hal-00154243), [pdf](https://hal.archives-ouvertes.fr/hal-00154243/document), [[MelliesFunctorialBoxesInStringDiagrams.pdf:file]])

On [[type refinement]]:

* [[Paul-André Melliès]], [[Noam Zeilberger]], _Type refinement and monoidal closed bifibrations_, [arXiv:1310.0263](http://arxiv.org/abs/1310.0263)

* [[Paul-André Melliès]], [[Noam Zeilberger]], _Functors are Type Refinement Systems_, ([pdf](http://noamz.org/papers/funts.pdf) , [HAL](https://hal.inria.fr/hal-01096910))

* [[Paul-André Melliès]], [[Noam Zeilberger]], _An Isbell Duality Theorem for Type Refinement Systems_, ([pdf](https://www.irif.fr/~mellies/papers/an-isbell-duality-theorem-for-type-refinement-systems.pdf))

On the [[local state monad]]:

* [[Paul-André Melliès]], *Local States in String Diagrams*, In: G. Dowek (eds.) *Rewriting and Typed Lambda Calculi* RTA TLCA 2014, Lecture Notes in Computer Science **8560** Springer (2014) [[doi:10.1007/978-3-319-08918-8_23](https://doi.org/10.1007/978-3-319-08918-8_23)]

On the [[Grothendieck construction for model categories]]:

* [[Pierre Cagne]], [[Paul-André Melliès]], *On bifibrations of model categories*, Advances in Mathematics
**370** (2020) 107205 [[arXiv:1709.10484](https://arxiv.org/abs/1709.10484), [doi:10.1016/j.aim.2020.107205](https://doi.org/10.1016/j.aim.2020.107205)]

On [[presheaves]] of [[modules]] as models for [[intuitionistic linear logic]]

* [[Paul-André Melliès]], _A Functorial Excursion Between Algebraic Geometry and Linear Logic_, in 37th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (LICS '22), August 2–5, 2022, Haifa, Israel. ACM, New York, ([doi:10.1145/3531130.3532488](https://doi.org/10.1145/3531130.3532488), [pdf](https://www.irif.fr/~mellies//papers/functorial-excursion.pdf))

category: people

[[!redirects Paul André Melliès]]


Paul-André Meyer

* [Wikipedia entry](https://en.wikipedia.org/wiki/Paul-Andr%C3%A9_Meyer)

## Selected writings

On [[quantum probability theory]]:

* {#Meyer95} [[Paul-André Meyer]], Section I.1.1 in: *Quantum Probability for Probabilists*,  Lecture Notes in Mathematics **1538**, Springer 1995 ([doi:10.1007/BFb0084701](https://link.springer.com/book/10.1007/BFb0084701))

category: people

[[!redirects Paul-Andre Meyer]]


Paul-Emile Paradan

* [webpage](http://www.math.univ-montp2.fr/~paradan/Menu.html)

## Related entries

* [[geometric quantization]]

* [[spin^c quantization]]

* [[metaplectic correction]]

* [[Guillemin-Sternberg geometric quantization conjecture]]

category: people


Paul-Eugène Parent

* [institute page](https://www.uottawa.ca/faculty-science/professors/paul-eugene-parent)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=132231)

## Selected writings

On the [[Adams-Hilton model]] for the [[Pontrjagin ring]]-structure on the [[singular chain complex]] of a [[based loop space]]:

* [[Kathryn Hess]], [[Paul-Eugène Parent]], [[Jonathan Scott]], [[Andrew Tonks]], *A canonical enriched Adams-Hilton model for simplicial sets*, Advances in Mathematics **207** 2 (2006) 847-875 [[doi:10.1016/j.aim.2006.01.013](https://doi.org/10.1016/j.aim.2006.01.013), [arXiv:math/0408216](https://arxiv.org/abs/math/0408216)]

category: people

[[!redirects Paul-Eugene Parent]]


Pauli exclusion principle

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], the _Pauli exclusion principle_ ([Pauli 25](#Pauli25)) states that no two [[spinors]] may inhabit the same [[quantum state]].
This is a consequence of the _[[spin-statistics theorem]]_, which states that spinors must be [[fermions]].

Mathematically this is the statement that fermion [[field (physics)|fields]] are the odd-graded elements of a [[superalgebra]] of functions in [[supergeometry]].

The Pauli exclusion principle implies that [[fermions]], as opposed to [[bosons]], may form [[solidity|solid bodies]] that do not necessarily collapse under their own [[gravity]] (as opposed to bosonic [[condensates]]): the exclusion principle induces an outward [[pressure]], the _[[degeneracy pressure]]_ (see also at _[[Fermi sea]]_ and _[[neutron star]]_). It is used in the explanation of the [[stability of matter]] of the second kind.

## Related concepts

* [[spin-statistics theorem]]

* [[Slater determinant]]

## References

### General

* {#Pauli25} [[Wolfgang Pauli]], *Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren*, Zeitschrift für Physik, **31** 1 (1925) 765-783 [[doi:10.1007/BF02980631](https://doi.org/10.1007/BF02980631)]

* [[Wolfgang Pauli]], *Über Gasentartung und Paramagnetismus*, Zeitschrift für Physik: A Hadrons and nuclei **41** (1927) 81–102 [[doi:10.1007/BF01391920](https://doi.org/10.1007/BF01391920)]

* Wikipedia, _[Pauli exclusion principle](http://en.wikipedia.org/wiki/Pauli_exclusion_principle)_

### Relation to s-rule for D-brane intersections

An interpretation of the [[s-rule]] for [[D-brane|D-]][[brane intersections]] with [[NS5-branes]] ([[Dp-D(p+2) brane intersections]] and [[Dp-D(p+4) brane intersections]]) as a version of the [[Pauli exclusion principle]] is discussed in:

* [[Constantin Bachas]], [[Michael Green]], Adam Schwimmer, Section 2.3 of: _$(8,0)$ Quantum mechanics and symmetry enhancement in type I' superstrings_, JHEP 9801 :006, 1998 ([arXiv:hep-th/9712086](https://arxiv.org/abs/hep-th/9712086))

* [[Constantin Bachas]], [[Michael Green]], _A Classical Manifestation of the Pauli Exclusion Principle_, JHEP 9801 (1998) 015 ([arXiv:hep-th/9712187](https://arxiv.org/abs/hep-th/9712187))


Pauli gate

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#### Computation
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=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[qbit]]-based [[quantum computation]], by the *Pauli gates* one means the [[linear basis]] of [[quantum gates]] on single [[qbits]], hence on the 2-dimensional [[Hilbert spaces]] $QBit \simeq \mathbb{C}^2$, which, in terms of the canonical [[quantum measurement]]-[[linear basis|basis]] $\mathbb{C}^2 \simeq Span\big( \{ \vert 0 \rangle ,\, \vert 1 \rangle\} \big)$, are given by the [[Pauli matrices]].

Explicitly this means that (in the conentional normalization) the:

1. *Pauli-X gate* (or quantum [[NOT]] gate) is given by the [[matrix]]

   $$
     X \;\;\coloneqq\;\;
     \left[
       \begin{array}{cc}
         0 & 1
         \\
         1 & 0
       \end{array}
     \right]
   $$

1. *Pauli-Y gate* is given by the [[matrix]]

   $$
     X \;\;\coloneqq\;\;
     \left[
       \begin{array}{cc}
         0 & - \mathrm{i}
         \\
         \mathrm{i} & 0
       \end{array}
     \right]
   $$

1. *Pauli-Z gate* is given by the [[matrix]]

   $$
     X \;\;\coloneqq\;\;
     \left[
       \begin{array}{cc}
         1 & 0
         \\
         0 & -1
       \end{array}
     \right]
   $$

## Properties

### Relation to Hadamard gates and the ZX-calculus

The [[Hadamard gate]] transforms the [[eigenstates]] $\vert 0 \rangle$, $\vert 1 \rangle$ of the Pauli Z-gate into those $\propto \vert 0 \rangle \pm \vert 1 \rangle$ of the Pauli-X gate, a relation that is elaborated on by the correspondingly named *[[ZX-calculus]]*.

## Related concepts

* [[quantum computation]]

* [[quantum gate]], [[quantum circuit]]

* [[Hadamard gate]]

* [[XOR]], [[CNOT]]

* [[Solovay-Kitaev theorem]]

## References

For example:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], p. xxx in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

[[!redirects Pauli gates]]

[[!redirects Pauli quantum gate]]
[[!redirects Pauli quantum gates]]

[[!redirects Pauli-X gate]]
[[!redirects Pauli-X gates]]
[[!redirects Pauli-Y gate]]
[[!redirects Pauli-Y gates]]
[[!redirects Pauli-Z gate]]
[[!redirects Pauli-Z gates]]

[[!redirects Pauli-X]]
[[!redirects Pauli-Y]]
[[!redirects Pauli-Z]]


Pauli group

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#### Group Theory
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[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The *Pauli group* $\mathcal{P}_1$ is the [[finite group]] of [[order of a group|order]] 16 which is [[isomorphism|isomorphic]] to the [[subgroup]] of the [[complex numbers|complex]] [[general linear group]] $GL(2,\mathbb{C})$ that consists of the multiples by $\pm 1$ and $\pm i$  of the four [[Pauli matrices]] $\sigma_{0,1,2,3}$:

\[
  \label{PauliGroupAsMatrixSubgroup}
  \mathcal{P}_1
  \;\coloneqq\;
  \big\{
    \pm \sigma_0,
    \,
    \pm \sigma_1,
    \,
    \pm \sigma_2,
    \,
    \pm \sigma_3,
    \,\;
    \pm i \sigma_0,
    \,
    \pm i \sigma_1,
    \,
    \pm i \sigma_2,
    \,
    \pm i \sigma_3,
  \big\}
  \;\subset\;
  GL(2,\mathbb{C})
  \,,
\]

where

$$
  \sigma_0
  \;\coloneqq\;
  \left[
  \array{
    1 & 0
    \\
    0 & 1
  }
  \right]
  ,\,\;
  \sigma_1
  \;\coloneqq\;
  \left[
  \array{
    0 & 1
    \\
    1 & 0
  }
  \right]
  ,\,\;
  \sigma_2
  \;\coloneqq\;
  \left[
  \array{
    1 & 0
    \\
    0 & -1
  }
  \right]
  ,\,\;
  \sigma_3
  \;\coloneqq\;
  \left[
  \array{
    0 & -i
    \\
    i & 0
  }
  \right]
  \,.
$$

In [[quantum information theory]] one often considers the higher Pauli group $\mathcal{P}_n$ whose elements are (multiples by $\pm 1$, $\pm i$) of $n$-fold [[tensor products of vector spaces|tensor products]] of the Pauli matrices.

## Properties

### Relation to the quaternion group
 {#RelationToQuaternionGroup}

The [[quaternion group]] $Q$ is the ([[normal subgroup|normal]]) [[subgroup]] of the Pauli group (eq:PauliGroupAsMatrixSubgroup) omitting the $\pm 1$-phases on nonidentity matrices:

\[
  \label{QuaternionGroupAsMatrixSubgroup}
  Q
  \;\simeq\;
  \big\{
    \pm \sigma_0,
    \,
    \pm i\sigma_1,
    \,
    \pm i\sigma_2,
    \,
    \pm i\sigma_3,
  \big\}
  \;\subset\;
  GL(2,\mathbb{C})
  \,,
\]

In fact, the Pauli group is a [[semidirect product group]] of the [[quaternion group]] with the [[cyclic group of order 2]]:

$$
  \mathcal{P}_1
  \;\simeq\;
  Q \rtimes \mathbb{Z}/2
  \,.
$$

## Related concepts

* [[stabilizer code]]

## References

See also:

* Wikipedia, *[Pauli group](https://en.wikipedia.org/wiki/Pauli_group)*

Discussion in the context of [[quantum error correction codes]]:

* Simeon Ball, Aina Centelles, Felix Huber, p. 8 of: *Quantum error-correcting codes and their geometries* ([arXiv:2007.05992](https://arxiv.org/abs/2007.05992))

[[!redirects Pauli groups]]


Pauli-Lubanski vector

## Related concepts

* [[angular momentum]]

* [[helicity]]

* [[twistor]]

## References

* Wikipedia, _[Pauli-Lubanski pseudovector](http://en.wikipedia.org/wiki/Pauli–Lubanski_pseudovector)_

[[!redirects Pauli-Lubanski vectors]]

[[!redirects Pauli-Lubanski pseudovector]]
[[!redirects Pauli-Lubanski pseudovectors]]


Pavel Aleksandrov

Па́вел Серге́евич Алекса́ндров (Pavel Sergeevič Aleksandrov/Alexandrov, published as Paul Alexandroff) was a Russian [[topology|topologist]].

* Wikipedia: [English](http://en.wikipedia.org/wiki/Pavel_Alexandrov), [German](http://de.wikipedia.org/wiki/Pawel_Sergejewitsch_Alexandrow), [Russian](http://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80%D0%BE%D0%B2,_%D0%9F%D0%B0%D0%B2%D0%B5%D0%BB_%D0%A1%D0%B5%D1%80%D0%B3%D0%B5%D0%B5%D0%B2%D0%B8%D1%87)
* math-net.ru [page](http://www.mathnet.ru/php/person.phtml?personid=21996&option_lang=eng)

## Selected writings

Introducing the notion of [[nerve]] of a [[cover]]:

* [[Paul Alexandroff]], _Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung_, Mathematische Annalen 98 (1928), 617–635 ([doi:10.1007/BF01451612](https://doi.org/10.1007/BF01451612)).

On [[Alexandroff spaces]] ([[specialization topology]]):

* [[Paul Alexandroff]], _Diskrete Räume_ ([web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5579&option_lang=eng)) (1973)

On [[one-point compactification]]:

* {#Aleksandrov24} [[Paul Alexandroff]], _Über die Metrisation der im Kleinen kompakten topologischen Räume_, Mathematische Annalen (1924) Volume: 92, page 294-301 ([dml:159072](https://eudml.org/doc/159072))

## Related $n$Lab entries

* [[Alexandroff space]] ([[specialization topology]])

* [[one-point compactification]]

category: people

[[!redirects Па́вел Серге́евич Алекса́ндров]]
[[!redirects Павел Сергеевич Александров]]
[[!redirects Павел С. Александров]]
[[!redirects П. С. Александров]]
[[!redirects Павел Александров]]
[[!redirects Pavel Sergeevič Aleksandrov]]
[[!redirects Pavel Sergeevich Aleksandrov]]
[[!redirects Pavel Sergejevich Aleksandrov]]
[[!redirects Pavel Sergeyevich Aleksandrov]]
[[!redirects Pavel S. Aleksandrov]]
[[!redirects P. S. Aleksandrov]]
[[!redirects Pavel Aleksandrov]]
[[!redirects Pavel Alexandrov]]
[[!redirects Paul Alexandrov]]
[[!redirects Paul Alexandroff]]
[[!redirects Pawel Alexandrow]]


Pavel Etinghof > history

* [webpage](http://www-math.mit.edu/~etingof/)

category: people


Pavel Etingof

__Pavel Etingof__ is a mathematician at MIT ([web](http://www-math.mit.edu/~etingof/)). He has been one of the main developers of the theory of [[fusion categories]].

## Selected writings

On [[orbifold cohomology]] of [[symplectic orbifolds]]:

* [[Vasily Dolgushev]], [[Pavel Etingof]], _Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology_, Int. Math. Res. Not. 2005, no. 27, 1657-1688 ([arXiv:math/0410562](https://arxiv.org/abs/math/0410562))

On [[tensor categories]]:

* [[Damien Calaque]], [[Pavel Etingof]], _Lectures on tensor categories_, IRMA Lectures in Mathematics and Theoretical Physics 12, 1-38 (2008) ([arXiv:math/0401246](https://arxiv.org/abs/math/0401246))

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _Topics in Lie theory and Tensor categories -- 9 Tensor categories_, Lecture notes (spring 2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf) [web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)$]$

On [[fusion categories]]:

* {#EtingofNikshychOstrik05} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On fusion categories*, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 ([arXiv:math/0203060](http://arxiv.org/abs/math/0203060), [jstor:20159926](https://www.jstor.org/stable/20159926))

* {#DrinfeldGelakiNikshychOstrik10} [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

On [[Frobenius exact categories]]:

* {#EtingofOstrik21} [[Pavel Etingof]], [[Victor Ostrik]], *On the Frobenius functor for symmetric tensor categories in positive characteristic*, J. für die reine und angewandte Mathematik **773** (2021) 165–198 $[$[arXiv:1912.12947](https://arxiv.org/abs/1912.12947)$]$

and in view of [[Deligne's theorem on tensor categories]]:

* {#CoulembierEtingofOstrik21} [[Kevin Coulembier]], [[Pavel Etingof]], [[Victor Ostrik]], _On Frobenius exact symmetric tensor categories_ $[$[arXiv:2107.02372](https://arxiv.org/abs/2107.02372)$]$

On Calogero-Moser systems and [[double affine Hecke algebra]]s:

* [[Pavel Etingof]], _Lectures on Calogero-Moser systems_, [pdf](http://www-math.mit.edu/~etingof/zlecnew.pdf)

category: people

[[!redirects P. Etingof]]
[[!redirects Etingof]]
[[!redirects Pavel Etinghof]]


Pavel Putrov

* [webpage](https://www.ictp.it/phonebook/person?id=3052)

## Selected writings

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]

* [[Mykola Dedushenko]], [[Sergei Gukov]], [[Pavel Putrov]], *Vertex algebras and 4-manifold invariants*, Chapter 11 in: *Geometry and Physics: Volume I*  (2018) 249–318 [[arXiv:1705.01645](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)]

and in relation to the [[M5-brane elliptic genus]]:

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)]

On [[generalized global symmetries]] hypothesized to play a role in the [[standard model of particle physics]]:

* {#PutrovWang23} [[Pavel Putrov]], [[Juven Wang]], *Categorical Symmetry of the Standard Model from Gravitational Anomaly* [[arXiv:2302.14862](https://arxiv.org/abs/2302.14862)]

category: people


Pavel Safronov

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

## Selected writings

On [[4d TQFTs]] from [[fully dualizable object|fully dualizable]] [[braided monoidal category|braided]] [[tensor categories]], via the [[cobordism hypothesis]]:

* [[Adrien Brochier]], [[David Jordan]], [[Pavel Safronov]], [[Noah Snyder]], _Invertible braided tensor categories_ ([arXiv:2003.13812](https://arxiv.org/abs/2003.13812))

On [[non-perturbative quantum field theory|non-perturbative]] aspects of the [[BV-formalism]]:

* [[Marco Benini]], [[Pavel Safronov]], [[Alexander Schenkel]], *Classical BV formalism for group actions* ([arXiv:2104.14886](https://arxiv.org/abs/2104.14886))

## Related $n$Lab entries

* [[topological twist]]

* [[topologically twisted D=4 super Yang-Mills theory]]

category: people


Pavel Urysohn

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

## Related entries

* [[compact space]]
* [[dimension]]
* [[Urysohn's lemma]]
* [[Urysohn metrization theorem]]

category: people

[[!redirects Pavel Samuilovich Urysohn]]
[[!redirects Pavel Samuelovich Urysohn]]
[[!redirects Paul Samuilovich Urysohn]]
[[!redirects Paul Samuelovich Urysohn]]
[[!redirects Pavel Samuilovich Uryson]]
[[!redirects Pavel Samuelovich Uryson]]
[[!redirects Paul Samuilovich Uryson]]
[[!redirects Paul Samuelovich Uryson]]
[[!redirects Павел Самуилович Урысон]]
[[!redirects P. S. Urysohn]]
[[!redirects P. S. Uryson]]
[[!redirects П. С. Урысон]]
[[!redirects P.S. Urysohn]]
[[!redirects P.S. Uryson]]
[[!redirects П.С. Урысон]]
[[!redirects P S Urysohn]]
[[!redirects P S Uryson]]
[[!redirects П С Урысон]]
[[!redirects PS Urysohn]]
[[!redirects PS Uryson]]
[[!redirects ПС Урысон]]
[[!redirects Pavel Urysohn]]
[[!redirects Paul Urysohn]]
[[!redirects Pavel Uryson]]
[[!redirects Paul Uryson]]
[[!redirects Павел Урысон]]
[[!redirects P. Urysohn]]
[[!redirects P. Uryson]]
[[!redirects П. Урысон]]
[[!redirects P Urysohn]]
[[!redirects P Uryson]]
[[!redirects П Урысон]]
[[!redirects Urysohn]]
[[!redirects Uryson]]
[[!redirects Урысон]]


Paventhan Vivekanandan

## Selected writings

On [[homotopy type theory]] for [[verified programming|verified]] [[cryptography]]:

* [[Paventhan Vivekanandan]], *A Homotopical Approach to Cryptography*, talk at [FCS 2018](https://www.andrew.cmu.edu/user/liminjia/events/fcs2018/papers/s33.pdf) ([pdf](https://www.andrew.cmu.edu/user/liminjia/events/fcs2018/papers/s33.pdf), [easychair:GLtQ#](https://easychair.org/smart-slide/slide/GLtQ#)), In: Charles Morisset and Limin Jia (eds.) FCS Informal Proceedings  **[55](http://t-news.cn/Floc2018/FLoC2018-pages/volume55.html)** (2018)

* [[Paventhan Vivekanandan]], *HoTT-Crypt: A Study in Homotopy Type Theory based on Cryptography*, Kalpa Publications in Computing Volume 9, 2018, Pages 75-90 ([doi:10.29007/tvpp](https://doi.org/10.29007/tvpp), [web slides](https://easychair.org/smart-slide/slide/WSST#), [[VivekanandanHoTTCryptography.pdf:file]])

category: people


Pavol Severa

Pavol Ševera is a mathematical physicist originally from Bratislava.

* [website](http://fyzikus.fmph.uniba.sk/typo/index.php?id=433)

## Selected writings

On [[Courant algebroids]]:

* [[Pavol Ševera]], _Letters to A. Weinstein_ ([web](http://sophia.dtp.fmph.uniba.sk/~severa/letters/), [arXiv:1707.00265](https://arxiv.org/abs/1707.00265))

On [[symplectic Lie n-algebroids]]:

* Pavol Ševera, _[[Some title containing the words "homotopy" and "symplectic", e.g. this one]]_, based on a talk at "Poisson 2000", CIRM Marseille, June 2000; ([arXiv:0105080](http://arxiv.org/abs/math/0105080))

On [[non-abelian T-duality]]:

* {#KlimcikSevera95} [[Ctirad Klimcik]], [[Pavol Ševera]], _Dual non-Abelian duality and the Drinfeld double_, Physics Letters B, Volume 351, Issue 4, 1 June 1995, Pages 455-462 (doi:10.1016/0370-2693(95)00451-P)

specifically applied to [[type II supergravity]]:

* [[Pavol Ševera]], [[Fridrich Valach]], _Courant algebroids, Poisson-Lie T-duality, and type II supergravities_ ([arXiv:1810.07763](https://arxiv.org/abs/1810.07763))

On [[duality in physics|duality]] for [[higher gauge theories]], subsuming [[non-abelian T-duality]] and [[electric-magnetic duality]], via [[boundary conditions]] for [[AKSZ sigma-models]]:

* [[Ján Pulmann]], [[Pavol Ševera]], [[Fridrich Valach]], _A non-abelian duality for (higher) gauge theories_ ([arXiv:1909.06151](https://arxiv.org/abs/1909.06151))

## Related $n$Lab entries

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

* [[Poisson-Lie T-duality]]

category: people

[[!redirects Pavol Ševera]]


Pawel Gajer

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=1241)

## Selected writings

On higher [[bundle gerbes]] ([[bundle 2-gerbes]], etc.) [[connection on a bundle gerbe|with connection]] ([[circle n-bundles with connection]]) as representing [[Deligne cohomology]]

* [[Pawel Gajer]], _Geometry of Deligne cohomology_, Invent. Math., 127(1):155-207 (1997) ([arXiv:alg-geom/9601025](http://arxiv.org/abs/alg-geom/9601025), [doi:10.1007/s002220050118](https://doi.org/10.1007/s002220050118))

category: people


Pawel Horodecki

* [Wikipedia entry](https://en.wikipedia.org/wiki/Pawe%C5%82_Horodecki)

* [Institute page](http://aqualung.mif.pg.gda.pl/index.php?mod=9&s=29&lang=en)

## Selected writings

Discussion of "[[noisy quantum operations]]" in the particular sense of [[quantum channels]] that admit an [environmental representation](quantum+channel#QuantumChannelsAndDecoherence) where the environment/bath system $\mathscr{B}$ is in its *maximally* [[mixed quantum state]]:

* {#HorodeckiHorodeckiOppenheim03} [[Michal Horodecki]], [[Pawel Horodecki]], [[Jonathan Oppenheim]], *Reversible transformations from pure to mixed states, and the unique measure of information*, Phys. Rev. A 67 062104 (2003) [[doi:10.1103/PhysRevA.67.062104](https://doi.org/10.1103/PhysRevA.67.062104), [arXiv;quant-ph/0212019](https://arxiv.org/abs/quant-ph/0212019)]

category: people


Paweł Duch

* [webpage](https://www.lqp2.org/user/1697)

## Selected writings

On [[infrared divergences]], notably in [[quantum electrodynamics]]:

* {#Duch17} [[Paweł Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))

* {#Duch19} [[Paweł Duch]], _Infrared problem in perturbative quantum field theory_ ([arXiv:1906.00940](https://arxiv.org/abs/1906.00940))

On [[renormalization]] of the [[Higgs field]]'s diphoton decay using [[causal perturbation theory]]:

* [[Pawel Duch]], [[Michael Duetsch]], [[Jose Gracia-Bondia]], _Diphoton decay of the higgs from the Epstein--Glaser viewpoint_ ([arXiv:2011.12675](https://arxiv.org/abs/2011.12675))

## Related $n$Lab entries

* [[adiabatic limit]]

* [[infrared divergency]]

* [[quantum electrodynamics]]

* [[quantization of Yang-Mills theory]]

category: people

[[!redirects Pawel Duch]]


Paweł Dłotko

* [research group page](https://dioscuri-tda.org/members/pawel.html)

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

* [personal page](https://www2.math.upenn.edu/~dlotko/) (old)

## Selected writings

On [[persistent homology]] and [[Morse theory]]:

* [[Paweł Dłotko]], [[Hubert Wagner]], _Computing homology and persistent homology using iterated Morse decomposition_ $[$[arxiv/1210.1429](http://arxiv.org/abs/1210.1429)$]$

category: people

[[!redirects Pawel Dlotko]]


Paweł Sobociński

* [webpage](https://www.ioc.ee/~pawel/)

## Selected writings

On [[van Kampen colimits]]:

* {#SobocinskiHeindel11} [[Pawel Sobocinski]], [[Tobias Heindel]], *Being Van Kampen is a universal property*, Logical Methods in Computer Science, **7** 1 (2011) [[arXiv:1101.4594](https://arxiv.org/abs/1101.4594), doi:10.2168/LMCS-7(1:14)2011]

On [[Functorial Semantics of Algebraic Theories]] generalized to [[relational theories]]:

* [[Filippo Bonchi]], [[Dusko Pavlovic]], [[Pawel Sobocinski]], _Functorial Semantics for Relational Theories_, ([arXiv:1711.08699](https://arxiv.org/abs/1711.08699))

On [[toposes]] as [[adhesive categories]]:

* {#LackSobocínski06} [[Stephen Lack]], [[Pawel Sobocínski]], Lemma 18 in:  *Toposes are adhesive*, in: *Graph Transformations* ICGT 2006,  Lecture Notes in Computer Science **4178**, Springer (2006) 184-198 [[doi:10.1007/11841883_14](https://doi.org/10.1007/11841883_14), [pdf](http://users.ecs.soton.ac.uk/ps/papers/toposesAdhesive.pdf)]

## Related pages

* [[strict initial object]]

category: people

[[!redirects Pawel Sobocinski]]


PBW theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

Given a [[Lie algebra]] $g$ which is [[flat module|flat]] as a $k$-[[module]] over a [[commutative ring|commmutative ground ring]] $k\supset\mathbb{Q}$ containing the [[rational numbers|rationals]], consider the [[universal enveloping algebra]] $U(g)=\cup_{i= 0}^\infty U^i(g)$ as a [[filtered algebra]] and a [[Hopf algebra]], where $g$ is the subspace of [[primitive elements]].

The __Poincaré–Birkhoff–Witt theorem__ (often abbreviated to PBW theorem) says that the [[associated graded algebra]] is canonically [[isomorphism|isomorphic]] to the [[symmetric algebra]] $Sym(g)$ as an algebra, and the projection $U(g)\to Gr U(g)\cong Sym(g)$ is an  isomorphism of $k$-[[coalgebras]].

The fact that associated graded algebra is isomorphic to the symmetric algebra is a weak form of the theorem and it is usually proved either by very explicit and long calculation constructing certain representation or by application of the [[zoranskoda:diamond lemma]]. The induced Poisson structure on $Sym(g)$ is the linear Poisson structure for the corresponding $g$.

The notions of Lie algebra, symmetric algebra, enveloping algebra, etc. can also be formulated in the context of a $k$-linear [[tensor category]] (with finite direct sums and split idempotents, and also with countable coproducts over which the tensor product distributes in case one wants to work with ungraded objects), and the PBW theorem remains valid in that context. In particular, the PBW theorem may be formulated and proven for [[superalgebra|super Lie algebras]]. Details may be found in [Deligne-Morgan](#DM).

## Abstract framework

An abstract framework for the Poincar\'e-Birkhoff-Witt theorem in the setting of [[natural transformations]] of [[monads]]
was developed by Dotsenko and Tamaroff.

## Related concepts

* [[Lie-Poisson structure]]

* [[Duflo isomorphism]]

* [[Wheels theorem]]

## References

* {#Quillen69} [[Daniel Quillen]], appendix B.3 of _Rational homotopy theory_, Annals of Math., 90(1969), 205–295 ([JSTOR](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

* wikipedia [Poincaré–Birkhoff–Witt theorem](http://en.wikipedia.org/wiki/Poincaré–Birkhoff–Witt_theorem), [[eom]]: [Birkhoff%E2%80%93Witt_theorem](http://www.encyclopediaofmath.org/index.php/Birkhoff%E2%80%93Witt_theorem)

* [[Pierre Deligne]], [[John Morgan]], _Notes on Supersymmetry, chapter I_. In _[[Quantum Fields and Strings]]: A Course for Mathematicians_ (vol. I), Amer. Math. Soc. 1999.
{#DM}

* Pierre-Paul Grivel, _Une histoire du théorème de Poincaré-Birkhoff-Witt_, Expo. Math. __22__ (2004), no. 2, 145–184 [MR2005b:17024](http://www.ams.org/mathscinet-getitem?mr=2056653) doi

* Vladimir Dotsenko, Pedro Tamaroff,
_Endofunctors and Poincaré-Birkhoff-Witt theorems_,
[arXiv:1804.06485](https://arxiv.org/abs/1804.06485).

[[!redirects Poincaré–Birkhoff–Witt theorem]]
[[!redirects Poincare-Birkhoff-Witt theorem]]


PCT theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

This page is about PCT theorems in [[quantum field theory]]. PCT stands for parity, charge and [[time-reversal symmetry]] (warning: the order of the letters P, C and T varies, some authors use CPT, for example). The laws of nature as described by [[quantum field theory]]s are believed to be invariant if one simultaneously reverses the arrow of time, conjugates all charges and reverses all chiral properties. PCT theorems try to make this belief precise by defining the appropriate operators and showing that certain expressions remain constant. Both the statements and the proofs depend on the framework for [[quantum field theory]] one uses.

Being "invariant" means that every process that can be observerd in our universe can be observed identically in the "mirror" universe, that is there is no experiment in our universe that cannot be duplicated in the mirror universe.

For some time physicists believed that subsets of the PCT symmetry are respected by nature, but today there are counterexamples known for every subset, for example:

* The [[weak force]] violates parity symmetry.

* Charge symmetry would violate the observation that the universe consists mostly of matter and not of matter and antimatter.

* For CP symmetry violation see Wikipedia: [CP violation](http://en.wikipedia.org/wiki/CP_violation)

* All known physical laws are time symmetric, but since PCT symmetry is believed to hold and CP symmetry does not hold, there has to be a process that breaks T symmetry. As of today there is no consenus about what that process may be (there are good reasons to believe that it has nothing to do with the second law of thermodynamics).

## Abstract ##
...

## Definition ##

### Definition in the Wightman approach ###

The PCT theorem for Wightman fields (see [[Wightman axioms]]) was proved by Res Jost, see references.

This proof clarified the different conditions one has to impose, these are:

1. Covariance of the theory under the (connected part of the) [[Poincare group]].

2. Positivity of the energy.

3. There are only fields, which transform with respect to finite dimensional representations of the [[Lorentz group]]. (Transformation of the index space.)

4. Locality, which means that for spacelike distances the Bose fields commute with all other fields and the Fermi fields anticommute with each other.

5. The [[Minkowski space]] has even dimensions.

6. To every field in the theory appears its conjugate complex partner.

### Definition in the Haag-Kastler approach ###

Let $\mathcal{M}(\mathcal{J})$ be a [[AQFT|Haag-Kastler net]] on [[Minkowski spacetime]].

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

A **PCT operator** $\Theta$ on the local net is an anti-linear [[automorphism]], that is for every local algebra $\mathcal{M}(\mathcal{O})$, elements $A, B \in \mathcal{M}(\mathcal{O})$ and $\lambda \in \mathbb{C}$ we have the relations

1. $\Theta(A B) = \Theta(A) \Theta(B)$;

1. $\Theta(\lambda A) = \overline \lambda \Theta(A)$;

1. $\Theta(\mathcal{M}(\mathcal{O})) = \mathcal{M}(-\mathcal{O})$;

such that for $(\Lambda,a) \mapsto U(\Lambda, a)$ the given [[representation]] of the [[Poincare group]] on $\mathcal{M}$ we have

1. $\Theta U(\Lambda, a) A = U(\Lambda, -a) \Theta A$;

1. $\Theta$ transforms every [[charge sector]] into its conjugate sector.

=--

A **PCT theorem** in this context is a theorem that states sufficient conditions such that a PCT operator $\Theta$ exists.

## Properties ##
...

## Examples ##
...

## Related concepts

* [[antiparticle]],

* [[charge sector]], [[DHR category]]

## References ##

Review for the [[standard model of particle physics]]:

* Robert S. Thorne, *The Standard Model Parity, Charge Conjugation and Time Reversal* [[pdf](https://www.hep.ucl.ac.uk/postgrad/teaching/sm/SMIIL4.pdf), [[Thorne-StandardModelPCT.pdf:file]]]

See also:

* Wikipedia: [CPT symmetry](http://en.wikipedia.org/wiki/CPT_symmetry)

Proof from the [[Wightman axioms]]:

* Res Jost: _Eine Bemerkung zum CPT Theorem_ Helv. Phys. Acta 30 (1957), p.409-416

Textbook accounts of this formulation in [[algebraic quantum field theory]]:

* [[Raymond F. Streater]], [[Arthur S. Wightman]], *PCT, Spin and Statistics, and All That*, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)]

* {#Strocchi13} [[Franco Strocchi]], §4.3 in: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

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

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

Proof for [[Lagrangian]] field theory (not falling back to the [[AQFT]] axiomatics) is in

* Hilary Greaves, [[Teruji Thomas]], _The CPT theorem_, Studies in History and Philosophy of Modern Physics 45 (2014) 46-66 ([arXiv:1204.4674](http://arxiv.org/abs/1204.4674))

See also:

* [[Juven Wang]], *C-P-T Fractionalization*, Phys. Rev. D **106** (2022) 105009 [[arXiv:2109.15320](https://arxiv.org/abs/2109.15320), [doi:10.1103/PhysRevD.106.105009](https://doi.org/10.1103/PhysRevD.106.105009)]

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