Albert Schwarz

__Albert Schwarz__ is a mathematician and a theoretical physicist born in Soviet Union and now Professor at University of California-Davis ([web](http://www.math.ucdavis.edu/~schwarz)). He was one of the pioneers of [[Morse theory]] and brought up a first example of a [[topological quantum field theory]]. Schwarz worked on some examples in [[noncommutative geometry]]. He is "S" of the famous [[AKSZ model]].

* German Wikipedia, _[Albert S. Schwarz](http://de.wikipedia.org/wiki/Albert_S._Schwarz)_

* {#Schwarz04} [[Albert Schwarz]], _My Life In Science_, 2004 ([pdf](https://www.math.ucdavis.edu/~schwarz/bion.pdf), [[AlbertSchwarzLifeInScience.pdf:file]])

## Selected writings

(See also the [list of arXiv articles](http://arxiv.org/find/hep-th/1/au:+Schwarz_A/0/1/0/all/0/1) of A. Schwarz.)

On [[noncommutative tori]]:

* [[Marc Rieffel]], [[Albert Schwarz]], _Morita equivalence of multidimensional noncommutative tori_, Int. J. Math. 10 (1999) 289-299 ([arXiv:math/9803057](https://arxiv.org/abs/math/9803057))

On [[Morita equivalence]] and [[duality in physics]]/[[duality in string theory]]:

* {#Schwarz98} [[Albert Schwarz]], _Morita equivalence and duality_ ([arXiv:hep-th/9805034](http://arxiv.org/abs/hep-th/9805034))

and specifically in relation to [[T-duality]]:

* {#Pioline99} B. Pioline, [[Albert Schwarz]], _Morita equivalence and T-duality (or $B$ versus $\Theta$)_ ([arXiv:hep-th/9908019](http://arxiv.org/abs/hep-th/9908019))

On [[partition functions]]

* [[Albert Schwarz]] _The partition function of a degenerate functional_, Commun. Math. Phys. __67__, 1 (1979)

Introducing the [[AKSZ sigma-model]]:

* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]], _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))

On [[supergeometry]] as taking place over the [[base topos]] on the [[site]] of [[super points]]

* [[Albert Schwarz]], _On the definition of superspace_, Teoret. Mat. Fiz. (1984)  Volume 60,  Number 1, Pages 37–42, ([russian original pdf](http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf))

* Anatoly Konechny and [[Albert Schwarz]],

  _On $(k \oplus l|q)$-dimensional supermanifolds_, in: [[Julius Wess]], V. Akulov (eds.) _Supersymmetry and Quantum Field Theory_ (D. Volkov memorial volume) Springer-Verlag, 1998 ,
Lecture Notes in Physics, 509 ([arXiv:hep-th/9706003](http://arxiv.org/abs/hep-th/9706003))

  _Theory of $(k \oplus l|q)$-dimensional supermanifolds_ Sel. math., New ser. 6 (2000) 471 - 486
  {#KonechnySchwarz98}

* [[Albert Schwarz]], I- Shapiro, _Supergeometry and Arithmetic Geometry_ ([arXiv:hep-th/0605119](http://arxiv.org/abs/hep-th/0605119))

On the [[Lie algebra cohomology]] of the [[super Poincaré Lie algebra]] ([[brane scan]] of [[Green-Schwarz sigma-models]]):

* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries_ ([arXiv:1011.4731](http://arxiv.org/abs/1011.4731))

* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra_, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 ([arXiv:1106.0335](http://arxiv.org/abs/1106.0335))

On [[D=10 super Yang-Mills theory]]:

* [[Mikhail Movshev]], [[Albert Schwarz]], _On maximally supersymmetric Yang-Mills theories_, Nucl.Phys. B681 (2004) 324-350 ([arXiv:hep-th/0311132](https://arxiv.org/abs/hep-th/0311132))

On [[supersymmetry]] and [[equivariant localization]]:

* [[Albert Schwarz]], [[Oleg Zaboronsky]], _Supersymmetry and localization_, Comm. Math. Phys. __183__, 2 (1997), 463-476 ([euclid:cmp/1158328185](http://projecteuclid.org/euclid.cmp/1158328185))

On [[BV-formalism]]

* _Geometry of [[Batalin-Vilkovisky quantization]]_, Commun. Math. Phys. __155__, 249 (1993), [euclid](http://projecteuclid.org/euclid.cmp/1104253279)

On the [[semiclassical approximation]] in [[BV-formalism]]:

* _Semiclassical approximation in Batalin-Vilkovisky formalism_, Comm. Math. Phys.  __158__ (1993), no. 2, 373--396, [euclid](http://projecteuclid.org/euclid.cmp/1104254246)

On [[quantum field theory]] and [[topology]]:

* monograph: *Quantum field theory and topology*, Grundlehren der Math. Wissen. __307__, Springer 1993. (translated from Russian original Kvantovaja teorija polja i topologija, *Nauka*, Moscow, 1989. 400 pp.)

* scientific reminiscences, [pdf](http://www.math.ucdavis.edu/~schwarz/bion.pdf)

* V. Kac, A. Schwarz, _Geometric interpretation of the partition function of $2$D gravity_, Phys. Lett. B  __257__  (1991),  no. 3-4, 329--334, [doi](http://dx.doi.org/10.1142/S0217732391000634)

* A. A. Belavin, A. M. Polyakov, A. S. Schwartz, Yu. S. Tyupkin, _Pseudoparticle solutions of the Yang-Mills equations_,  Phys. Lett. B  __59__  (1975), no. 1, 85--87, http://dx.doi.org/10.1016/0370-2693(75)90163-X

* S. N. Dolgikh, A. A. Rosly, A. S. Schwarz, _Supermoduli spaces_,  Comm. Math. Phys.  135  (1990),  no. 1, 91--100, [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104201921)

* V. N. Romanov, A. S. Švarc, _Anomalies and elliptic operators_, (Russian)  Teoret. Mat. Fiz. __41__  (1979), no. 2, 190--204.

* Математические основы квантовой теории поля, Atomizdat, Moscow, 1975. 368 pp.

On 1-[[twisted de Rham cohomology]]:

* [[Albert Schwarz]], [[Ilya Shapiro]], *Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"*, Nucl. Phys. B **809** (2009) 547-560 [[arXiv:0809.0086](https://arxiv.org/abs/0809.0086), [[doi:10.1016/j.nuclphysb.2008.10.005](https://doi.org/10.1016/j.nuclphysb.2008.10.005)]

On [[quantum mechanics]] and [[quantum field theory]]:

* Igor Frolov, [[Albert Schwarz]], *Quantum mechanics and quantum field theory. Algebraic and geometric approaches*, lecture notes [[arXiv:2301.03804](https://arxiv.org/abs/2301.03804)]

On [[super Riemann surfaces]] and [[fat graphs]]:

* [[Albert S. Schwarz]], [[Anton M. Zeitlin]], *Super Riemann surfaces and fatgraphs* [[arXiv:2307.02706](https://arxiv.org/abs/2307.02706)]

## Related entries

* [[supergeometry]]

* [[10d super Yang-Mills theory]], [[IKKT matrix model]]

category: people

[[!redirects Albert S. Schwarz]]


Alberto Cattaneo

* [homepage](http://user.math.uzh.ch/cattaneo)
* publication list: [pdf](http://user.math.uzh.ch/cattaneo/allpub.pdf)

## Selected writings

On [[3-manifolds]]:

* {#BottCattaneo98} [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))

On [[deformation quantization]]:

* [[Alberto Cattaneo]], _From topological field theory to deformation quantization and reduction_, ICM 2006. ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/icm.pdf))

On higher order [[Vassiliev invariants]] as [[Chern-Simons theory]]-[[correlators]], hence as [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Chern-Simons propagators]] assigned to [[edges]] of [[Feynman diagrams]] in the [[graph complex]]:

* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))

* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218))

On the [[Poisson sigma-model]]:

* {#CattaneoFelder} [[Alberto Cattaneo]], [[Giovanni Felder]], _A path integral approach to the Kontsevich quantization formula_, Commun. Math. Phys. 212, 591--611 (2000) [doi](http://dx.doi.org/10.1007/s002200000229), [math.QA/9902090](http://arxiv.org/abs/math/9902090).

* [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and deformation quantization_, Mod. Phys. Lett. A 16, 179--190 (2001) [hep-th/0102208](http://arxiv.org/abs/hep-th/0102208)

* [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and symplectic groupoids_ , (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198, 61--93 (Birkhäuser, 2001)
[math.SG/0003023](http://arxiv.org/abs/math/0003023).

* [[Alberto Cattaneo]], [[Giovanni Felder]], _On the AKSZ formulation of the Poisson sigma model_, Lett. Math. Phys. 56, 163--179 (2001) [math.QA/0102108](http://arxiv.org/abs/math/0102108).

On [[BV-BRST formalism]] compatible with decomposing [[spacetime]] into pieces ([[BV-BFV formalism]]):

* {#CattaneoMnevReshetikhin12} [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical BV theories on manifolds with boundary_, [arXiv:1201.0290](http://arxiv.org/abs/1201.0290); _Classical and quantum Lagrangian field theories with boundary_, [arXiv:1207.0239](http://arxiv.org/abs/1207.0239);  _Perturbative quantum gauge theories on manifolds with boundary_, [arxiv/1507.01221](http://arxiv.org/abs/1507.01221)

* [[Alberto S. Cattaneo]], [[Pavel Mnev]], [[Michele Schiavina]], *BV Quantization*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2307.07761](https://arxiv.org/abs/2307.07761)]

On the local formulation (allowing decomposition and re-gluing of [[spacetime]]) of the [[phase space]] of [[Einstein gravity]] in [[first-order formulation of gravity|first-order formulation]] via [[BV-BFV formalism]]:

* {#CattaneoSchiavina15} [[Alberto Cattaneo]], [[Michele Schiavina]], _BV-BFV approach to General Relativity, Einstein-Hilbert action_, J. Math. Phys. **57** 023515 (2016) [[arXiv:1509.05762](https://arxiv.org/abs/1509.05762), [doi:10.1063/1.4941410](https://doi.org/10.1063/1.4941410)]

* {#CattaneoSchiavina17a} [[Alberto Cattaneo]], [[Michele Schiavina]], _The reduced phase space of Palatini-Cartan-Holst theory_, Ann. Henri Poincaré **20** (2019) 445 [[arXiv:1707.05351](https://arxiv.org/abs/1707.05351), [doi:10.1007/s00023-018-0733-z](https://doi.org/10.1007/s00023-018-0733-z)]

* {#CattaneoSchiavina17b} [[Alberto Cattaneo]], [[Michele Schiavina]], *BV-BFV approach to General Relativity: Palatini-Cartan-Holst action*, Adv. Theor. Math. Phys. **23**  (2019) 1801-1835 [[arXiv:1707.06328](https://arxiv.org/abs/1707.06328), [doi:10.4310/ATMP.2019.v23.n8.a3](https://doi.org/10.4310/ATMP.2019.v23.n8.a3)]

* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

and for flux observables:

* [[Alberto S. Cattaneo]], Alejandro Perez, *A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity*, Class. Quant. Grav. **34** (2017) 107001 [[arXiv:1611.08394](https://arxiv.org/abs/1611.08394), [doi:10.1088/1361-6382/aa69b4](https://doi.org/10.1088/1361-6382/aa69b4)]

and relation to [[BF-theory]]:

* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

On the [[relativistic particle]], and related systems, in [[BV-BRST formalism]]:

* [[Alberto Cattaneo]], [[Michele Schiavina]], _On time_, Lett. Math. Phys. **107** (2017) 375-408 [[doi:10.1007/s11005-016-0907-x](https://doi.org/10.1007/s11005-016-0907-x), [arXiv:1607.02412](http://arxiv.org/abs/1607.02412)]

## Related entries

* [[Poisson sigma-model]]

* [[deformation quantization]]

* [[BV-quantization]]

* [[first-order formulation of gravity]]

[[!redirects A. Cattaneo]]
[[!redirects A. S. Cattaneo]]
[[!redirects Alberto S. Cattaneo]]
[[!redirects Cattaneo]]

category: people


Alberto Elduque

* [personal page](https://personal.unizar.es/elduque/)

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

## Selected writings

On [[composition algebras]]:

* {#Elduque21} [[Alberto Elduque]], *Composition algebras*, Chapter 2 in:  Abdenacer Makhlouf (ed.), *Algebra and Applications I: Non-associative Algebras and Categories*, Sciences-Mathematics, ISTEWiley (2021) 27-57 [[arXiv:1810.09979](https://arxiv.org/abs/1810.09979), [ISBN:978-1-789-45017-0](https://www.wiley.com/en-ae/Algebra+and+Applications+1:+Non+associative+Algebras+and+Categories-p-9781789450170)]

category: people


Alberto Facchini

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=153516)

editor of *[Journal of Algebra and its Applications](https://www.worldscientific.com/worldscinet/jaa)*

## Related entries

* [[list of journals publishing homotopy theory and category theory]]

category: people


Alberto Gandolfi

* [institute page](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/alberto-gandolfi.html)

## Selected writings

Relating the [[Brownian loop soup]] to [[conformal field theory]]:

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *Scalar conformal primary fields in the Brownian loop soup* [[arXiv:2109.12116](https://arxiv.org/abs/2109.12116)]

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *The Brownian loop soup stress-energy tensor* [[arXiv:2112.00074](https://arxiv.org/abs/2112.00074), doi:10.1007/JHEP11(2022)009]

category: people


Alberto García Raboso

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

category: people

[[!redirects Alberto Garcia Raboso]]
[[!redirects Alberto Raboso]]


Alberto Marchisio

* [institute page](https://ti.tuwien.ac.at/ecs/people/marchisio)

* [GoogleScholar page](https://scholar.google.com/citations?user=6QPlLnAAAAAJ&hl=en)

## Selected writings

On [[quantum computation|quantum]] [[quantum machine learning|machine learning]]:

* Kamila Zaman, [[Alberto Marchisio]], Muhammad Abdullah Hanif, [[Muhammad Shafique]], *A Survey on Quantum Machine Learning: Current Trends, Challenges, Opportunities, and the Road Ahead* [[arXiv:2310.10315](https://arxiv.org/abs/2310.10315)]

category: people


Alberto Verjovsky

* [Institute page](https://www.matem.unam.mx/fsd/alberto)

## Selected writings

On [[Hopf fibrations]] and [[twistor fibrations]]:

* [[Bonaventure Loo]] and [[Alberto Verjovsky]], _On quotients of Hopf fibrations_, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 103-108 ([hdl:10077/4637](	http://hdl.handle.net/10077/4637), [pdf](https://www.openstarts.units.it/bitstream/10077/4637/1/LooVerjovskyRendMat26.pdf))

On [[twistor fibrations]]:

* {#SeadeVerjovsky02} [[José Seade]], [[Alberto Verjovsky]], _Higher dimensional complex Kleinian groups_,  Math Ann 322, 279–300 (2002) ([doi:10.1007/s002080100247](https://doi.org/10.1007/s002080100247))

and also on the equivariant [[Arnold-Kuiper-Massey theorem]]:

* Le, [[José Seade]], [[Alberto Verjovsky]], Section 4 of: _Quadrics, orthogonal actions and involutions in complex projective space_, L'Enseignement Mathématique, t.  49 (2003) ([e-periodica:001:2003:49::488](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488))

category: people


Alberto Vezzani

* [personal page](http://users.mat.unimi.it/users/vezzani/)

* [GoogleScholar page](https://scholar.google.it/citations?user=9jKNgDMAAAAJ&hl=en)

## Selected writings

On [[Grothendieck's yoga of six operations]] for [[motives]] in [[rigid analytic geometry]]:

* [[Joseph Ayoub]], [[Martin Gallauer]], [[Alberto Vezzani]], *The six-functor formalism for rigid analytic motives*, Forum of Mathematics, Sigma **10** (2022) E61 [[doi:10.1017/fms.2022.55](https://doi.org/10.1017/fms.2022.55), [arXiv:2010.15004](https://arxiv.org/abs/2010.15004)]

category: people


Alberto Zaffaroni

* [webpage](https://virgilio.mib.infn.it/~zaffaron/)

## Selected writings

On embedding [[D=3 N=4 super Yang-Mills theory]] in [[M-theory]]:

* M. Porrati, [[Alberto Zaffaroni]], _M-Theory Origin of Mirror Symmetry in Three Dimensional Gauge Theories_, Nucl. Phys. B490 (1997) 107-120 ([arXiv:hep-th/9611201](https://arxiv.org/abs/hep-th/9611201))

[[(p,q)5-brane webs]] intersected with [[orientifolds]]:

* [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242))

On [[M-theory on S1/G_HW times H/G_ADE]]

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

On [[D=3 N=4 super Yang-Mills theory]]:

* [[Stefano Cremonesi]], [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories_, JHEP 01 (2014) 005 ([arXiv:1309.2657](https://arxiv.org/abs/1309.2657))

category: people


Albrecht Bertram

* [personal page](https://www.math.utah.edu/~bertram/)

## Selected writings

On [[Gromov-Witten theory]]:

* {#Bertram02} [[Albrecht Bertram]], *Stable Maps and Gromov-Witten Invariants*, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 ([pdf](http://users.ictp.it/~pub_off/lectures/lns019/Bertram/Bertram.pdf), [[Bertram_StableMaps.pdf:file]])

category: people


Albrecht Dold

* [website](http://www.mathunion.org/Publications/Bulletins/38/IMU.EC/DoldA.html)

* [Wikipedia entry](http://de.wikipedia.org/wiki/Albrecht_Dold)

**Albrecht Dold** (1928-2011) was a German mathematician who made important contributions in [[algebraic topology]]. He worked at Princeton, Columbia University, ETH Zürich and held a chair at Heidelberg University from 1963 to his retirement in 1996. Besides numerous research articles he published the textbook _Lectures on Algebraic Topology_ (1972) that became a widely used classic in the field.

Some biographical notes:

* Klaus Volkert: _Vier Heidelberger Topologen 1935-1996_, Jahresbericht der Deutschen Mathematiker-Vereinigung **124**:215–238 (2022). ([doi](https://doi.org/10.1365/s); [pdf](https://link.springer.com/content/pdf/10.1365/s13291-022-00254-8.pdf))

## Selected writings

Introducing the [[Chern-Dold character]]:

* [[Albrecht Dold]], _Relations between ordinary and extraordinary homology_, Matematika, 9:2 (1965), 8–14; Colloq. algebr. Topology, Aarhus Universitet, 1962, 2–9 ([mathnet:mat350](http://mi.mathnet.ru/eng/mat350)), reprinted in: J. Adams & G. Shepherd (Authors), _Algebraic Topology: A Student's Guide_ (London Mathematical Society Lecture Note Series, pp. 166-177). Cambridge: Cambridge University Press 1972 ([doi:10.1017/CBO9780511662584.015](https://doi.org/10.1017/CBO9780511662584.015))

On [[dualizable objects]], [[traces]] and the [[Becker-Gottlieb transfer]]:

* [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)]

On [[algebraic topology]]:

* {#Dold95} [[Albrecht Dold]], *Lectures on Algebraic Topology*, Springer 1995 ([doi:10.1007/978-3-642-67821-9](https://www.springer.com/gp/book/9783540586609), [pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf))

category: people


Albrecht Fröhlich

* [Wikipedia entry](https://en.wikipedia.org/wiki/Albrecht_Fr%C3%B6hlich)

## Selected writings

On [[algebraic number theory]]:

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

category: people

[[!redirects Albrecht Froehlich]]


Albrecht Klemm

* [webpage](http://www.th.physik.uni-bonn.de/people/aklemm/)

## Selected writings

On [[geometric engineering of quantum field theory]]:

* {#KatzKlemmVafa97} [[Sheldon Katz]], [[Albrecht Klemm]], [[Cumrun Vafa]], _Geometric Engineering of Quantum Field Theories_, Nucl.Phys. B497 (1997) 173-195 ([arXiv:hep-th/9609239](http://arxiv.org/abs/hep-th/9609239))

On [[toroidal orbifolds]] in [[string theory]]:

* Jens Erler, [[Albrecht Klemm]], _Comment on the Generation Number in Orbifold Compactifications_, Commun. Math. Phys. 153:579-604, 1993 ([arXiv:hep-th/9207111](https://arxiv.org/abs/hep-th/9207111))

On the [[M5-brane elliptic genus]]:

* Murad Alim, [[Babak Haghighat]], Michael Hecht, [[Albrecht Klemm]], Marco Rauch, Thomas Wotschke, *Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes*, Comm. Math. Phys. **339** (2015) 773–814 [[arXiv:1012.1608](https://arxiv.org/abs/1012.1608), [doi:10.1007/s00220-015-2436-3](https://doi.org/10.1007/s00220-015-2436-3)]

On [[E-strings]] in [[F-theory]]:

* Jie Gu, [[Babak Haghighat]], [[Albrecht Klemm]], Kaiwen Sun, Xin Wang, _Elliptic Blowup Equations for 6d SCFTs. III: E--strings, M--strings and Chains_ ([arXiv:1911.11724](https://arxiv.org/abs/1911.11724))

On the [[mirror map]]:

* S. Hosono, [[Albrecht Klemm]], [[Stefan Theisen]], [[Shing-Tung Yau]], _Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces_, Commun. Math. Phys. __167__ (1995) 301-350 [[doi:10.1007/BF02100589](https://doi.org/10.1007/BF02100589), [arXiv:hep-th/9308122](https://arxiv.org/abs/hep-th/9308122)]

On [[topological string theory]] and [[integrable systems]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]], *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]

## Related entries

* [[topological recursion]]

* [[topological string]]

category: people


Alcides Buss

* [webpage](http://mtm.ufsc.br/~alcides/)

category: people


Aldridge Bousfield

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

## Selected writings

On [[homotopy limits]], [[completion of a space|completions]] and [[localization of a space|localizations]] (such as [[p-completion]] and [[rationalization]]):

* {#BousfieldKan71} [[Aldridge Bousfield]], [[Daniel Kan]], *Localization and completion in homotopy theory*, Bull. Amer. Math. Soc. **77** 6 (1971) 1006-1010 [[doi:10.1090/S0002-9904-1971-12837-9](https://doi.org/10.1090/S0002-9904-1971-12837-9), [pdf](https://www.ams.org/journals/bull/1971-77-06/S0002-9904-1971-12837-9/S0002-9904-1971-12837-9.pdf)]

On [[orthogonal factorization systems]]:

* [[Aldridge Bousfield]], *Constructions of factorization systems in categories*, Journal of Pure and Applied Algebra **9** 2-3 (1977) 207-220 [doi:10.1016/0022-4049(77)90067-6]

Introducing what came to be known as *[[Bousfield localization]]* ([[Bousfield localization of spectra|of spectra]] via that of [[Bousfield localization of model categories|model categories]]):

* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_, Topology Volume 18 Issue 4 (1979) (doi:10.1016/0040-9383(79)90018-1, [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

* [[Daniel M. Kan]] and [[Aldridge K. Bousfield]], *[[Homotopy Limits, Completions and Localizations]]*, Lecture Notes in Mathematics 304 (1972; 1987), Springer ([doi:10.1007/978-3-540-38117-4](https://link.springer.com/book/10.1007/978-3-540-38117-4))

On the [[core of a ring]]:

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _The core of a ring_, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (doi:10.1016/0022-4049(72)90023-0)

On [[rational homotopy theory]] and the [[fundamental theorem of dg-algebraic rational homotopy theory]]:

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

Establishing the [[Bousfield-Friedlander model structure]], in particular the [[stable model category|stable]] [[model structure on topological sequential spectra]]:

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])

On the [[universal Kan fibration]] (origin of the [[univalence axiom]]):

* {#Bousfield06} [[Aldridge Bousfield]], [email to VV from 01 May 2006 10:10:30 CDT](https://groups.google.com/g/homotopytypetheory/c/K_4bAZEDRvE/m/YSQz-jJ_AAAJ) $[$[[BousfieldOnUnivalence.jpg:file]]$]$

## Related $n$Lab entries

* [[rational homotopy theory]]

* [[Bousfield-Friedlander model structure]]

* [[Bousfield localization of spectra]]

* [[Bousfield localization of model categories]]

* [[Bousfield-Friedlander theorem]]

* [[proper model category]]

* [[semi-left-exact left Bousfield localization]]

category: people

[[!redirects A. K. Bousfield]]
[[!redirects Aldridge K. Bousfield]]
[[!redirects Aldridge Knight Bousfield]]
[[!redirects Bousfield]]


ALE space

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

#Contents#
* table of contents
{:toc}

## Idea

An _asymptotically locally Euclidean space_ or _ALE space_ for short is a solution to the Euclidean [[Einstein equations]] which is a [[blow-up]] of an [[ADE-orbifold]] singularity $\mathbb{C}^2/\Gamma$ for [[finite group|finite]] [[subgroup]] $\Gamma \hookrightarrow SU(2)$.

[[!include KK-monopole geometries -- table]]

## In string theory

* In [[M-theory]]: [[KK-monopole]]

  In [[F-theory]]: [[D7-brane]]

## References

An [[ADE classification]] of 4d ALE-spaces is due to

* {#Kronheimer89} [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([Euclid](https://projecteuclid.org/euclid.jdg/1214443066))

In

* {#IntriligatorSeiberg96} [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_ ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))

this result is interpreted physically as describing the [[moduli space]] of [[vacua]] of [[gauge theories]] with [[spontaneously broken symmetry]] ("Higgs branches"). See at _[[3d mirror symmetry]]_ for more on this.

For application in [[string theory]] see at _[[KK-monopole]]_ and see

* {#JohnsonMyers96} [[Clifford Johnson]], [[Robert Myers]], _Aspects of Type IIB Theory on ALE Spaces_, Phys.Rev. D55 (1997) 6382-6393 ([arXiv:hep-th/9610140](https://arxiv.org/abs/hep-th/9610140))

[[!redirects ALE spaces]]

[[!redirects asymptotically locally Euclidean space]]
[[!redirects asymptotically locally Euclidean spaces]]


Alejandro Adem

* [webpage](http://www.math.ubc.ca/~adem/)

## Selected writings

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

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

On [[braid group representations]] seen in the [[topological K-theory]] of the [[classifying space]] of the [[braid group]]:

* [[Alejandro Adem]], [[Daniel C. Cohen]], [[Frederick R. Cohen]], *On representations and K-theory of the braid groups*, Math. Ann. **326** (2003) 515-542 ([arXiv:math/0110138](https://arxiv.org/abs/math/0110138), [doi:10.1007/s00208-003-0435-8](https://doi.org/10.1007/s00208-003-0435-8))

On [[free action|free]] [[group actions on n-spheres]] (or rather on [[product topological space|product spaces]] of [[n-spheres]]):

* [[Alejandro Adem]], _Constructing and deconstructing group actions_, in [[Paul Goerss]], [[Stewart Priddy]], *Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory*, Contemporary Mathematics **346** AMS 2004 ([arXiv:0212280](http://arxiv.org/abs/math/0212280), [doi:10.1090/conm/346](http://dx.doi.org/10.1090/conm/346))

On [[group cohomology]] of [[finite groups]]:

* [[Alejandro Adem]], _Lectures on the cohomology of finite groups_ ([pdf](http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf))

*  [[Alejandro Adem]], [[R. James Milgram]], _Cohomology of Finite Groups_, Springer 2004

On [[orbifolds]]:

* {#AdemKlaus} [[Alejandro Adem]], Michele Klaus, _Lectures on orbifolds and group cohomology_ ([pdf](http://www.math.ubc.ca/~adem/hangzhou.pdf), [[AdemKlausOrbifolds.pdf:file]])

On [[orbifolds]], [[orbifold cohomology]] and specifically on [[Chen-Ruan cohomology]] and [[equivariant K-theory]]:

* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:

* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))

* [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64.  ([arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf))

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

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

## Related $n$Lab entries

* [[orbifold]], [[Riemannian orbifold]]

* [[group actions on spheres]]

* [[equivariant K-theory]]

* [[Bredon cohomology]]

* [[string topology]]

* [[sphere]]

* [[Borel construction]]

* [[finite group]]

* [[transgression]]

category: people


Alejandro Corichi

* [InSpire page](https://inspirehep.net/authors/1013037)

* [GoogleScholar page](https://scholar.google.com/citations?user=OO8beHMAAAAJ)

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

## Selected writings

On [[Fock space]]-[[quantization]] of the [[electromagnetic field]] based on careful analysis of its [[covariant phase space]]:

* {#Corichi98} [[Alejandro Corichi]], *Introduction to the Fock Quantization of the Maxwell Field*, Rev. Mex. Fis. **44**  4  (1998) 402-412 [[arXiv:physics/9804018](https://arxiv.org/abs/physics/9804018)]

and generalizing to the case of [[spacetimes]] [[manifold with boundary|with boundaries]]:

* [[Alejandro Corichi]], Juan D. Reyes, Tatjana Vukasinac, *On covariant and canonical Hamiltonian formalisms for gauge theories* [[arXiv:2312.10229]](https://arxiv.org/abs/2312.10229)]

category: people


Alejandro Uribe

* [webpage](http://www.math.lsa.umich.edu/~uribe/)

## related $n$Lab pages

* [[theta function]], [[quantization of 3d Chern-Simons theory]]

category: people


Aleks Kissinger

* [website](http://web.comlab.ox.ac.uk/people/Aleks.Kissinger/)

## Selected writings

On [[classical structures]] in the [[quantum information theory via dagger-compact categories]] and their graphical formalization in the [[ZX-calculus]]:

* [[Aleks Kissinger]], *Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories*, MSc thesis, Oxford (2008) [[pdf](https://www.cs.ox.ac.uk/people/bob.coecke/Aleks.pdf), [[Kissinger-CLassicalStructures.pdf:file]]]

* [[Aleks Kissinger]], *Exploring a Quantum Theory with Graph Rewriting and Computer Algebra*, in: *Intelligent Computer Mathematics. CICM 2009*, Lecture Notes in Computer Science **5625** (2009) 90-105 [[doi:10.1007/978-3-642-02614-0_12](https://doi.org/10.1007/978-3-642-02614-0_12)]

On [[quantum information theory via dagger-compact categories]]:

* [[Bob Coecke]], [[Aleks Kissinger]], *Picturing Quantum Processes -- A First Course in Quantum Theory and Diagrammatic Reasoning*, Cambridge University Press (2017) [[ISBN:9781107104228](https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/picturing-quantum-processes-first-course-quantum-theory-and-diagrammatic-reasoning?format=HB&isbn=9781107104228)]

Relating the [[ZX-calculus]] to [[braided fusion categories]] for [[anyon]] [[braiding]]:

* [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], *Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics* $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$

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

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

## Related entries

* [[quantum information]]
* [[rewriting]]

category: people


Aleksandar Mikovic

Aleksandar Mikovic is Associate Professor at the Department of Mathematics, Lusófona University, Lisbon. He is a member of the Grupo de Física Matemática of the Universidade de Lisboa.

* [webpage](http://gfm.cii.fc.ul.pt/people/amikovic/)

##Selected writings

On [[DBI-action]] and [[Green-Schwarz action functional]] for [[D-branes]]:

* [[Martin Cederwall]], Alexander von Gussich, [[Aleksandar Mikovic]], [[Bengt Nilsson]], Anders Westerberg, _On the Dirac-Born-Infeld Action for D-branes_,  Phys.Lett.B390:148-152, 1997 ([arXiv:hep-th/9606173](https://arxiv.org/abs/hep-th/9606173))

On [[spin networks]]:

* [[João Faria Martins]], [[Aleksandar Mikovic]], _Invariants of spin networks embedded in three-manifolds_, Comm. Math. Phys. (2006) 279, pp. 381-399 ([arXiv:gr-qc/0612137](https://arxiv.org/pdf/gr-qc/0612137.pdf), [doi:10.1007/s00220-008-0422-8](https://doi.org/10.1007/s00220-008-0422-8))

On [[piecewise flat spacetimes]] and [[quantum gravity]]:

* [[Aleksandar Mikovic]], _Piecewise Flat Metrics and Quantum Gravity_ ([arXiv:2001.11439](https://arxiv.org/abs/2001.11439))

category:people


Aleksandar Nanevski

* [webpage](http://software.imdea.org/~aleks/)

category: people


Aleksandr Aleksandrov

Алекса́ндр Дани́лович Алекса́ндров (Aleksandr Danilovič Aleksandrov) was a Russian [[geometry|geometer]].

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

## Related articles

* [[Alexandrov space]]

category: people

[[!redirects Алекса́ндр Дани́лович Алекса́ндров]]
[[!redirects Александр Данилович Александров]]
[[!redirects Александр Д. Александров]]
[[!redirects А. Д. Александров]]
[[!redirects Александр Александров]]
[[!redirects Aleksandr Danilovič Aleksandrov]]
[[!redirects Aleksandr Danilovich Aleksandrov]]
[[!redirects Aleksandr D. Aleksandrov]]
[[!redirects A. D. Aleksandrov]]
[[!redirects Aleksandr Aleksandrov]]
[[!redirects Aleksander Aleksandrov]]
[[!redirects Alexander Alexandrov]]


Aleksei Starobinsky

Alexei Starobinsky (1948 - 2023)

* [personal page](http://www.itp.ac.ru/en/persons/starobinsky-aleksei-aleksandrovich/)

* [InSpire page](https://inspirehep.net/authors/987711)

* *[‘A Physicist Has to Be a Romantic’](https://www.hse.ru/en/news/305811947.html)*, HSE University News (Sep 2019)

## Selected writings

Introducing the [[Starobinsky model of cosmic inflation]] (and with it the field of [[cosmic inflation]] in the first place):

* {#Starobisnky80} [[Aleksei Starobinsky]], *A new type of isotropic cosmological models without singularity*, Phys. Lett. B **91**  (1980) 99-102 \[doi:10.1016/0370-2693(80)90670-X\]

* [[Aleksei Starobinsky]], *Isotropization of arbitrary cosmological expansion given an effective cosmological constant*, JETP Lett. **37** (1983) 66-69 [[spire:187801](https://inspirehep.net/literature/187801)]

* [[Aleksei Starobinsky]], *The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy*, Sov. Astron. Lett. **9** (1983) 302 [[spire:199078](https://inspirehep.net/literature/199078)]

## Related entries

* [[Starobinsky model of cosmic inflation]]

* [[eternal inflation]]

category: people


aleph

An aleph (ℵ) is the [[cardinality]] of an [[infinite set|infinite]] [[well-ordered set|well-ordered]] (or well-orderable) [[set]].  Assuming the [[axiom of choice]] (in the guise of the [[well-ordering theorem]]), every [[cardinal number]] is either a [[natural number]] or an aleph.

The alephs are themselves well-ordered; for an [[ordinal number]] $\mu$, we denote the $\mu$th aleph as $\aleph_\mu$.

In particular $\aleph_0$ is the the cardinality of the set of [[natural numbers]].

[[!redirects aleph]]
[[!redirects alephs]]
[[!redirects ℵ]]
[[!redirects ℵs]]


Alessandra Buonanno

* [webpage](http://www.aei.mpg.de/1282509/Homepage_of_Alessandra_Buonanno)

Buonanno made the theoretical prediction ([Buonanno-Damour 98](#BuonannoDamour98), see also [Damour 10](Thibault+Damour#Damour10)) of the [[gravitational wave]]-signal emitted by inspiralling [[relativistic binaries]] which has later been observed by [[LIGO]].

## Selected writings

On the theoretical prediction of the [[gravitational wave]]-signal produced by [[relativistic binaries]] (as later detected by [[LIGO]]):

* {#BuonannoDamour98} [[Alessandra Buonanno]], [[Thibault Damour]], _Effective one-body approach to general relativistic two-body dynamics_, Phys.Rev. D59 (1999) 084006 ([arXiv:gr-qc/9811091](http://arxiv.org/abs/gr-qc/9811091))

* {#BuonannoDamour00} [[Alessandra Buonanno]], [[Thibault Damour]], _Binary black holes coalescence: transition from adiabatic inspiral to plunge_, IX Marcel Grossmann Meeting in Rome, July 2000 ([arXiv:gr-qc/0011052](http://arxiv.org/abs/gr-qc/0011052))

* [[Alessandra Buonanno]], _The making of high-precision gravitational waves_, talk at ICTS 2019 ([home page](https://www.icts.res.in/lectures/gravitationalwaves2019))

## Related $n$Lab entries

* [[gravitational wave]]

* [[relativistic binary]]

category: people


Alessandra Di Pierro

* [GoogleScholar page](https://scholar.google.ae/citations?user=t-1lOBUAAAAJ&hl=zh-CN)

## Selected writings

On [[kernel methods]] in [[topological data analysis]] via [[quantum computation]]:

* Massimiliano Incudini, Francesco Martini, [[Alessandra Di Pierro]], *Higher-order topological kernels via quantum computation*, 2023 IEEE International Conference on Quantum Computing and Engineering, QCE **1** (2023) [[arXiv:2307.07383](https://arxiv.org/abs/2307.07383), [doi:10.1109/QCE57702.2023.00076](https://doi.ieeecomputersociety.org/10.1109/QCE57702.2023.00076)]

category: people


Alessandro Luongo

* [personal page](https://luongo.pro/)

## Selected writings

On [[quantum computation|quantum]] [[algorithms]] for [[data analysis]]:

* [[Alessandro Luongo]], *[Quantum algorithms for data analysis](https://quantumalgorithms.org)* [[quantumalgorithms.org](https://quantumalgorithms.org)]

category: people


Alessandro Nagar

* [webpage](http://www.ihes.fr/~nagar/Home.html)

## related $n$Lab entries

* [[gravitational wave]]

category: people


Alessandro Strumia

* [Institute page Pisa](https://sites.google.com/a/unipi.it/pisa-theory-group/staff/strumia-alessandro)

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

## Selected writings

On [[dark matter]] [[model (in theoretical physics)|models]]:

* Marco Cirelli, Nicolao Fornengo, [[Alessandro Strumia]], _Minimal Dark Matter_, Nucl.Phys.B753:178-194, 2006 ([arXiv:hep-ph/0512090](https://arxiv.org/abs/hep-ph/0512090))

On the [[Higgs field]] [[vacuum stability]]:

* {#BDGGSSS13} Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, [[Alessandro Strumia]], section 7 of _Investigating the near-criticality of the Higgs boson_, JHEP12(2013)089 ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))

* {#EGMRSST15} Jose R. Espinosa, [[Gian Giudice]], Enrico Morgante, Antonio Riotto, Leonardo Senatore, [[Alessandro Strumia]], Nikolaos Tetradis, _The cosmological Higgstory of the vacuum instability_ ([arXiv:1505.04825](https://arxiv.org/abs/1505.04825))

* {#Strumia17} [[Alessandro Strumia]], _Higgs and Vacuum (In)Stability_, talk at GGI 2017 ([pdf](https://indico.cern.ch/event/660870/contributions/2746200/attachments/1538374/2411246/2017-HiggsDecay.pdf))

and implications for [[split supersymmetry]]:

* [[Gian Giudice]], [[Alessandro Strumia]], _Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements_, Nuclear Physics B Volume 858, Issue 1, 1 May 2012, Pages 63-83 ([arXiv:1108.6077](https://arxiv.org/abs/1108.6077))

On [[flavour anomalies]]:

* {#StrumiaEtAl17} Guido D'Amico, Marco Nardecchia, Paolo Panci, Francesco Sannino, [[Alessandro Strumia]], Riccardo Torre, Alfredo Urbano,

  _Flavour anomalies after the $R_{K^\ast}$ measurement_,

  J. High Energ. Phys. (2017) 2017 ([arXiv:1704.05438](https://arxiv.org/abs/1704.05438))

## related $n$Lab entries

* [[Higgs field]]

* [[flavour anomaly]]

* [[asymptotic safety]] (or not)

category: people


Alessandro Tomasiello

* [webpage](http://moby.mib.infn.it/~atom/)

## Selected writings

On [[D8-branes]], [[D6-branes]], and [[D6-D8-brane bound states]] in [[massive type IIA supergravity]]/[[massive type IIA string theory]]:

* Fabio Apruzzi, [[Marco Fazzi]], Dario Rosa, [[Alessandro Tomasiello]],  _All $AdS_7$ solutions of type II supergravity_, JHEP 04 (2014) 064 ([arxiv:1309.2949](https://arxiv.org/abs/1309.2949))

On [[M-theory on S1/G_HW times H/G_ADE]] and [[D=6 N=(1,0) SCFT]]:

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP12(2014)003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

  (and on [[D6-D8 brane intersections]] and their [[fuzzy funnel]] [[noncommutative geometry]])

* {#DHTV14} [[Michele Del Zotto]], [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

* [[Ibrahima Bah]], Achilleas Passias, [[Alessandro Tomasiello]], _$AdS_5$ compactifications with punctures in massive IIA supergravity_, JHEP11 (2017)050 ([arXiv:1704.07389](https://arxiv.org/abs/1704.07389))

category: people


Alessandro Torrielli

Alessandro Torrielli works on [[integrable systems]], especially in the context of [[AdS-CFT]] and [[Yangian]] symmetry, at Surrey University.

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

## Selected writings

* Alessandro Torrielli, _Yangians, S-matrices and AdS/CFT_,	  J. Phys. A44:263001, 2011 ([arXiv:1104.2474](http://arxiv.org/abs/1104.2474))

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation opetator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:

* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))

On the [[sine-Gordon equation]], the [[Thirring model]] and their [[duality in physics|duality]]:

* [[Alessandro Torrielli]], *LonTI Lectures on Sine-Gordon and Thirring* [[arXiv:2211.01186](https://arxiv.org/abs/2211.01186)]

category: people


Alessandro Valentino

* [website](http://www.uni-math.gwdg.de/sandro/)

## Selected writings

On [[QFTs with defects]] via [[higher algebra|higher]] [[categorical algebra]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], *Bicategories for boundary conditions and for surface defects in 3-d TFT*, Commun. Math. Phys. **321**  (2013) 543–575 [[arXiv:1203.4568](https://arxiv.org/abs/1203.4568), [doi:10.1007/s00220-013-1723-0](https://doi.org/10.1007/s00220-013-1723-0)]

category: people


Alessandro Vichi

## Selected writings

On the [[conformal bootstrap]]:

* [[David Poland]], [[Slava Rychkov]], [[Alessandro Vichi]], _The Conformal Bootstrap: Numerical Techniques and Applications_, Rev. Mod. Phys. 91, 15002 (2019) ([arXiv:1805.04405](https://arxiv.org/abs/1805.04405))

Application of [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))


Alessio Lomuscio

[[!redirects A. Lomuscio]]
* [Home page](http://www.doc.ic.ac.uk/~alessio/index.html)

category:people


Alessio Marrani

* [Inspire page](https://inspirehep.net/authors/1029392)

# Selected writings

On [[magic supergravity]]:

* {#CacciatoriCerchiaiMarrani12} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Squaring the Magic_ [[arXiv:1208.6153](http://arxiv.org/abs/1208.6153)]

* {#CacciatoriCerchiaiMarrani2013} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Magic coset decompositions_, Adv. Theor. Math. Phys. **17** 5 (2013) 1077-1128 [[doi:10.4310/ATMP.2013.v17.n5.a4](https://doi.org/10.4310/ATMP.2013.v17.n5.a4)]

On generalizations of [[exceptional structures]], including [[E8]], [[octonions]] and the [[exceptional Jordan algebra]]:

* [[Piero Truini]], [[Michael Rios]], [[Alessio Marrani]], *The Magic Star of Exceptional Periodicity*, J. Phys.: Conf. Ser. **1194** (2019) 012106 [[arXiv:1711.07881](https://arxiv.org/abs/1711.07881), [doi:10.1088/1742-6596/1194/1/012106](https://iopscience.iop.org/article/10.1088/1742-6596/1194/1/012106)]

On [[D=14 supersymmetry]]:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D 99, 046004 (2019) ([arXiv:1811.06101](https://arxiv.org/abs/1811.06101))

On [[12-dimensional supergravity]], [[D=14 supersymmetry]] et al. and further indications that [[M-theory]] in 10+1 dimensions may be understood as the [[KK-compactification]] on Cayley-plane [[fibers]] of some kind of [[bosonic M-theory]] in 26+1 dimensions:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D **99** (2019) 046004  [[arXiv:1811.06101](https://arxiv.org/abs/1811.06101), [doi:10.1103/PhysRevD.99.046004](https://doi.org/10.1103/PhysRevD.99.046004)]

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]

reviewed in:

* [[Alessio Marrani]], *Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory*, talk at *[M-Theory and mathematics 2023](https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan, 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Marrani2023)]

Relation to the [[Monster group]], [[Moonshine]] and the [[Monster vertex operator algebra]]:

* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], *Monstrous M-theory*, Symmetry **15** 2 (2023) 490; [[doi:10.3390/sym15020490](https://doi.org/10.3390/sym15020490), [arXiv:2008.06742](https://arxiv.org/abs/2008.06742)]

On the work of [[Mike Duff]]:

* [[Leron Borsten]], [[Alessio Marrani]], [[Christopher N. Pope]], [[Kellogg Stelle]], *Introduction to the special issue dedicated to Michael J. Duff FRS on the occasion of his 70th birthday*, Proceedings of the Royal Society A,
**478** (2022) 2259 [[doi:10.1098/rspa.2022.0166](https://doi.org/10.1098/rspa.2022.0166)]

category: people


Alex Arvanitakis

* [webpage](http://www.damtp.cam.ac.uk/user/asa49/)

## Selected writings

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))

On [[branes]] in [[exceptional generalized geometry]]:

* [[Alex Arvanitakis]], Chris D. A. Blair, _Type II strings are Exceptional_ ([arXiv:1712.07115](https://arxiv.org/abs/1712.07115))

* [[Alex Arvanitakis]], Chris Blair, _The Exceptional Sigma Model_ ([arXiv:1802.00442](https://arxiv.org/abs/1802.00442))

On [[higher WZW terms]] for [[branes]]:

* [[Alex Arvanitakis]], _Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 5 ([arXiv:1804.07303](https://arxiv.org/abs/1804.07303), [doi:10.4310/ATMP.2019.v23.n5.a1](https://dx.doi.org/10.4310/ATMP.2019.v23.n5.a1))

On [[L-infinity algebra]]-[[structure]] in [[perturbative quantum field theory]]:

* {#Arvanitakis19} [[Alex Arvanitakis]], _The $L_\infty$-algebra of the S-matrix_ ([arXiv:1903.05643](https://arxiv.org/abs/1903.05643))

On [[Lagrangian correspondence]]s

* Alex S. Arvanitakis, _Topological defects as lagrangian correspondences_, Phys. Rev. D 107, 066016, 2023 [arXiv:2301.04143](https://arxiv.org/abs/2301.04143) [doi](https://doi.org/10.1103/PhysRevD.107.06601)

## Related $n$Lab entries

* [[exceptional generalised geometry]]

* [[higher WZW term]], [[branes]]

* [[L-infinity algebra]], [[perturbative quantum field theory]]

categories: people


Alex Cole

* [webpage](https://grad.wisc.edu/2020/05/06/alexander-cole/)

## Selected writings

Application of [[topological data analysis]] ([[persistent homology]]) to analysis of [[cosmological structure formation]]:

* Matteo Biagetti, [[Alex Cole]], [[Gary Shiu]], _The Persistence of Large Scale Structures I: Primordial non-Gaussianity_ ([arXiv:2009.04819](https://arxiv.org/abs/2009.04819))

and to analysis of [[phase transitions]]:

* [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology_ ([arXiv:2009.14231](https://arxiv.org/abs/2009.14231))

On the [[weak gravity conjecture]]:

* Lars Aalsma, [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]],
_A New Spin on the Weak Gravity Conjecture_ ([arXiv:2011.05337](https://arxiv.org/abs/2011.05337))

category: people


Alex Eskin

* [webpage](https://www.math.uchicago.edu/~eskin/)

## Selected writings

On [[pillowcase orbifolds]]:

* {#EskinOkounkov05} [[Alex Eskin]], [[Andrei Okounkov]], _Pillowcases and quasimodular forms_, In: Ginzburg V. (ed.) _Algebraic Geometry and Number Theory_, Progress in Mathematics, vol 253. Birkhäuser 2006
 ([arXiv:math/0505545](https://arxiv.org/abs/math/0505545), [doi:10.1007/978-0-8176-4532-8_1](https://doi.org/10.1007/978-0-8176-4532-8_1))

category: people


Alex Heller

Alex Heller, (born on July 9, 1925, died January 31,
2008), was professor of mathematics at the Graduate School and University Center, CUNY in New York. He was a PhD student of [[Sammy Eilenberg]] and made some very important contributions to [[homotopy theory]] and to [[category theory]].

* [mathematics genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=4773)

* [[Alex-Heller.pdf|Obituary:file]]  prepared by [[Noson Yanofsky]].

##References

A complete MathSciNet listing of his publications is given here in [[Alex-Heller-pub.pdf|pdf-format:file]] and in [[Alex-Heller-bibtex.bib|bibtex-format:file]].  Some selected papers of direct relevance to the themes of the nLab are listed below.

* Alex Heller, _Homotopy resolutions of semi-simplicial complexes_,Trans. Amer. Math. Soc. 80 (1955), 299-344, (available [here](http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0075588-0/).

* Alex Heller, _On the representability of homotopy functors_
J. London Math. Soc. (2) 23 (1981), no. 3, 551–562

* Alex Heller, _Homotopy theories_ , Memoirs of the American Mathematical Society, Vol. 71, No 383 (1988).

* Alex Heller, _Stable homotopy theories and stabilization_ , [MR](http://www.ams.org/mathscinet-getitem?mr=1431157)

* Alex Heller, Robert A Di Paola, _Dominical categories: recursion theory without elements_, J. Symbolic Logic 52 (1987), no. 3, 594-635.

* Alex Heller, _Stable homotopy categories_,
Bull. Amer. Math. Soc. 74 (1968) 28-63

* Alex Heller, _On stochastic processes derived from Markov chains
Ann. Math. Statist. 36 (1965) 1286-1291

* Alex Heller, _The loop-space functor in homological algebra_,
Trans. Amer. Math. Soc. 96 1960 382-394

* Alex Heller, _Homological algebra in abelian categories_,
Ann. of Math. (2) 68 1958 484-525

* Alex Heller, _Relative homotopy_,
J. London Math. Soc. (2) 44 (1991), no. 3, 537-552.

category: people


Alex Hoffnung

 chains.  Here we will sketch the idea and eventually should formalize this on the page for Borel-Moore homology.  In the meantime the Wikipedia article plus its references and the Chriss-Ginzburg book `Representation Theory and Complex Geometry` provide plenty of information.

$$ H_\bullet(Y) = (possibly\; unbounded)\; i-chains\; in\; Y\;(with\; coefficients\; in \;\mathbb{C})$$

$$ H_i(Y) = i-dim\; subspace\;of\; Y \;without\; boundary \;(possibly \;singular\; or\; unbounded)$$

The construction starts by triangulating the space and considering the vector space of formal sums of $i$-simplices.

Of course, one wants independence of triangulation so we need to take a direct limit of these vectors spaces over all refinements of the triangulation.  The boundary map then comes from simplicial homology and we have a chain complex.  The homology of this complex is the Borel-Moore homology.

+--{.query}

To anyone reading this: These are notes from the University of Ottawa/Fields Institute summer school which took place last week (if you are reading this before Friday July 3).  Adam Katz and I are trying to make our way through the notes from several of the classes.  Anyone is welcome to add comments, corrections, insights or just join our discussion.  Actually it would be great if we had some more people involved.  I think the plan is something like: 1) Understand Kamnitzer's lectures on Borel-Weil, Ginzburg construction and geometric Satake, then move to Savage's lectures relating the geometry from Kamnitzer and the combinatorics from Kang's lectures.  So Kang's lectures will hopefully serve as background reading to understand Savage's lectures.

Alex
=--

+--{.query}
Other projects:

1) Understand q-Schur algebras and relationship to representation theory of Hecke algebras

2) Learn lots of things in Ginzburg-Chriss.

3) Write my thesis!

4) Write everything else I should be writing.

5) Get a job! :)

I guess 3) and 5) are taken care of now.

If anyone wants to talk about the first item that would be great.  I guess this is my new recruiting station to get people to talk to me about math.
=--

category: people

[[!redirects Alexander Hoffnung]]


Alex Kavvos

* [personal page](https://seis.bristol.ac.uk/~tz20861/)

## Selected writings

On [[directed homotopy type theory]]:

* [[Alex Kavvos]], *A quantum of direction* (2019) [[pdf](https://seis.bristol.ac.uk/~tz20861/papers/meio.pdf)]

On [[modal type theory]]:

* [[Daniel Gratzer]], [[G. Alex Kavvos]], [[Andreas Nuyts]], [[Lars Birkedal]]: _Multimodal Dependent Type Theory_, Logical Methods in Computer Science **17** 3 (2021) lmcs:7713 [[arXiv:2011.15021](https://arxiv.org/abs/2011.15021), doi:10.46298/lmcs-17(3:11)2021]

category: people

[[!redirects G. Alex Kavvos]]
[[!redirects G. A. Kavvos]]


Alex Kehagias

* [webpage](http://www.physics.ntua.gr/~kehagias/)

## Selected writings

Discussion of [[pp-wave spacetimes]] as [[Penrose limits]] of [[anti de Sitter spacetime|AdSp]] [[product manifold|x]] [[n-sphere|S^q]] spacetimes:

* E. Floratos, [[Alex Kehagias]],
_Penrose Limits of Orbifolds and Orientifolds_, JHEP 0207 (2002) 031 ([arXiv:hep-th/0203134](https://arxiv.org/abs/hep-th/0203134))

On [[Horava-Witten theory]] fopr [[7d supergravity]]:

* {#GherghettaKehagias02} Tony Gherghetta, [[Alex Kehagias]], _Anomaly Cancellation in Seven-Dimensional Supergravity with a Boundary_, Phys.Rev. **D68** (2003), 065019, ([arXiv:hep-th/0212060](http://arxiv.org/abs/hep-th/0212060))

On the [[Starobinsky model of cosmic inflation]]:

*  {#KehagiasDizgahRiotto13} [[Alex Kehagias]], Azadeh Moradinezhad Dizgah, Antonio Riotto, _Comments on the Starobinsky Model of Inflation and its Descendants_, Phys. Rev. D 89, 043527 (2014) ([arXiv:1312.1155](http://arxiv.org/abs/1312.1155))

* {#FKR13} [[Fotis Farakos]], [[Alex Kehagias]], A. Riotto, _On the Starobinsky Model of Inflation from Supergravity_, Nucl. Phys. B 876, 187 (2013) ([arXiv:1307.1137](http://arxiv.org/abs/1307.1137))

* [[Sergio Ferrara]], [[Alex Kehagias]], Antonio Riotto, _The Imaginary Starobinsky Model and Higher Curvature Corrections_ ([arXiv:1405.2353](http://arxiv.org/abs/1405.2353))

* {#FerrarKehagias14} [[Sergio Ferrara]], [[Alex Kehagias]], _Higher Curvature Supergravity, Supersymmetry Breaking and Inflation_ ([arXiv:1407.5187](http://arxiv.org/abs/1407.5187))

* {#DFKRU14} [[Ioannis Dalianis]], [[Fotis Farakos]], [[Alex Kehagias]], A. Riotto, [[Rikard von Unge]], _Supersymmetry Breaking and Inflation from Higher Curvature Supergravity_ ([arXiv:1409.8299](http://arxiv.org/abs/1409.8299))

* [[Luis Alvarez-Gaume]], [[Alex Kehagias]], [[Costas Kounnas]], [[Dieter Luest]], Antonio Riotto, _Aspects of Quadratic Gravity_ ([arXiv:1505.07657](http://arxiv.org/abs/1505.07657))

## Related $n$Lab entries

* [[7d supergravity]]

* [[Starobinsky model of cosmic inflation]]

category: people


Alex Simpson

* [webpage](https://www.fmf.uni-lj.si/si/imenik/32646/)
* [old webpage](http://homepages.inf.ed.ac.uk/als/)

## Selected writings

On [[exact real computer arithmetic]]:

* Dave Plume (supervised by [[Martín Escardó]], [[Alex Simpson]]): *A Calculator for Exact Real Number Computation*, University of Edinburgh  (1998) [[web](https://www.dcs.ed.ac.uk/home/mhe/plume/)]

On [[measure theory]], [[probability theory]], and [[locales]]:

* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))

On [[probability theory]] in [[topos theory]]:

* _Probability sheaves_, talk at Topos à l'IHÉS, November 2015 ([YouTube](https://www.youtube.com/watch?v=IMGoluu1mgc))

On [[cartesian closed category|cartesian closed]] [[convenient categories of topological spaces]], such as [[compactly generated topological spaces]]:

* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))

On [[first-order set theory]] and [[categorical logic]]:

* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], *Relating first-order set theories, toposes and categories of classes*, Annals of Pure and Applied Logic **165** 2 (2014) 428-502 [[doi:10.1016/j.apal.2013.06.004](https://doi.org/10.1016/j.apal.2013.06.004)]

## Related $n$Lab entries

* [[Banach-Tarski paradox]]

category: people


Alexander A. Migdal

* [Institute page](https://as.nyu.edu/faculty/alexander-migdal.html)

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

## Selected writings

Early discussion of [[flux tubes]]/[[Wilson lines]] as effective [[strings]] in [[Yang-Mills theory]] ([Gauge/String duality](AdS-CFT+correspondence#PolyakovGaugeStringDualityReferences)):

* {#MakeenkoMigdal81} [[Yuri Makeenko]], [[Alexander A. Migdal]], *Quantum chromodynamics as dynamics of loops*, Nuclear Physics B **188** 2 (1981) 269-316 [doi:10.1016/0550-3213(81)90258-3]

  > "So the [[world sheet]] of [[string]] should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric [[flux tube]] in the [[monopole]] plasma."

On the [[large-N limit]]:

* [[Alexander A. Migdal]], _Loop equations and $1/N$ expansion_, Physics Reports **102** (4) 199-290 (1983) [doi]

category: people

[[!redirects Alexander Migdal]]
[[!redirects Alexander Arkadyevich Migdal]]


Alexander B. Atanasov

[[!redirects Alexander B. Atanosov]]

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

## Selected writings

On the [[geometric Langlands correspondence]] with emphasis on [[Yang-Mills monopoles]]:

* [[Alexander B. Atanasov]], *Magnetic Monopoles, 't Hooft Lines, and the Geometric Langlands Correspondence*, 2018 ([pdf](http://abatanasov.com/Files/Thesis.pdf), [slides](http://abatanasov.com/Files/Thesis%20Presentation.pdf))

category: people

[[!redirects Alexander Atanasov]]
[[!redirects Alex Atanasov]]


Alexander Beilinson

**Alexander (or Sasha) Beĭlinson** is currently a professor at University of Chicago. He was student of [[Yuri Manin]] at Moscow State University, with main works in algebraic geometry. He has made visionary contributions to the study of algebraic cycles,  automorphic forms and L-functions, [[algebraic K-theory]], Hodge theory, [[motive]]s and [[motivic cohomology]]. He conjectured the category of motivic sheaves with remarkable cohomological properties, what is often called the Beĭlinson dream. Some of his works, especially those in collaboration with [[Vladimir Drinfeld|Vladimir Drinfel'd]] are of large importance to mathematical physics, especially on their concept of **[[chiral algebra]]s** which are an approach to a chiral part of the [[conformal field theory]] on a curve, which is a geometric counterpart of the theory of [[vertex operator algebra]]s. In late 1980s Beĭlinson proposed a geometric analogue of the Langlands program, now called [[geometric Langlands]] program, which has been continued in his collaboration with Drinfel'd and also by Ed Frenkel, Dennis Gaitsgory, [[Ivan Mirković]], Kari Vilonen, and more recently taken up by mathematical physics community led by [[Edward Witten|Witten]].

Beĭlinson has substantial contributions to [[geometric representation theory]], which has been revolutionized after two discoveries: his proof (with input from Bernstein) of Kazhdan-Lusztig's connjectures in 1980, and his paper with Bernstein on what is now called [[Beĭlinson-Bernstein localization]] theorem, which lead to influx of the algebraic methods involving algebraic D-modules to representation theory. With Kazhdan, Beĭlinson has used D-modules in the proof of Jantzen's conjecture, where he introduced the notion of [[D-affinity]] and the geometric viewpoint via D-schemes. In the work on Hitchin fibration and Hitchin integrable system (with Drinfel'd) much of technique of algebraic geometry on ind-schemes, including study of D-modules is developed and used. In similar spirit to D-modules, he was also using perverse modules; with [[Ofer Gabber]], Bernstein and Deligne he developed their basic theory including deep and extremely powerful theorem for usage in representation theory, the **decomposition theorem** (see a survey: [pdf](http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf)). On technical side, he also described appropriate gluing procedure for the derived categories of perverse sheaves, involving t-structures.

Beĭlinson has shown a remarkable structure of the bounded derived category of coherent sheaves on projective spaces, and its connections to quivers. This work, together with subsequent work with [[Joseph Bernstein]] and also later works of [[Mikhail Kapranov]] and [[Alexei Bondal]] marked the birth of the [[derived noncommutative algebraic geometry]]. With [[Victor Ginzburg]], [[Yuri Manin|Manin]], [[Wolfgang Soergel]] and others, Beĭlinson introduced a wide picture of "Koszul duality patterns" in [[representation theory]].

His other works concentrated on [[motives]], [[higher regulators]], [[epsilon-factors]], and so on.

* [geometric Langlands homepage](http://www.math.uchicago.edu/~mitya/langlands.html)

* wikipedia: [Alexander Beilinson](http://en.wikipedia.org/wiki/Alexander_Beilinson)

## Selected writings

Introducing the notion of [[perverse sheaves]] (and of [[t-structures]] on [[triangulated categories]]):

* [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], *Faisceaux pervers*, Astérisque **100** (1982) [[ISBN:978-2-85629-878-7](https://smf.emath.fr/publications/faisceaux-pervers), [pdf](https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf), [MR86g:32015](http://www.ams.org/mathscinet-getitem?mr=751966)]

See also:

* {#Beilinson85} [[Alexander Beilinson]] _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))

* {#Beilinson80} [[Alexander Beilinson]], _Higher regulators of curves_, Funct. Anal. Appl. 14 (1980), 116-118, [mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1800&option_lang=eng).

* {#Beilinson87} [[Alexander Beilinson]], _Height pairing between algebraic cycles_, in _K-Theory, Arithmetic and Geometry_, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, [DOI](http://dx.doi.org/10.1007/BFb0078364).
* A. Beilinson, J. Bernstein, _Localisations de $\mathfrak{g}$–modules_, C. R. Acad. Sci. Paris __292__ (1981), 15–18.

* A. A. Beilinson, [[V. Drinfeld]], _[[Chiral Algebras]]_, AMS 2004 (a preprint in various forms since around 1995, cf. [here](http://www.math.uchicago.edu/~mitya/langlands.html)).

* A. A. Beilinson, V. Ginzburg, W. Soergel, _Koszul duality patterns in representation theory_, J. Amer. Math. Soc. __9__ (2): 473–527 (1996).

* A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, _Koszul duality_,  J. Geom. Phys. __5__  (1988),  no. 3, 317--350.

* A. Beilinson, [[Victor Ginzburg|V. Ginzburg]], _[[wall crossing|Wall-crossing]] functors and $D$-modules_, Representation Theory __3__ (electronic), 1--31 (1999)

* A. Beĭlinson, [[J. Bernstein]], _A proof of Jantzen conjectures_, I. M. Gelʹfand Seminar,  1--50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf)

## Related entries

* [[Deligne-Beilinson cohomology]]

* [[Beilinson regulator]]

* [[Beilinson conjecture]]

* [[derived noncommutative algebraic geometry]]

* [[p-adic Hodge theory]]

[[!redirects Beilinson]]
[[!redirects A. A. Beilinson]]
[[!redirects A.A. Beilinson]]
[[!redirects A. Beilinson]]
[[!redirects Sasha Beilinson]]

[[!redirects Beĭlinson]]
[[!redirects A. A. Beĭlinson]]
[[!redirects A. Beĭlinson]]

category: people


Alexander Belavin

* [webpage](http://www.itp.ac.ru/en/persons/belavin-aleksander-abramovich/)

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

## Selected writings

Introducing [[conformal field theory]]:

* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (doi:10.1016/0550-3213(84)90052-X)

On [[gravitational instantons]]:

* {#BelavinBurlankov76} [[Alexander Belavin]], D. Burlankov, _The renormalisable theory of gravitation and the Einstein equations_, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (doi:10.1016/0375-9601(76)90530-2)

## Related $n$Lab entries

* [[conformal bootstrap]], [[operator product expansion]]

category: people


Alexander Belopolsky

* [mathematics genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=112023)

## related entries

* [[picture changing operator]]

* [[differential form on a supermanifold]]

* [[superstring field theory]]

category: people


Alexander Berenbeim

Blog on [[cohesive homotopy type theory]]:

* Alexander Berenbeim, _Explorations of the interesection of Cohesive Homotopy Type Theory, Allegory Theory, and Model Categories_ ([web](http://topostheorist.wordpress.com))

category: people


Alexander Berglund

* [webpage](http://staff.math.su.se/alexb/)

## Selected writings

On [[rational models of mapping spaces]] via [[L-∞ algebra]]:

* [[Alexander Berglund]], _Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras_,  Homology, Homotopy and Applications, Volume 17 (2015) Number 2 ([arXiv:1110.6145](https://arxiv.org/abs/1110.6145), [doi:10.4310/HHA.2015.v17.n2.a16]( http://dx.doi.org/10.4310/HHA.2015.v17.n2.a16))

## Related $n$Lab entries

* [[nilpotent L-∞ algebra]]

category: people


Alexander Bobenko

* [webpage](http://page.math.tu-berlin.de/~bobenko/)

category: people


Alexander Braverman

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

## Selected writings

On a mathematical definition of [[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II_, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 ([arXiv:1601.03586](http://arxiv.org/abs/1601.03586))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))

  (relation to [[Hilbert schemes]])

category: people


Alexander Campbell

Alexander Campbell is an Australian category theorist.

* [webpage](https://acmbl.github.io/)

## Selected writings

On [[gerbe (in nonabelian cohomology)|gerbes]] in [[nonabelian cohomology]] via [[tricategories]]:

* [[Alexander Campbell]], _A higher categorical approach to Giraud's non-abelian cohomology_, PhD thesis, Macquarie University (2016) [[hdl:1959.14/1261186](http://hdl.handle.net/1959.14/1261186)]

On [[double categories]]:

* {#CampbellGreg} [[Alexander Campbell]], *The gregarious model structure for double categories* (2020) [[talk slides pdf](https://acmbl.github.io/greg_slides.pdf)]

On the [[straightening theorem]]:

* [[Alexander Campbell]], *A modular proof of the straightening theorem*, talk at Macquarie University (2020) [[pdf](https://acmbl.github.io/straight_slides.pdf), [[Campbell-ModularProofofStraightening.pdf:file]]]

category: people


Alexander Efimov

[[!redirects Alexander I. Efimov]]

__Alexander I. Efimov__ is a mathematician at Steklov Mathematical Institute in Moscow and has involvement into the laboratory for algebraic geometry ([Efimov page](http://www.hse.ru/en/org/persons/26335030)) which is lead by [[Fedor Bogomolov]].

* _Cohomological Hall algebra of a symmetric quiver_, Compositio Mathematica __148__:4 (2012) 1133-1146 [doi](https://doi.org/10.1112/S0010437X12000152) [arXiv:1103.2736](https://arxiv.org/abs/1103.2736)

* other [arXiv papers](http://arxiv.org/find/math/1/au:+Efimov_A/0/1/0/all/0/1)

[[!redirects Alexander I. Efimov]]


Alexander Engel

* [webpage](http://www.uni-regensburg.de/mathematik/mathematik-engel/)

## Selected writings

On ([[equivariant K-theory|equivariant]]) [[KK-theory]] as the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (infinity,1)-category|stable $(\infty,1)$-category]]:

* {#BunkeEngelLand21} [[Ulrich Bunke]], [[Alexander Engel]], [[Markus Land]], *A stable $\infty$-category for equivariant KK-theory* $[$[arXiv:2102.13372](https://arxiv.org/abs/2102.13372)$]$

## Related entries

* [[bornological coarse space]]

category: people


Alexander Givental

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

## Selected writings

On [[quantum cohomology rings]]:

* [[Alexander Givental]], *A tutorial on Quantum Cohomology* [[pdf](https://math.berkeley.edu/~giventh/papers/lqc.pdf), [[Givental-QuantumCohomologyTutorial.pdf:file]]]

Introducing a [[quantum K-theory]]-analog of [[quantum cohomology rings]]:

* [[Alexander B. Givental]], *On the WDVV-equation in quantum K-theory*, Michigan Math. J. **48** 1 (2000) 295-304 [[arXiv:math/0003158](https://arxiv.org/abs/math/0003158), [doi:10.1307/mmj/1030132720](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-48/issue-1/On-the-WDVV-equation-in-quantum-K-theory/10.1307/mmj/1030132720.full)]

On [[quantum K-theory rings]] for some form of [[equivariant K-theory]]:

* [[Alexander B. Givental]], *Permutation-equivariant quantum K-theory* [[webpage](https://math.berkeley.edu/~giventh/perm/perm.html)]

  *I. Definitions. Elementary K-theory of $\overline{\mathcal{M}}_{0,n}/S_n$* [[arXiv:1508.02690](https://arxiv.org/abs/1508.02690)]

  *II. Fixed point localization* [[arXiv:1508.04374](https://arxiv.org/abs/1508.04374)]

  *III. Lefschetz' formula on $\overline{\mathcal{M}}_{0,n}/S_n$ and adelic characterization* [[arXiv:1508.06697](https://arxiv.org/abs/1508.06697)]

  *IV. $D_q$-modules* [[arXiv:1508.06697](https://arxiv.org/abs/1508.06697)]

  *V. Toric $q$-hypergeometric functions* [[arXiv:1509.03903](https://arxiv.org/abs/1509.03903)]

  *VI. Mirrors* [[arXiv:1509.07852](https://arxiv.org/abs/1509.07852)]

  *VII. General theory* [[arXiv:1510.03076](https://arxiv.org/abs/1510.03076)]

  *VIII. Explicit reconstruction* [[arXiv:1510.06116](https://arxiv.org/abs/1510.06116)]

  *IX. Quantum Hirzebruch-Riemann-Roch in all genera* [[arXiv:1709.03180](https://arxiv.org/abs/1709.03180)]

  *X. Quantum Hirzebruch-Riemann-Roch in genus 0*, SIGMA **16** 031 (2020) [[arXiv:1710.02376](https://arxiv.org/abs/1710.02376), [doi:10.3842/SIGMA.2020.031](https://doi.org/10.3842/SIGMA.2020.031)]

  *XI. Quantum Adams-Riemann-Roch* [[arXiv:1711.04201](https://arxiv.org/abs/1711.04201)]

category: people

[[!redirects Alexander B. Givental]]


Alexander Gnedin

* [institute page](https://www.qmul.ac.uk/maths/profiles/gnedina.html)

* [Mathematics Genealogy page](https://www.qmul.ac.uk/maths/profiles/gnedina.html)

## Selected writings

On [[character of a linear representation|characters]] on the [[symmetric group]] depending only on [[Cayley distance]]:

* {#GnedinGorinKerov11} [[Alexander Gnedin]], [[Vadim Gorin]], [[Sergei Kerov]], *Block characters of the symmetric groups*, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 ([arXiv:1108.5044](https://arxiv.org/abs/1108.5044), [doi:10.1007/s10801-012-0394-9](https://doi.org/10.1007/s10801-012-0394-9))

category: people


Alexander Goncharov

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

category: people


Alexander Gorsky

* [InSpire page](https://inspirehep.net/authors/1007902)

## Selected writings

On [[Seiberg-Witten theory]] in  relation to [[integrable systems]]:

* [[Alexander Gorsky]], [[Igor Krichever]], [[Andrei Marshakov]], [[Andrei Mironov]], [[Andrey Morozov]], *Integrability and Seiberg-Witten Exact Solution*, Phys. Lett. B **355** (1995) 466-474 [[arXiv:hep-th/9505035](https://arxiv.org/abs/hep-th/9505035), doi:10.1016/0370-2693(95)00723-X]

category: people


Alexander Green

* [haskellers page](https://www.haskellers.com/user/alexandersgreen)

* [github page](https://github.com/alexandersgreen)

## Selected writings

Introducing the [[quantum IO monad]]:

* [[Alexander Green]], *The Quantum IO Monad*, Nottingham  (2007) [[pdf](http://drinkupthyzider.co.uk/asg/pdfs/bctcs07.pdf), [[Green-QIOMonad.pdf:file]] ]

* {#AltenkirchGreen10} [[Thorsten Altenkirch]], [[Alexander Green]], *The quantum IO monad*, Ch. 5 of: Simon Gay, Ian Mackie (eds.): *Semantic Techniques in Quantum Computation* (2010) 173-205 [[pdf](http://www.cs.nott.ac.uk/~txa/publ/qio-chapter.pdf), [doi:10.1017/CBO9781139193313.006](https://doi.org/10.1017/CBO9781139193313.006)]

Implementation in [[Haskell]]:

* [[Alexander Green]], *[hackage.haskell.org/package/QIO](https://hackage.haskell.org/package/QIO)*

On [[software verification|formally verified]] [[quantum programming languages]] like [[QML]]:

* [[Alexander Green]], *Towards a formally verified functional quantum programming language* (2010) [[eprints:11457](http://eprints.nottingham.ac.uk/11457)]

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

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

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

category: people


Alexander Grothendieck

The European mathematician __Alexander Grothendieck__ (in French sometimes Alexandre Grothendieck), created a very influential body of work foundational for ([[algebraic geometry|algebraic]]) [[geometry]] but also for modern [[mathematics]] more generally. He is widely regarded as a singularly important figure of 20th century mathematics and his ideas continue to be highly influential in the 21st century.

* [Wikipedia article](http://en.wikipedia.org/wiki/Alexander_Grothendieck)

Initially working on [[topological vector spaces]] and [[analysis]], Grothendieck then made revolutionary advances in [[algebraic geometry]] by developing [[sheaf and topos theory]] and [[abelian sheaf cohomology]] and formulating algebraic geometry in these terms ([[locally ringed spaces]], [[schemes]]). Later [[topos theory]] further developed independently and today serves as the foundation also for other kinds of [[geometry]]. Notably its [[homotopy theory|homotopy theoretic]] refinement to [[higher topos theory]] serves as the foundation for modern [[derived algebraic geometry]].

#Contents#
* table of contents
{:toc}

## Selected writings

Grothendieck's geometric work is documented in texts known as [[EGA]] (with [[Dieudonné]]), an early account [[FGA]], and the many volume account [[SGA]] of the seminars at l'IHÉS, Bures-sur-Yvette, where he was based at the time. (See the [wikipedia article](http://en.wikipedia.org/wiki/Alexander_Grothendieck) for some indication of the story from there until the early 1980s.)

On [[fiber bundles]] and [[nonabelian cohomology|nonabelian]] [[Čech cohomology]]:

* [[Alexander Grothendieck]], *A General Theory of Fibre Spaces With Structure Sheaf*, University of Kansas, Report No. 4 (1955, 1958) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf), [[Grothendieck-FibreSpaces.pdf:file]]]

In the late 1970s and early 1980s Grothendieck wrote several documents that have been of outstanding importance in the origins of the theory that underlies the [[nPOV]].  These include

* _La Longue Marche à travers la Théorie de Galois_ (1600 manuscript pages written between January and June 1981, plus addenda etc. which double its length!) (see [[Long March]] for some discussion of the ideas.)

* _[[Esquisse d'un programme]]_, (January 1984), in which Grothendieck sketches out a vaste programme of research, incorporating many of the ideas from [[Long March]]. A copy is available [here](http://www.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf). It is discussed in brief at [[Esquisse d'un programme|Grothendieck's Esquisse]].

*  _À la poursuite des Champs_ (also entitled ''[[Pursuing Stacks]]'').
It starts with a short (12 page) letter to Quillen, dated 19 Feb. 1983, but then discusses a wide ranging vision of homotopy theory and its applicability to problems in algebraic and arithmetic geometry.

*  _[[Les Dérivateurs]]_ (another 2000 page manuscript taking  up some of the themes in Pursuing Stacks, section 69) Dating from the end of 1990 and the start of 1991.

In the same time he also wrote voluminous intellectual memoirs _Récoltes et Semailles_.

En Guise de Programme [p I](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Guise1.jpg) [p II](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Guise2.jpg), a text written by Grothendieck as a course description while teaching in Montpellier "Introduction à la recherche".

A chronological bibliography of Grothendieck's published mathematical writings ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GrothBiblio.pdf)).

## Texts about Grothendieck

For an account of his work, including some of the work published in the 1980s, see the [English Wikipedia entry](http://en.wikipedia.org/wiki/Alexander_Grothendieck).

The video of a talk by W. Scharlau on his life can be seen [here](http://www.dailymotion.com/video/x8juek_colloque-grothendieck-winfried-scha_tech).

* Reminiscences of Grothendieck, 2007 conversations of [[Sasha Beilinson]], [[Luc Illusie]], [[Vladimir Drinfel'd]], [[Spencer Bloch]],  ([pdf](http://www.math.uchicago.edu/~mitya/langlands/reminiscences1.pdf))

* [[Pierre Cartier]], _Alexander Grothendieck: A Country Known Only by Name_ ([pdf](http://www.ams.org/notices/201504/rnoti-p373.pdf))

* [[Luca Barbieri-Viale]], _Alexander Grothendieck: Enthusiasm and creativity_ ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/barbierieng.pdf))

* [[Pierre Deligne]], _Quelques idées maîtresses de l'œuvre de A Grothendieck_ ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/Mathbiographies/Deligne.pdf))

* [[Luc Illusie]], _Alexandre Grothendieck, le magicien des foncteurs_ ([pdf](http://www.cnrs.fr/insmi/IMG/pdf/Alexandre-Grothendieck.pdf))

* _Une entrevue avec [[Jean Giraud]], à propos d'Alexandre Grothendieck_ ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/giraud.pdf)).

A  recent article in French on Grothendieck is to be found [here](http://images.math.cnrs.fr/Alexandre-Grothendieck.html).

There were two articles on Grothendieck's life and work in the Notices AMS in 2004:

* Allyn Jackson, _Comme Appelé du Néant_, As If Summoned from the Void:
The Life of Alexandre Grothendieck, Part 1, [Notices AMS](http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf)

* Allyn Jackson, _Comme Appelé du Néant_, As If Summoned from the Void:
The Life of Alexandre Grothendieck, Part 2, [Notices AMS](http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf)

Grothendieck obituary in the Notices AMS (Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors):

* "_The obituary begins here with a brief sketch of Grothendieck's life,
followed by a description of some of his most outstanding work in mathematics._" Alexandre Grothendieck 1928–2014, ([Part 1] (http://www.ams.org/journals/notices/201603/rnoti-p242.pdf))

* "_set of reminiscences by some of the many mathematicians who knew Grothendieck and were influenced by him._" Alexandre Grothendieck 1928–2014, ([Part 2](http://www.ams.org/publications/journals/notices/201604/rnoti-p401.pdf)).

## Students

* [[Pierre Gabriel]]
* [[Michel Demazure]]
* [[Jean Giraud]]
* [[Jean-Louis Verdier]]
* [[Monique Hakim]]
* [[Michel Raynaud]]
* [[Jean-Pierre Jouanolou]]
* [[Luc Illusie]]
* [[William Messing]]
* [[Pierre Berthelot]]
* [[Pierre Deligne]]
* [[Michèle Raynaud]]
* [[Neantro Saavedra-Rivano]]
* [[Hamet Seydi]]
* [[Hoàng Xuân Sính]]
* [[Yves Ladegaillerie]]
* [[Marcus Wanderley]]
* [[Carlos Contou-Carrère]]

See _[Mathematics Genealogy for Grothendieck](http://www.genealogy.ams.org/id.php?id=31245)_

## Correspondence

[The Grothendieck-Serre correspondence](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/GS.pdf)

[The Grothendieck-Mumford correspondence](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/AGMumford.pdf)

## External links

[Grothendieck circle](http://www.grothendieckcircle.org/)

[Grothendieck's Angle] (https://www.dropbox.com/s/9u7n6xkwytocogf/grothendieckangle.pdf?dl=0) by G. Aiello

[A. Grothendieck](http://matematicas.unex.es/~navarro/res/) by J. A. Navarro

[A. Grothendieck](https://agrothendieck.github.io/) by M. Carmona

A. Grothendieck, una guía a la obra matemática y filosófica ([pdf](http://matematicas.unex.es/~navarro/res/zalamea.pdf)) por F. Zalamea

## Quotes

On [[K-theory]]:

> The way I first visualized a K-group was as a group of "classes of objects" of an abelian (or more generally, additive) category, such as coherent sheaves on an algebraic variety, or vector bundles, etc. I would presumably have called this group $C(X)$ ($X$ being a variety or any other kind of "space"), $C$ the initial letter of 'class', but my past in functional analysis may have prevented this, as $C(X)$ designates also the space of continuous functions on $X$ (when $X$ is a topological space). Thus, I reverted to $K$ instead of $C$, since my mother tongue is German, Class = Klasse (in German), and the sounds corresponding to $C$ and $K$ are the same.

from Grothendieck's letter to Bruce Magurn, on 9th February 1985, quoted after:

* {#Bak87} A. Bak, Editorial, K-theory 1 (1987), 1 ([doi:10.1007/BF00533984](https://access.portico.org/stable?au=pgg197frtxd))

## Related $n$Lab entries

* [[The Rising Sea]]

* [[Grothendieck group]]

category: people

[[!redirects Grothendieck]]
[[!redirects A. Grothendieck]]
[[!redirects Alexander Grothendieck]]
[[!redirects Alexandre Grothendieck]]


Alexander Haupt

* [webpage](https://www.math.uni-hamburg.de/home/haupt/)

## Selected writings

On [[M-theory on Calabi-Yau 5-folds]] and [[supersymmetric quantum mechanics]]:

* [[Alexander Haupt]], [[Andre Lukas]], [[Kellogg Stelle]], _M-theory on Calabi-Yau Five-Folds_, JHEP 0905:069, 2009 ([arXiv:0810.2685](https://arxiv.org/abs/0810.2685))

category: people


Alexander Jahn

* [Institute page](https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert/people/jahn/index.html)

## Selected writings

Introducing [[Majorana dimer codes]] ([[quantum error correcting codes]] induced by [[tensor networks]] exhibiting [[holographic entanglement entropy]]):

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Majorana dimers and holographic quantum error-correcting code*, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

based on:

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Holography and criticality in matchgate tensor networks*, Science Advances 5, eaaw0092 (2019) ([arXiv:1711.03109](https://arxiv.org/abs/1711.03109), [doi:10.1126/sciadv.aaw0092](https://advances.sciencemag.org/content/5/8/eaaw0092))

Review:

* [[Alexander Jahn]], [[Jens Eisert]], _Holographic tensor network models and quantum error correction: A topical review_ ([arXiv:2102.02619](https://arxiv.org/abs/2102.02619))

Discussion of [[asymptotic boundaries]] of [[hyperbolic space|hyperbolic]] [[tensor networks]] as conformal [[quasicrystals]] (see also at [[AdS/CFT in solid state physics]]):

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Tensor network models of AdS/qCFT_ ([arXiv:2004.04173](https://arxiv.org/abs/2004.04173))

More on holographic [[quantum error correcting codes]]:

* Matthew Steinberg, Sebastian Feld, [[Alexander Jahn]], *Holographic Codes from Hyperinvariant Tensor Networks* [[arXiv:2304.02732](https://arxiv.org/abs/2304.02732)]

category: people


Alexander K. Zvonkin

[website](https://www.labri.fr/perso/zvonkin/)

## Related entries

* [[embedded graph]]
* [[combinatorial map]]
* [[child's drawing]]
* [[chord diagram]]


Alexander Kahle

* [website](http://www.uni-math.gwdg.de/kahle/)

category: people


Alexander Karabegov

* [InSpire page](https://inspirehep.net/authors/2332229)

## Selected writings

On [[Fedosov deformation quantization]] for [[almost Kähler structures]]:

* {#KarabegovSchlichenmaier01} [[Alexander Karabegov]], [[Martin Schlichenmaier]],  _Almost-Kähler deformation quantization_, Letters in Mathematical Physics, **57** 2 (2001) 135–148  [[doi:10.1023/A:1017993513935](https://doi.org/10.1023/A:1017993513935), [arXiv:0102169](https://arxiv.org/abs/math/0102169)]

category: people


Alexander Kirillov

* [webpage](http://www.math.sunysb.edu/~kirillov/)

category: people


Alexander Kupers

[[!redirects Sander Kupers]]

* [website](http://math.harvard.edu/~kupers/)

## Selected writings

On the [[h-principle]] and [[microflexible sheaf|microflexible sheaves]]:

* [[Alexander Kupers]], Section 2 of: *Three applications of delooping to H-principles*, Geom Dedicata **202** (2019) 103–151  ([arXiv:1701.06788](https://arxiv.org/abs/1701.06788), [doi:10.1007/s10711-018-0405-7](https://doi.org/10.1007/s10711-018-0405-7))

## Related pages

* [[string topology]]

category: people


Alexander Kurz

Alexander Kurz is a Professor  in the department of Computer Science at Chapman University, Orange, Southern California.. He has written extensively on the applications of [[coalgebra]] techniques in [[modal logic]].

* [Faculty Webpage](https://www.chapman.edu/engineering/about/faculty/program-faculty/alex-kurz.aspx).

Publication list (from previous post at Leicester University, UK.):

* [Publication List](http://www.cs.le.ac.uk/people/akurz/works.html)

###References

* C. Cirstea, A. Kurz, D. Pattinson, L. Schröder, Y. Venema: Modal Logics are Coalgebraic. [BCS Visions in Computer Science 2008](www.cs.le.ac.uk/people/akurz/Papers/BCS08/ModalCoalg.pdf).

*  A. Kurz :  _Coalgebras and Modal Logic._ Course Notes for ESSLLI 2001, Version of October 2001. Appeared on the CD-Rom ESSLLI'01, Department of Philosophy, University of Helsinki, Finland, available from [site](http://www.cs.le.ac.uk/people/akurz/works.html).

category:people

[[!redirects A. Kurz]]


Alexander Kuznetsov

**Alexander Kuznetsov** is a former student of [[Alexei Bondal]].

* [website](http://www.mi.ras.ru/~akuznet/)

* [[derived categories of coherent sheaves]]
* [[derived noncommutative algebraic geometry]]

category: people


Alexander Körschgen

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

## Selected writings

On [[orbispaces]] in [[global equivariant homotopy theory]]:

* {#Koerschgen16} [[Alexander Körschgen]], _A Comparison of two Models of Orbispaces_, Homology, Homotopy and Applications, vol. 20(1), 2018, pp. 329--358 ([arXiv:1612.04267](https://arxiv.org/abs/1612.04267))

category: people


Alexander L. Fetter

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

* [GoogleScholar page](https://scholar.google.com/citations?user=RKnDQh4AAAAJ)

## Selected writings

A textbook on [[quantum theory]]/[[quantum field theory]] of many-particle systems with application to [[condensed matter theory]]:

* [[Alexander L. Fetter]], [[John Dirk Walecka]], *Quantum theory of many-particle systems*, Mcgraw-Hill (1991); Dover (2003) [[archive.org](https://archive.org/details/quantum-theory-of-many-particle-systems-by-alexander-l.-fetter-john-dirk-walecka-physics-z-lib.org/mode/thumb)]

[[!redirects Alexander Fetter]]

category: people


Alexander Lenz

* [webpage](https://www.ippp.dur.ac.uk/profile/lenz)

* [webpage](https://www.dur.ac.uk/research/directory/staff/?mode=staff&id=10956)

## Selected writings

Experimental constraints on the existence of a [[4th generation of fermions]]:

* {#Lenz13} [[Alexander Lenz]], _Constraints on a fourth generation of fermions from Higgs Boson searches_, Adv. High Energy Phys. 2013 (2013) 910275 ([doi:10.1155/2013/910275](https://doi.org/10.1155/2013/910275))

* [[Alexander Lenz]], _How to really kill a physics model_, talk notes 2012 ([pdf](https://www.ippp.dur.ac.uk/~lenz/Talk_Bern_Lenz.pdf), [[Lenz4thGenerationConstraints.pdf:file]])

category: people


Alexander Merkurjev

* [personal page](https://www.math.ucla.edu/~merkurev/)

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

## Selected writings

On [[quadratic forms]]:

* [[Richard Elman]], [[Nikita Karpenko]], [[Alexander Merkurjev]], *Algebraic and Geometric Theory of Quadratic Forms*, Colloquium Publication **56**, AMS (2008) [[ams:coll-56](https://bookstore.ams.org/coll-56), [pdf](https://www.math.ucla.edu/~rse/old_book/Kniga.pdf)]

category: people


Alexander Odesskii

[[!redirects A. Odesskii]]
[[!redirects Aleksandr Odesskii]]
__Alexander/Aleksandr V. Odesskii__ is a mathematical physicist.

* A. V. Odesskii, B.L. Feigin, _Sklyanin’s elliptic algebras_, Funkc. Anal. i Pril. 23 (3) (1989) 45--54
* A. V. Odesskii, _Elliptic algebras_, Uspekhi Mat. Nauk (2002) 57:6 (348) 87--122 (Russian) [doi](https://doi.org/10.4213/rm573); Engl. transl. Russian Math. Surveys 2002, 57:6, 1127--1162 [doi](https://doi.org/10.1070/RM2002v057n06ABEH000573)
* A. Odesskii, [[V. Rubtsov]], V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935)
*  A. Odesskii, [[V. Rubtsov]], V. Sokolov, _Parameter-dependent associative Yang–Baxter equations and Poisson brackets_, Int. J. Geom. Meth. Mod. Phys. __11__:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 [doi](https://doi.org/10.1142/S0219887814600366)
* A. V. Odesskii, [[V. Rubtsov|V. N. Rubtsov]], V. V. Sokolov, _Bi-hamiltonian ordinary differential equations with matrix variables_, Theor. and Math. Phys. 171(1): 442--447 (2012)
* [[M. Kontsevich]], A. Odesskii, _$p$-Determinants and monodromy of differential operators_, Sel. Math. New Ser. __28__, 52 (2022) [doi](https://doi.org/10.1007/s00029-022-00770-6) [arXiv:2009.12159](https://arxiv.org/abs/2009.12159)
* [[M. Kontsevich]], A. Odesskii, _Multiplication kernels_, Lett. Math. Phys. __111__, 152 (2021) [arxiv:2105.04238](https://arxiv.org/abs/2105.04238) [doi](https://doi.org/10.1007/s11005-021-01491-1)

category: people


Alexander Ostrowski

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

category: people


Alexander Polishchuk

* [Webpage](http://pages.uoregon.edu/apolish/)

## Selected writings

On [[t-structures]] on [[derived categories of coherent sheaves]]:

* [[Dan Abramovich]], [[Alexander Polishchuk]], *Sheaves of t-structures and valuative criteria for stable complexes*, J. reine angew. Math. **590** (2006) 89-130 [[arXiv:math/0309435](https://arxiv.org/abs/math/0309435), [doi:10.1515/CRELLE.2006.005](https://doi.org/10.1515/CRELLE.2006.005)]

* [[Alexander Polishchuk]], *Constant families of t-structures on derived categories of coherent sheaves*, Moscow Math. J. __7__ (2007) 109-134 [[arXiv:math/0606013](https://arxiv.org/abs/math/0606013)]

Discussion of traditional [[algebraic geometry]]  for [[super-schemes]]:

* [[Ugo Bruzzo]], Daniel Hernandez Ruiperez, [[Alexander Polishchuk]], _Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes_ ([arXiv:2008.00700](https://arxiv.org/abs/2008.00700))

category: people