Michael Atiyah

**Sir Michael Atiyah** was a British-Lebanese mathematician, a Fields' medalist, and Abel prize winner (with [[Isadore Singer]]). He was professor of [[mathematics]] at Edinburgh University.

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

* Notices of the AMS, _Memories of Sir Michael Atiyah_ ([pdf](https://www.ams.org/journals/notices/201911/rnoti-p1834.pdf), [[MemoriesOfAtiyah.pdf:file]])

  * [[Bernd Schroers]], _Michael Atiyah and Physics: the Later Years_,  ([arXiv:1910.10630](https://arxiv.org/abs/1910.10630))

* [[Alain Connes]], Joseph Kouneiher, _Sir Michael Atiyah, a Knight Mathematician A tribute to Michael Atiyah, an inspiration and a friend_ ([arxiv:1910.07851](https://arxiv.org/abs/1910.07851))

* [[Nigel Hitchin]], _Sir Michael Atiyah OM. 22 April 1929–11 January 2019_, Biographical Memoirs of the FRS, September 2020 ([doi:10.1098/rsbm.2020.0001](https://doi.org/10.1098/rsbm.2020.0001))

* {#Hitchin20} [[Nigel Hitchin]], _Michael Atiyah: Geometry and Physics_, 2020 ([pdf](https://cmsa.fas.harvard.edu/wp-content/uploads/2021/01/Geometry-and-physics.pdf), [[HitchinOnAtiyahGeometryPhysics.pdf:file]])

## Selected writings

Introducing [[topological K-theory]]:

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Riemann-Roch theorems for differentiable manifolds_, Bull.  Amer. Math Soc. vol. 65 (1959) pp. 276-281 ([euclid:bams/1183523205](https://projecteuclid.org/euclid.bams/1183523205))

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Vector bundles and homogeneous spaces_, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. (doi:10.1142/9789814401319_0008, [web](http://hirzebruch.mpim-bonn.mpg.de/87/), [MR 0139181](http://www.ams.org/mathscinet-getitem?mr=0139181))

*  {#Atiyah67}  [[Michael Atiyah]], _K-theory_, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin 1967 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf), [[AtiyahKTheory.pdf:file]])

Introducing [[cobordism cohomology theory]] [[MSO]]:

* [[Michael Atiyah]], _Bordism and Cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 2, April 1961, pp. 200 - 208 ([doi:10.1017/S0305004100035064](https://doi.org/10.1017/S0305004100035064), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahb.pdf))

Introducing the [[Atiyah-Bott-Shapiro orientation]] [[MSpin]]$\to$[[KO]] and [[MSpin^c|MSpinc]]$\to$[[KU]]:

* {#AtiyahBottShapiro64} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (doi:10.1016/0040-9383(64)90003-5, [pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))

Introducing what came to be known as [[Karoubi K-theory]]:

* [[Michael Atiyah]], [[Isadore Singer]], *Index theory for skew-adjoint Fredholm operators*, Publications Mathématiques de l'IHÉS, Tome **37** (1969) 5-26 $[$[numdam:PMIHES_1969__37__5_0](http://www.numdam.org/item/PMIHES_1969__37__5_0)$]$

On [[Bott periodicity]]:

* [[Michael Atiyah]], _Bott periodicity and the index of elliptic operators_, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 ([doi:10.1093/qmath/19.1.113](https://doi.org/10.1093/qmath/19.1.113))

On [[commutative algebra]]:

* [[Michael Atiyah]], [[Ian G. Macdonald]], _Introduction to commutative algebra_, (1969, 1994) $[$[pdf](http://math.univ-lyon1.fr/~mathieu/CoursM2-2020/AMD-ComAlg.pdf), [ISBN:9780201407518](https://www.routledge.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518)$]$

On the [[KO-theory]] of [[complex projective 3-space]]:

* [[Michael Atiyah]], E. Rees, _Vector bundles on projective 3-space_ Invent. Math.35, 131–153 (1976) ([[AtiyahReesVectorBundlesOnCP3.pdf:file]])

Introducing the [[ADHM construction]] for [[Yang-Mills instantons]]:

* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65**  3 (1978) 185-187  [doi:10.1016/0375-9601(78)90141-X]

On [[quantum anomalies]] via [[index theory]]:

* [[Michael Atiyah]], [[Isadore Singer]], _Dirac operators coupled to vector potentials_, Proc. Nat. Acad. Sci. USA __81__, 2597-2600 (1984) ([doi:10.1073/pnas.81.8.2597]( https://doi.org/10.1073/pnas.81.8.2597), [jstor:23378](https://www.jstor.org/stable/23378))

On [[moment maps]], [[equivariant de Rham cohomology]] and [[equivariant localization]]:

* {#AtiyahBott84} [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology 23, 1 (1984) (doi:10.1016/0040-9383(84)90021-1, [pdf](https://www.math.stonybrook.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf))

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

* [[Michael Atiyah]], _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 ([numdam]( http://www.numdam.org/item?id=PMIHES_1961__9__23_0))

On the [[Atiyah-Hirzebruch spectral sequence]]

* [[Michael Atiyah]], [[Friedrich Hirzebruch]], _Vector bundle and homogeneous spaces_, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 ([[AtiyahHirzebruch61.pdf:file]])

On [[KR-theory]]

* {#Atiyah66} [[Michael Atiyah]], _K-theory and reality_, The Quarterly Journal of Mathematics. Oxford. Second Series 17 (1) (1966),: 367–386, ISSN 0033-5606, MR 0206940 ([doi:10.1093/qmath/17.1.367](https://doi.org/10.1093/qmath/17.1.367), [[AtiyahKReal.pdf:file]])

On [[equivariant K-theory]] and the [[Atiyah-Segal completion theorem]]:

* [[Michael Atiyah]], [[Graeme Segal]], _Equivariant $K$-theory and completion_,  J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18. ([Euclid](http://projecteuclid.org/euclid.jdg/1214428815))

On [[Yang-Mills instantons]]:

* {#AtiyahJones78} [[Michael Atiyah]], [[John David Stuart Jones]], _Topological aspects of Yang-Mills theory_, Comm. Math. Phys. Volume 61, Number 2 (1978), 97-118 ([arXiv:1103904210](https://projecteuclid.org/euclid.cmp/1103904210))

* {#Atiyah84} [[Michael Atiyah]], *Instantons in two and four dimensions*, Commun. Math. Phys. 93, 437–451 (1984) ([doi:10.1007/BF01212288](https://doi.org/10.1007/BF01212288))

On [[Yang-Mills monopoles]]:

* [[Michael Atiyah]], [[Nigel Hitchin]], J. T. Stuart and M. Tabor, _Low-Energy Scattering of Non-Abelian Magnetic Monopoles_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 315, No. 1533, New Developments in the Theory and Application of Solitons (Aug. 13, 1985), pp. 459-469 ([jstor:37546](https://www.jstor.org/stable/37546))

On the [[moduli spaces of Yang-Mills monopoles]]:

* [[Michael Atiyah]], [[Nigel Hitchin]], _The geometry and dynamics of magnetic monopoles_  M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 ([jstor:j.ctt7zv206](https://www.jstor.org/stable/j.ctt7zv206))

On [[knot theory]] and [[Chern-Simons theory]]:

* [[Michael Atiyah]], _The Geometry and Physics of Knots_,  Cambridge University Press 1990 ([doi:10.1017/CBO9780511623868](https://doi.org/10.1017/CBO9780511623868))

On [[skyrmions]] from [[Kaluza-Klein theory|KK-reduction]] of [[instantons]] in [[D=5 Yang-Mills theory]] ([[hadron Kaluza-Klein theory]]):

* {#AtiyahManton89} [[Michael Atiyah]], [[Nicholas Manton]], _Skyrmions from instantons_, Phys.  Lett. B, 222(3) 438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

and in a variant on [[hyperbolic space]]:

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

On the [Clay Millennium Problems](https://www.claymath.org/millennium-problems/)

* {#Atiyah2000} [[Michael Atiyah]], *The millennium problems II*, talk at *[CMI Millennium Event at Collège de France](https://www.msri.org/realvideo/ln/hosted/cmi/2000/cmiparis/index-tate.html)* (24-25 May 2000) [[web](https://www.claymath.org/lectures/the-millennium-prize-problems-ii/), [YT](https://youtu.be/trxjCOmg8GY), discussion of [Yang-Mills&MassGap](mass+gap#ReferencesMassGapProblem) starting at: [29:40](https://youtu.be/trxjCOmg8GY?t=1764)]

On [[M-theory on G2-manifolds]]:

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

Introducing the [[Atiyah-Sutcliffe conjecture]]:

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

* [[Michael Atiyah]], [[Joseph Malkoun]], *The Relativistic Geometry and Dynamics of Electrons*, Found Phys 48, 199–208 (2018) ([doi:10.1007/s10701-018-0139-2](https://doi.org/10.1007/s10701-018-0139-2))

On the [[Arnold-Kuiper-Massey theorem]] and its further generalization to the [[13-sphere]] being an [[Sp(1)]]-quotient of the [[octonionic projective plane]]:

* {#AtiyahBerndt02} [[Michael Atiyah]], [[Jürgen Berndt]], *Projective planes, Severi varieties and spheres*, in:  Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))

On [[twisted K-theory]] and [[twisted equivariant K-theory]]:

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

On [[Raoul Bott]]:

* [[Michael Atiyah]], *[Raoul Harry Bott: 24 September 1923 – 20 December 2005](https://celebratio.org/Bott_R/article/296/)*, Celebratio Mathematica (2007)

On [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]:

* {#AtiyahDijkgraafHitchin10} [[Michael Atiyah]], [[Robbert Dijkgraaf]], [[Nigel Hitchin]], _Geometry and physics_, Phil. Trans. R. Soc. A 13 March 2010 vol. 368 no. 1914 913-926 ([pdf](http://rsta.royalsocietypublishing.org/content/368/1914/913.full.pdf), [doi;10.1098/rsta.2009.0227](https://royalsocietypublishing.org/doi/10.1098/rsta.2009.0227))

On patterns in the [[characteristic classes]] ([[Chern classes]]) of [[complex surfaces]], together with some speculations about an anlogy with [[atomic number]] of [[atomic nuclei]] (inspired by [[hadron Kaluza-Klein theory]]):

* [[Michael Atiyah]], [[Nicholas Manton]], [[Bernd Schroers]], _Geometric Models of Matter_, Proceedings of the Royal Society A  ([arXiv:1108.5151](https://arxiv.org/abs/1108.5151), [doi:10.1098/rspa.2011.0616](https://doi.org/10.1098/rspa.2011.0616))

* {#AtiyahManton16}  [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_,  International Journal of Modern Physics AVol. 33, No. 24, 1830022 (2018) ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816), [doi:10.1142/S0217751X18300223](https://doi.org/10.1142/S0217751X18300223))

* [[Michael Atiyah]], _Geometric Models of Helium_,  Modern Physics Letters AVol. 32, No. 14, 1750079 (2017) ([arXiv:1703.02532](https://arxiv.org/abs/1703.02532), [doi:10.1142/S0217732317500791](https://doi.org/10.1142/S0217732317500791))

On [[twistors]]:

* [[Michael Atiyah]], Maciej Dunajski, Lionel Mason, _Twistor theory at fifty: from contour integrals to twistor strings_,  Proc. R. Soc. A473: 20170530 ([arXiv:1704.07464](https://arxiv.org/abs/1704.07464), [doi:10.1098/rspa.2017.0530](http://dx.doi.org/10.1098/rspa.2017.0530))

## Selected quotes

> The aim of theory really is, to a great extent, that of systematically organizing past experience in such a way that the next generation, our students and their students and so on, will be able to absorb the essential aspects in as painless a way as possible, and this is the only way in which you can go on cumulatively building up any kind of scientific activity without eventually coming to a dead end.

> (M.F. Atiyah, "How research is carried out", Bull. IMA., 10:232–234, 1974)

## Related entries

* [[equivariant de Rham cohomology]]

* [[topological K-theory]]

* [[equivariant K-theory]]

* [[Atiyah-Bott-Shapiro orientation]]

* [[Atiyah-Hirzebruch spectral sequence]]

* [[Atiyah-Singer index theorem]]

* [[Atiyah-Segal completion theorem]]

* [[topological field theory]]

* [[Kaluza-Klein compactification]]

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

* [[configuration space of points]]

* [[instanton]], [[magnetic monopole]], [[Skyrmion]]

* [[moduli space of monopoles]]

* ...many more...

category: people

[[!redirects Michael F. Atiyah]]

[[!redirects M. F. Atiyah]]
[[!redirects M. Atiyah]]
[[!redirects Atiyah]]
[[!redirects Sir Michael Atiyah]]


Michael Bachtold

# Michael Bächtold

I'm interested in nonlinear PDEs from the algebro-geometric point of view, i.e. the language of diffieties and D-schemes, and how this may be used for understanding QFT.

You may reach me at "firstnamelastname" at gmail (where my last name is written without the two dots over the a)

category: people

[[!redirects Michael Bachtold]]
[[!redirects Michael Bächtold]]
[[!redirects Michael+Bächtold]]


Michael Barr

Michael Barr is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University. Although his earlier work was in [[homological algebra]], his principal research area for a number of years has been [[category theory]].

He is on the editorial boards of Mathematical Structures in Computer Science and the electronic journal *Homology, Homotopy and Applications*, and is editor of the electronic journal Theory and Applications of Categories. Michael Barr has much advocated the methods of his late student [[Jon Beck]], involving [[monads]], especially [[monadicity theorem|monadicity]] criteria and [[monadic cohomology]].

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

## Selected writings

See also:

* personal list of publications: [web page](https://www.math.mcgill.ca/barr/)

On [[Hochschild cohomology]] of [[commutative algebras]]:

* [[Michael Barr]], *Cohomology of commutative algebra I*, Ph.D. Thesis, University of Pennsylvania (1962), Retyped with a few corrections and notes (2003) [[pdf](https://www.math.mcgill.ca/barr/papers/thes.pdf), [[Barr-CohomologyThesis.pdf:file]]]

see also

* [[Michael Barr]], *A cohomology theory for commutative algebra I*, Proc. Amer. Math. Soc. **16** (1965) 1379–1384 [[doi:1965-016-06/S0002-9939-1965-0193119-3](https://www.ams.org/journals/proc/1965-016-06/S0002-9939-1965-0193119-3), [pdf](https://www.math.mcgill.ca/barr/papers/comcoh1.pdf), [[Barr-CohomologyCommutativeI.pdf:file]]]

* [[Michael Barr]], *A cohomology theory for commutative algebra II*, Proc. Amer. Math. Soc. **16** (1965) 1385–1391 [[jstor:2035937](https://www.jstor.org/stable/2035937), [pdf](https://www.math.mcgill.ca/barr/papers/comcoh2.pdf), [[Barr-CohomologyCommutativeII.pdf:file]]]

and in relation to [[monadic cohomology]]:

* [[Michael Barr]], *Cohomology in tensored categories*, [[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]], Springer (1966) 344–354 [[doi:10.1007/978-3-642-99902-4_17](https://doi.org/10.1007/978-3-642-99902-4_17), [pdf](https://www.math.mcgill.ca/barr/papers/ctc.pdf), [[Barr-CohomologyTensored.pdf:file]]]

* [[Michael Barr]], *Shukla cohomology and triples*, J. Algebra **5** 2 (1967) 222–231 [doi:10.1016/0021-8693(67)90036-1, [pdf](https://www.math.mcgill.ca/barr/papers/shukla.pdf), [[Barr-ShuklaCohomology.pdf:file]]]

* [[Michael Barr]], *Harrison homology, Hochschild homology and triples* J. Algebra **8** 3 (1968) 314–323 [doi:10.1016/0021-8693(68)90062-8, [pdf](https://www.math.mcgill.ca/barr/papers/harr-hoch.pdf), [[Barr-HarrHochAndTriples.pdf:file]]]

* [[Michael Barr]], *Cohomology and obstructions: commutative algebras*, in: _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 357–374 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf), [pdf](https://www.math.mcgill.ca/barr/papers/obstruct.pdf)]

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On [[cohomology]] of [[algebra|algebraic]]-[[structures]] (such as [[Lie algebra cohomology]]) via [[monads]] ("triples", cf. *[[monadic cohomology]]* and *[[canonical resolution]]*):

* [[Michael Barr]], [[Jon Beck]], _Homology and Standard Constructions_, in: _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 186-248 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

* [[Michael Barr]], *Cartan-Eilenberg cohomology and triples*, J. Pure Applied Algebra **112** 3 (1996) 219–238 [doi:10.1016/0022-4049(95)00138-7, [pdf](https://www.math.mcgill.ca/barr/papers/coho.pdf), [[Barr-CECohomology.pdf:file]]]

* [[Michael Barr]], *Algebraic cohomology: the early days*, in *Galois Theory, Hopf Algebras, and Semiabelian Categories*, Fields Institute Communications **43** (2004) 1–26 [[doi:10.1090/fic/043](https://doi.org/10.1090/fic/043), [pdf](https://www.math.mcgill.ca/barr/papers/algcohom.pdf), [[Barr-AlgebraicCohomology.pdf:file]]]

On [[monadicity|comonadicity]] of free [[algebra over a monad|monadic algebra]]-functors:

* [[Michael Barr]], *Coalgebras in a category of algebras*, in: *Category Theory, Homology Theory and their Applications I*, Lecture Notes in Mathematics **86**, Springer (1969) 1-12 [[doi:10.1007/BFb0079381](https://doi.org/10.1007/BFb0079381), [pdf](https://www.math.mcgill.ca/barr/papers/coalgs.pdf), [[Barr-CoalgInAlg.pdf:file]]]

Introducing [[Barr-exact categories]] and [[regular categories]]:

* {#BarrGrilletvonOsdool} [[Michael Barr]],  [[Pierre Grillet]], [[Donovan van Osdol]], *Exact Categories and Categories of Sheaves*,  Lec. Notes in Math. **236**, Springer (1971) [[doi:10.1007/BFb0058579](https://link.springer.com/book/10.1007/BFb0058579)]

* [[Michael Barr]], _Exact categories_, in: *Exact categories and categories of sheaves*, Springer Lec. Notes in Math. __236__ (1971) 1-120 [[doi:10.1007/BFb0058580](http://dx.doi.org/10.1007/BFb0058580), [pdf](https://www.math.mcgill.ca/barr/papers/exact.pdf), [[Barr-ExactCategories.pdf:file]]]

On [[coalgebras]] over a [[commutative ring]] (cf. [[CocommCoalg]]):

* [[Michael Barr]], *Coalgebras over a commutative ring*, Journal of Algebra **32** 3 (1974) 600-610 [doi:10.1016/0021-8693(74)90161-6, [pdf](https://www.math.mcgill.ca/barr/ftp/pdffiles/coalgebra.pdf), [[Barr-Coalgebras.pdf:file]]]

On [[right exact functors]]:

* [[Michael Barr]], *Right exact functors*, J. Pure Appl. Algebra **4** (1974) 1–8 [doi:10.1016/0022-4049(74)90025-5, [pdf](https://www.math.mcgill.ca/barr/papers/right.exact.pdf), [[Barr-ExactFunctors.pdf:file]]]

On [[topoi]] without [[point of a topos|topos points]]:

* [[Michael Barr]], *Toposes without points*, J. Pure Appl. Algebra **5** (1974) 265-280 [[pdf](https://www.math.mcgill.ca/barr/papers/top.no.pt.pdf), [[Barr-ToposesWithoutPoints.pdf:file]]]

On the abstract construction principle behind [[cartesian closed category|cartesian closed]] [[convenient categories of topological spaces]] (such as [[compactly generated topological spaces]]):

* [[Michael Barr]], *Building closed categories*, [[Cahiers|Cahiers Topologie Géométrie Différentielle]] **19** (1978) 115–129 [[numdam:CTGDC_1978__19_2_115_0](http://www.numdam.org/item/?id=CTGDC_1978__19_2_115_0)]

Introducing [[star-autonomous categories]]:

* [[Michael Barr]], _$\ast$-Autonomous Categories_, Lecture Notes in Mathematics **752**, Springer (1979) [[doi:10.1007/BFb0064579](https://doi.org/10.1007/BFb0064579)]

On [[Galois theory]]:

* [[Michael Barr]], *Abstract Galois Theory*, J. Pure Appl. Algebra **19** (1980) 21–42 [[pdf](https://www.math.mcgill.ca/barr/papers/galois.pdf), [[Barr-AbstractGaloisI.pdf:file]]]

* [[Michael Barr]], *Abstract Galois Theory II*, J. Pure Appl. Algebra **25** 3 (1982) 227–247 \[doi:10.1016/0022-4049(82)90080-9, [pdf](https://www.math.mcgill.ca/barr/papers/galois2.pdf), [[Barr-AbstractGaloisII.pdf:file]]]

On [[atomic toposes]]:

* [[Michael Barr]], [[Radu Diaconescu]], *Atomic Toposes*, J. Pure Appl. Algebra **17** (1980) 1-24 [doi:10.1016/0022-4049(80)90020-1, [pdf](http://www.math.mcgill.ca/barr/papers/atom.top.pdf), [[BarrDiaconescu-Atomictoposes.pdf:file]]]

On [[toposes]], [[monads]] ("triples") and [[algebraic theories]]:

*  [[Michael Barr]], [[Charles Wells]], _[[Toposes, Triples, and Theories]]_, Grundlehren der math. Wissenschaften **278** Springer (1983), [[TAC reprints series|Reprints in Theory and Applications of Categories]] **12** (2005) 1-287 [[ftp](ftp://ftp.math.mcgill.ca/pub/barr/ttt/), [web](http://www.cwru.edu/artsci/math/wells/pub/ttt.html), [pdf](http://www.case.edu/artsci/math/wells/pub/pdf/ttt.pdf), [tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]

Exposition of [[sheaf toposes]]:

* [[Michael Barr]], [[Colin McLarty]], [[Charles Wells]],  *Variable set theory*, prepared for *Scientific American* but unpublished (~1985) [[pdf](https://www.math.mcgill.ca/barr/papers/vst.pdf), [[BarrMcLartyWells-VariableSetTheory.pdf:file]]]

On [[models]] of [[Horn theories]]:

* [[Michael Barr]], *Models of Horn theories*, in: *Categories in Computer Science and Logic*, Contemporary Math. **92**, Amer. Math. Soc. (1989) 1–7 [[pdf](https://www.math.mcgill.ca/barr/papers/horn.pdf), [[Barr-ModelsOfHornTheories.pdf:file]]]

On [[domain theory]] [[internalization|internal to]] [[cartesian closed categories]]:

* [[Michael Barr]], *Fixed points in cartesian closed categories*, Theoretical Comp. Sci. **70** (1990) 65–72 [doi:10.1016/0304-3975(90)90152-8, [pdf](https://www.math.mcgill.ca/barr/papers/dom.pdf), [[Barr-FixedPointsInCCCs.pdf:file]]]

On [[star-autonomous categories]] and the [[Chu construction]] as [[categorical semantics]] for [[linear logic]]:

* [[Michael Barr]], *$\ast$-Autonomous categories and linear logic*, Math. Structures Comp. Sci. **1** 2 (1991) 159–178 [[doi:10.1017/S0960129500001274](https://doi.org/10.1017/S0960129500001274), [pdf](https://www.math.mcgill.ca/barr/papers/scatll.pdf), [[Barr-AutomomousCatAndLinLogic.pdf:file]]]

and via [[accessible categories]]:

* [[Michael Barr]], *Accessible categories and models of linear logic*, J. Pure Appl. Algebra **69** (1990) 219–232 [doi:10.1016/0022-4049(91)90020-3[pdf](https://www.math.mcgill.ca/barr/papers/accll.pdf), [[Barr-AccCatLinearLogic.pdf:file]]]

On [[coalgebras for an endofunctor]]:

* [[Michael Barr]], *Terminal coalgebras for endofunctors on sets*,  Theoretical Comp. Sci. **114** (1993) 299–315 [[pdf](https://www.math.mcgill.ca/barr/papers/trmclg.pdf), [[Barr-TerminalCoalgebras.pdf:file]]]

On [[fuzzy logic]]

in relation to [[topos theory]]:

* [[Michael Barr]], *Fuzzy set theory and topos theory*, Canad. Math. Bull. **29** 4 (1986) 501-508 [[doi:10.4153/CMB-1986-079-9](https://doi.org/10.4153/CMB-1986-079-9), [pdf](https://www.math.mcgill.ca/barr/papers/fuzzy.pdf), [[Barr-FuzzySetToposTheory.pdf:file]]]

and in relation to [[linear logic]]:

* [[Michael Barr]], *Fuzzy models of linear logic*, Math. Structures Comp. Sci. **6** 3 (1996) 301–312 [[doi:10.1017/S096012950000102X](https://doi.org/10.1017/S096012950000102X), [pdf](https://www.math.mcgill.ca/barr/papers/fzymod.pdf), [[Barr-FuzzyLinearLogic.pdf:file]]]

On the [[HSP theorem]]:

* [[Michael Barr]], *HSP type theorems in the category of posets*, in: *Proc. 7th International Conf. Mathematical Foundation of Programming Language Semantics*, Lecture Notes in Computer Science **598** (1992) 221–234 [[doi:10.1007/3-540-55511-0_11](https://doi.org/10.1007/3-540-55511-0_11), [[Barr-HSPinPosets.pdf:file]]]

* [[Michael Barr]], *Functorial semantics and HSP type theorems*, Algebra Universalis **31** (1994) 223–251 [[doi:10.1007/BF01236519](https://doi.org/10.1007/BF01236519), [pdf](https://www.math.mcgill.ca/barr/papers/hsp.pdf), [[Barr-FuncSemanticsHSP.pdf:file]]]

* [[Michael Barr]], *HSP subcategories of Eilenberg-Moore algebras*, Theory Appl. Categories **10** 18 (2002), 461–468 [[tac:10-18](http://www.tac.mta.ca/tac/volumes/10/18/10-18abs.html)]

On [[category theory]] in [[computer science]],
via [[star-autonomous categories]] and [[Chu spaces]]:

* [[Michael Barr]], [[Charles Wells]], *Category theory for computing science*, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories **22** (2012) 1-538 [[pdf](http://www.math.mcgill.ca/barr/papers/ctcs.pdf), [tac:tr22](http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html)]

On [[acyclic objects]] in [[categories of chain complexes]]:

* [[Michael Barr]], *Acyclic models*, Canadian J. Math. **48** 2 (1996) 258–273 [[doi:10.4153/CJM-1996-013-x](https://doi.org/10.4153/CJM-1996-013-x), [pdf](https://www.math.mcgill.ca/barr/papers/acmod.pdf), [[Barr-AcyclicModels1996.pdf:file]]]

* [[Michael Barr]], *Acyclic models*, CRM Monograph Series **17** (2002) [[ams:crmm-17](https://bookstore.ams.org/crmm-17), [pdf](https://www.math.mcgill.ca/barr/papers/acycmod.pdf), [[Barr-AcyclicModels2002.pdf:file]]]

On the [[Chu construction]]:

* [[Michael Barr]], *The Chu construction*, Theory Appl. Categories **2** 2 (1996) 17–35 [[tac:2-02](http://www.tac.mta.ca/tac/volumes/1996/n2/2-02abs.html)]

On the [[Chu construction]] on [[Vect]]:

* [[Michael Barr]], *Separability of tensor in Chu categories of vector spaces*, Math. Structures Comp. Sci. **6** (1996) 213–217 [[doi:10.1017/S0960129500000955](https://doi.org/10.1017/S0960129500000955), [pdf](https://www.math.mcgill.ca/barr/papers/sepvec.pdf), [[Barr-ChuOfVect.pdf:file]]]

On [[sketches]]:

* [[Michael Barr]], *Notes on Sketches*, Technical report of the Electrotechnical Laboratory (computer language section), Tsukuba, Japan (1997) [[pdf](https://www.math.mcgill.ca/barr/papers/sketch.pdf), [[Barr-NotesOnSketches.pdf:file]]]

On [[relational structures]]:

* [[Michael Barr]], *Note on a theorem of Putnam's*, Theory Appl. Categories, **3** 3 (1997) 45–49 [[tac:3-03](http://www.tac.mta.ca/tac/volumes/1997/n3/3-03abs.html)]

On the [[Chu construction]] and [[star-autonomous categories]]:

* [[Michael Barr]], *The separated extensional Chu category* Theory Appl. Categories **4** 6 (1998) 127–137 [[tac:4-06](http://www.tac.mta.ca/tac/volumes/1998/n6/4-06abs.html)]

On [[star-autonomous categories]] of [[unit balls]] in [[Banach spaces]]:

* [[Michael Barr]], [[Heinrich Kleisli]], *Topological balls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]], **40** 1 (1999) 3–20 [[numdam:CTGDC_1999__40_1_3_0](http://www.numdam.org/item/?id=CTGDC_1999__40_1_3_0)]

More on [[star-autonomous categories]]:

* [[Michael Barr]], _$\ast$-Autonomous categories: once more around the track_, Theory and Applications of Categories **6** 1 (1999) 5-24 [[tac:6-01](http://www.tac.mta.ca/tac/volumes/6/n1/6-01abs.html)]

On topological [[star-autonomous categories]]:

* [[Michael Barr]], *Topological $\ast$-autonomous categories*, Theory Appl. Categories, 16 (2006) 700–708 [[pdf](https://www.math.mcgill.ca/barr/papers/mack.pdf), [[Barr-TopologicalStarAutonomous.pdf:file]]]

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On $\ast$-autonomous categories of topological modules*, Theory Appl. Categories, **24** 14 (2010) 278–293 [[tac:24-14](http://www.tac.mta.ca/tac/volumes/24/14/24-14abs.html)]

* [[Michael Barr]], *Topological $\ast$-autonomous categories, revisited*, Tbilisi Math. J. **10** 3 (2017) 51–64 [[arXiv:1609.04241](https://arxiv.org/abs/1609.04241)]

English translation of [[Alexander Grothendieck]]'s [[Tohoku]]-article on [[homological algebra]]:

* Marcia L. Barr & [[Michael Barr]] (2011), Engl. translation of: [[Alexander Grothendieck]], *Some aspects of homological algebra* [[pdf](https://www.math.mcgill.ca/barr/papers/gk.pdf), [[BarrTranslOf-GrothedieckTohoku.pdf:file]]]

On an improved context for [[Pontrjagin duality]]:

* [[Michael Barr]], *On duality of topological abelian groups* [[pdf](http://www.math.mcgill.ca/barr/ftp/pdffiles/abgp.pdf), [[Barr_PontrjaginDuality.pdf:file]]]

  > Did you know that there is a [[star-autonomous category|*-autonomous category]] of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin?  The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology.  It is not obvious that such a finest exists, but it does and that is the key.

On the [[Chu construction]]:

* [[Michael Barr]], *The Chu construction: history of an idea*, TAC **17** 1 (2006) 10-16 [[tac:17-01](http://www.tac.mta.ca/tac/volumes/17/1/17-01abs.html), [pdf](http://www.math.mcgill.ca/barr/papers/chu-hist.pdf)]

On [[product topological space|product spaces]] with [[Lindelöf topological spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On productively  Lindelöf spaces*, Scient. Math. Japon. **65** (2007) 23–36 [[pdf](https://www.math.mcgill.ca/barr/papers/alster.pdf), [[BKR-ProductivelyLindeloef.pdf:file]]]

On [[computability]] and [[formal languages]]:

* [[Joachim Lambek]] (with [[Michael Barr]]), *Programs, Grammars, Arguments*, lecture notes (2007) [[pdf](https://www.math.mcgill.ca/barr/papers/pga.pdf), [[LambekBarr-ProgramsGrammarArguments.pdf:file]]]

On [[Isbell duality]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Isbell Duality* Theory and Applications of Categories **20** 15 (2008) 504-542 [[tac:20-15](http://www.tac.mta.ca/tac/volumes/20/15/20-15abs.html)]

and specifically for [[modules]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Isbell duality for for modules*, Theory and Applications of Categories **22** 17 (2009) 401-419 [[tac:22-17](http://www.tac.mta.ca/tac/volumes/22/17/22-17abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/rmod.pdf)]

On discrete [[dynamical systems]] (here called "flows") on [[compact Hausdorff spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Flows: cocyclic and almost cocyclic*, Theory Appl. Categories **25** 18 (2010) 490–507 [[tac:25-18](http://www.tac.mta.ca/tac/volumes/25/18/25-18abs.html)]

On [[coherent spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Countable meets in coherent spaces with applications to the cyclic spectrum*, Theory Appl. Categories **25** 19 (2011), 508–232 [[tac:25-19](http://www.tac.mta.ca/tac/volumes/25/19/25-19abs.html)]

On [[injective hulls]] of [[partial order|partially ordered]] [[monoids]]:

* [[Joachim Lambek]], [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Injective hulls of partially ordered monoids*, Theory Appl. Categories **26** 13 (2012) 338–348 [[tac:16-13](http://www.tac.mta.ca/tac/volumes/26/13/26-13abs.html)]

On [[star-autonomous categories]] of [[suplattice|sup]]-[[semilattices]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *The $\ast$-autonomous category of uniform sup semi-lattices*,  Theory Appl. Categories **27** (2012) 222–241 [[tac:27-11](http://www.tac.mta.ca/tac/volumes/27/11/27-11abs.html)]

On [[limits]] of [[integral domains]] among [[commutative rings]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Limit closures of classes of commutative rings*, Theory Appl. Categories **30** (2015) 229–304 [[tac:30-08](http://www.tac.mta.ca/tac/volumes/30/8/30-08abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/domrefl.pdf)]

On [[limits]] of [[integral domains]] among [[commutative rings]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Limit closures of classes of commutative rings*, Theory Appl. Categories **30** (2015) 229–304 [[tac:30-08](http://www.tac.mta.ca/tac/volumes/30/8/30-08abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/domrefl.pdf)]

On [[reflective subcategory|reflective]] and [[coreflective subcategories]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On reflective and coreflective hulls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]] **56** (2015) 162–208 [[pdf](https://www.math.mcgill.ca/barr/papers/biref.pdf), [[BarrKennisonRaphael-Reflective.pdf:file]]]

On contractibility of [[simplicial objects]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Contractible simplicial objects*, Comm. Math. Univ. Carol. **60** 4 (2019) 473–495 [[pdf](https://math.mcgill.ca/barr/papers/cso.pdf), [eudml:295068](https://eudml.org/doc/295068)]

On [[coequalizers]] and [[free monads]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Coequalizers and free triples, II*, Theory Appl. Categories **34** (2019) 662–683 [[tac:34-23](http://www.tac.mta.ca/tac/volumes/34/23/34-23abs.html)]

## Related entries

*  [[Barr-exact category]]

*  [[Barr embedding theorem]]

category: people

[[!redirects Michael Barr]]
[[!redirects Mike Barr]]
[[!redirects M. Barr]]
[[!redirects Barr]]


Michael Barratt

Michael G. Barratt, Professor of Mathematics from 1974-99 at the Northwestern University, Illinois, and Professor of Pure Mathematics from 1964-74 at the University of Manchester. He was a student of [[Henry Whitehead]].  His research interests were in algebraic topology.

* [Mathematics genealogy page](http://www.genealogy.ams.org/id.php?id=12765)

* [obituary](https://lists.lehigh.edu/pipermail/algtop-l/2015q4/002298.html) by Bill Richter

His  students included [[Ronnie Brown]] and [[Peter Eccles]].

## Selected writings

On [[twisting functions]] and ([[semisimplicial set|semi]]-)[[simplicial fiber bundles]]:

* [[Michael G. Barratt]], [[Victor K.A.M. Gugenheim]], [[John C. Moore]], _On semisimplicial fibre-bundles_, Amer. J. Math. __81__ (1959) 639-657 [[doi:10.2307/2372920](https://doi.org/10.2307/2372920), [jstor:2372920](https://www.jstor.org/stable/2372920), MR0111028]

On the [[group completion theorem]]:

* [[Michael Barratt]], [[Stewart Priddy]], _On the homology of non-connected monoids and their associated groups_, Commentarii Mathematici Helvetici (1972) 47: 1 ([doi:10.1007/BF02566785](https://doi.org/10.1007/BF02566785))

## Related entries

* [[Barratt-Eccles operad]]

category:people

[[!redirects M. Barratt]]
[[!redirects Michael G. Barratt]]


Michael Batanin

* [website](http://www.ics.mq.edu.au/gen/person/mbatanin.html)

* [articles](http://web.science.mq.edu.au/~mbatanin/papers.html)

* Baez on Batanin's work in *This Week Finds*, [Week 258](http://math.ucr.edu/home/baez/week258.html)

## Selected writings

On [[monoidal enriched categories]]:

* {#BataninMarkl12} [[Michael Batanin]], [[Martin Markl]], Section 2 of *Centers and homotopy centers in enriched monoidal categories*, Advances in Mathematics **230** 4–6 (2012) 1811-1858 [[doi:10.1016/j.aim.2012.04.011](https://doi.org/10.1016/j.aim.2012.04.011), [arXiv:1109.4084](https://arxiv.org/abs/1109.4084)]

Proving the [[biequivalence]] between [[Feynman categories]] and [[colored operads]]:

* {#BataninKockWeber15} [[Michael Batanin]], [[Joachim Kock]], [[Mark Weber]], _Regular patterns, substitudes, Feynman categories and operads_, Theory Appl. Categ. 33 (2018), 148--192 ([arXiv:1510.08934](https://arxiv.org/abs/1510.08934))

## Related $n$Lab entries

* [[Batanin ∞-category]]

* [[globular operad]]

[[!redirects M. Batanin]]
[[!redirects Batanin]]

category: people


Michael Ben-Or

* [personal page](https://www.cs.huji.ac.il/w~benor/)

* [GoogleScholar page](https://scholar.google.com/citations?user=22dxhNQAAAAJ&hl=en)

## Selected writings

Mathematical argument that [[quantum computation]] with [[quantum error correction]] is practically feasible:

* [[Dorit Aharonov]], [[Michael Ben-Or]], *Fault-Tolerant Quantum Computation With Constant Error Rate*, SIAM J. Comput., **38** 4 (2008) 1207–1282 [[arXiv:quant-ph/9906129](https://arxiv.org/abs/quant-ph/9906129), [doi:10.1007/978-3-642-32512-0_31](https://doi.org/10.1007/978-3-642-32512-0_31)]

category: people


Michael Berry

* Wikipedia entry

## Selected writings

Introducing the [[Berry phase]]:

* {#Berry84} [[Michael V. Berry]], *Quantal phase factors accompanying adiabatic changes*, Proc. R. Soc. Lond. A **392** (1984) 45–57 ([doi:10.1098/rspa.1984.0023](https://doi.org/10.1098/rspa.1984.0023), [jstor:2397741](https://www.jstor.org/stable/2397741))

* [[Michael Berry]], *The quantum phase, five years after*, in: *Geometric phases in physics*,  Adv. Ser. Math. Phys. **5**, World Scientific (1989) 7--28 ([[Berry-FiveYearsAfter.pdf:file]], [doi:10.1142/0613](https://doi.org/10.1142/0613))

On a geometric proof of the [[spin-statistics theorem]] (leading to the [[Atiyah-Sutcliffe conjecture]]):

* {#BerryRobbins97} [[Michael V. Berry]], [[Jonathan M. Robbins]], *Indistinguishability for quantum particles: spin, statistics and the geometric phase*, Proceedings of the Royal Society A **453** 1963 (1997) 1771-1790 ([doi:10.1098/rspa.1997.0096](https://doi.org/10.1098/rspa.1997.0096))

On [[asymptotic series]]:

* Michael Berry, C. J. Howls, _Hyperasymptotics_,  The Royal Society Proceedings: Mathematical and Physical Sciences __430__:1880 (1990) 653--668 [jstor](https://www.jstor.org/stable/79960)

* Michael Berry, _Asymptotics, superasymptotics, hyperasymptotics_, in  H. Segur, S. Tanveer, and H. Levine, (eds.)  _Asymptotics Beyond All Orders_,  Plenum, Amsterdam, 1991, 1--14 [doi](https://doi.org/10.1007/978-1-4757-0435-8_1)

## Related $n$Lab entries

* [[Berry phase]]

* [[spin-statistics theorem]]

* [[asymptotic series]]

category: people

[[!redirects Michael V. Berry]]


Michael Bershadsky

* [inspire page](https://inspirehep.net/authors/1016550)

## Selected writings

On how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]]:

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

category: people


Michael Boardman

**John Michael Boardman** (1938-2021) was a British-born mathematician, specialised in [[algebraic topology]] and [[differential topology]]. He received his Ph.D. from the University of Cambridge in 1964. His thesis advisor was [[C. T. C. Wall]]. He was formerly at the University of Cambridge, England; later a full professor at Johns Hopkins University in Baltimore, USA.

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

* [JH University's obituary](https://hub.jhu.edu/2021/03/19/michael-boardman-obituary/)

* [homepage](http://www.math.jhu.edu/~jmb/)

## Selected writings

Boardman introduced the [[stable homotopy theory|stable homotopy category]] in 1969. His notes on this subject were never formally published, but [[Rainer Vogt]] gave a course on this subject in Aarhus in 1969:

* [course notes](http://openlibrary.org/b/OL5743316M/Boardman%27s_stable_homotopy_category)

He coauthored with [[Vogt]] the famous book

* [[Michael Boardman]], [[Rainer Vogt]], *Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics **347** Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]

This book introduced the notion of *[[weak Kan complex]]* that was later popularized by [[André Joyal]] under the name *[[quasi-categories]]* as a natural basis for the  [[higher category theory]] of [[(∞,1)-category|(∞,1)-categories]].

See also: _Homotopy invariant algebraic structures_: _a conference in honor of J. Michael Boardman_ edited by Jean-Pierre Meyer, [[Jack Morava]], and [[W. Stephen Wilson]]. AMS, 1999, CONM/239.

Introducing the [[Boardman homomorphism]]:

* [[J. Michael Boardman]], _The eightfold way to BP-operations_, in: _Current trends in algebraic topology_, pp. 187–226, Canadian Mathematical Society Proceedings, 2, Part 1.  Providence 1982 ([ISBN:978-0-8218-6003-8](https://bookstore.ams.org/cmsams-2))

On stable [[cohomology operations]], but mostly reviewing the background of [[stable homotopy theory]], [[Whitehead-generalized cohomology theory]], [[ring spectra]], [[multiplicative cohomology theory]], [[complex oriented cohomology]] and the examples of [[KU]], [[MU]], [[BP]]:

* [[John Michael Boardman]], _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

On unstable [[cohomology operations]] (hence: in [[non-abelian cohomology]]):

* [[John Michael Boardman]], [[David Copeland Johnson]], [[W. Stephen Wilson]], _Unstable Operations in Generalized Cohomology_ ([pdf](https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf)), in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

## Related entries

* [[stable homotopy category]], [[smash product of spectra]]

* [[Boardman homomorphism]]

* [[quasi-category]]

* [[Boardman-Vogt resolution]]

* [[Boardman-Vogt tensor product]]

## References

Some of the above material is taken from [[joyalscatlab:Michael Boardman|Joyal's CatLab -- Michael Boardman]].

* Tribute to Mike Boardman from [[Peter May]], [web](http://hopf.math.purdue.edu/May/boardman.txt).

and Rainer Vogt's article (with self explanatory title):

* {#VogtMemory} [[Rainer Vogt]], _My time as Mike Boardman's student and our work on infinite loop spaces_ ([[VogtMemory.pdf:file]])

category:people

[[!redirects Michael Boardman]]
[[!redirects Boardman]]
[[!redirects J. M. Boardman]]
[[!redirects John Michael Boardman]]
[[!redirects J. Michael Boardman]]


Michael C. Crabb

## Selected writings

On [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant homotopy theory]] (such as concerning [[KR-theory]] and [[topological Hermitian K-theory]]):

* {#Crabb80} [[Michael C. Crabb]], *$\mathbb{Z}/2$-Homotopy Theory*, Lond. Math. Soc. Lecture Notes **44**, Cambridge University Press (1980) [[ark:/13960/t3jx7bg4w](https://archive.org/details/zz2homotopytheor0000crab), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabb.pdf)]

On [[parameterized homotopy theory]]:

* [[Michael C. Crabb]], [[Ioan Mackenzie James]]: *Fiberwise homotopy theory*, Springer Monographs in Mathematics, Springer (1998) $[$[doi:10.1007/978-1-4471-1265-5](https://doi.org/10.1007/978-1-4471-1265-5), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabbjames.pdf) ,[pdf](https://www.gbv.de/dms/ilmenau/toc/243540361.PDF)$]$

On [[surgery theory]] in relation to [[Hopf invariants]]:

* [[Michael Crabb]], [[Andrew Ranicki]], *The Geometric Hopf Invariant and Surgery Theory*,  Springer Monographs in Mathematics, Springer (2017) $[$[doi:10.1007/978-3-319-71306-9](https://doi.org/10.1007/978-3-319-71306-9), [slides pdf](https://www.maths.ed.ac.uk/~v1ranick/slides/hopfbeam.pdf), [[CrabbRanicki-GeometricHopf.pdf:file]]$]$

category: people

[[!redirects Michael Crabb]]
[[!redirects Michael Charles Crabb]]


Michael C. McCord

Michael Campbell McCord was an American mathematician probably best known for his work on [[finite topological spaces]] and on the [[nerve theorem]].

## References

On [[homotopy groups]] and [[weak homotopy equivalences]] of [[finite topological spaces]]:

* {#McCord} [[Michael C. McCord]], _Singular homology groups and homotopy groups of finite topological spaces_, Duke Math. J. Volume 33, Number 3 (1966), 465-474. ([euclid.dmj/1077376525](http://projecteuclid.org/euclid.dmj/1077376525))

On the [[nerve theorem]]:

* {#McCord67} [[Michael C. McCord]], _Homotopy type comparison of a space with complexes associated with its  open covers_, Proc. Amer. Math. Soc. 18 (1967), 705–708 ([jstor:2035443](https://www.jstor.org/stable/2035443))

On the construction of [[classifying spaces]] using (and introducing) [[compactly generated weakly Hausdorff topological spaces]]:

* [[Michael C. McCord]], *Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298  ([jstor:1995173](https://www.jstor.org/stable/1995173), [pdf](https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf))

## related $n$Lab entries

* [[Quillen's theorem A]]

category:people

[[!redirects Michael McCord]]


Michael Ching

* [webpage](https://www3.amherst.edu/~mching/)

## Selected writings

On [[Goodwillie calculus]]:

* {#AroneChing19} [[Gregory Arone]], [[Michael Ching]], *Goodwillie Calculus*, in: *[[Handbook of Homotopy Theory]]* Taylor & Francis (2019) [[arXiv:1902.00803](https://arxiv.org/abs/1902.00803),[doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]

On an extension of the concept of [[tangent bundle categories]] to [[(infinity,1)-categories|$(\infty, 1)$-categories]]:

* {#BauerBurkeChing21} [[Kristine Bauer]], [[Matthew Burke]], [[Michael Ching]], _Tangent $\infty$-categories and Goodwillie calculus_ [[arXiv:2101.07819](https://arxiv.org/abs/2101.07819)]

Two further examples of tangent structures on an $(\infty,1)$-category, on [[(∞,1)-Topos]] and its [[opposite (infinity,1)-category|opposite]], are given in:

* [[Michael Ching]], _Dual tangent structures for infinity-toposes_ [[arXiv:2101.08805](https://arxiv.org/abs/2101.08805)]

## Related entries

* [[Goodwillie calculus]]

* [[Goodwillie-Taylor tower]]

* [[Goodwillie derivatives of the identity functor]]

* [[configuration space (mathematics)]]

* [[L-infinity operad]]

category: people


Michael Cole

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=9370)

## Selected writings

On [[equivariant complex oriented cohomology theory]] and [[equivariant formal group laws]]:

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _Equivariant Formal Group Laws_, Proceedings of the LMS, Volume81, Issue2 (2000)([doi:10.1112/S0024611500012466]( https://doi.org/10.1112/S0024611500012466))

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _The universality of equivariant complex bordism_, Math Z 239, 455–475 (2002) ([doi:10.1007/s002090100315](https://doi.org/10.1007/s002090100315))

category: people


Michael Creutz

* [webpage](http://thy.phy.bnl.gov/~creutz/)

## Selected writings

Introducing [[Monte Carlo methods]] to [[lattice field theory]] and in particular to lattice [[QCD]] and the [[confinement]]-problem:

* {#Creutz79} [[Michael Creutz]], *Confinement and the Critical Dimensionality of Space-Time*, Phys. Rev. Lett. **43**  (1979) 553 [[doi:10.1103/PhysRevLett.43.553](https://doi.org/10.1103/PhysRevLett.43.553)]

* [[Michael Creutz]], _Monte Carlo study of quantized $SU(2)$ gauge theory_, Phys. Rev. D **21** (1980) 2308-2315 [[doi:10.1103/PhysRevD.21.2308](https://doi.org/10.1103/PhysRevD.21.2308), [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub037.pdf)]

* [[Michael Creutz]], *Monte Carlo study of renormalization in lattice gauge theory* Phys.Rev. D **23** (1981) 1815  [[doi:10.1103/PhysRevD.23.1815](https://doi.org/10.1103/PhysRevD.23.1815), [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub045.pdf)&rbrack

* [[Michael Creutz]], Laurence Jacobs, Claudio Rebbi, *Monte Carlo computations in lattice gauge theories*, Physics Reports **95** 4 (1983) 201-282 [doi:10.1016/0370-1573(83)90016-9, [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub068.pdf)]

## Related pages

* [[lattice gauge theory]]

category: people


Michael Dine

* [webpage](http://scipp.ucsc.edu/~dine/)

## Selected writings

On [[duality in string theory]]:

* [[Michael Dine]], _String Theory Dualities_ [[arXiv:hep-th/9609051](https://arxiv.org/abs/hep-th/9609051)]

On [[supersymmetry]] and [[string theory]] beyond the [[standard model of particle physics]] ([[MSSM]]):

* [[Michael Dine]], _Supersymmetry and string theory: beyond the standard model_, Cambridge University Press (2006, 2007) [[ISBN:9781009290920](https://www.cambridge.org/ae/academic/subjects/physics/particle-physics-and-nuclear-physics/supersymmetry-and-string-theory-beyond-standard-model-2nd-edition?format=HB), [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/dine.pdf)]

On [[confinement]] in [[super Yang-Mills theory]] and in plain [[Yang-Mills theory]] ([[QCD]]):

* [[Michael Dine]], *On the Possibility of Demonstrating Confinement in Non-Supersymmetric Theories by Deforming Confining Supersymmetric Theories* [[arXiv:2211.17134](https://arxiv.org/abs/2211.17134)]

category: people


Michael Douglas

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

## Selected writings

On the [[CP^N sigma-model|$\mathbb{C}P^N$ sigma-model]] and its [[quantum cohomology ring]]:

* {#BourdeauDouglas94} [[M. F. Bourdeau]], [[Michael R. Douglas]], *Topological-antitopological fusion and the large $N$ $\mathbb{C}P^N$ model*, Nucl. Phys. B **420** (1994) 243-267 [[doi:hep-th/9312095](https://arxiv.org/abs/hep-th/9312095), doi:10.1016/0550-3213(94)90380-8]

On the [[AdS/CFT correspondence]]:

* [[Tom Banks]], [[Michael Douglas]], [[Gary Horowitz]], [[Emil Martinec]], _AdS Dynamics from Conformal Field Theory_ ([arXiv:hep-th/9808016](https://arxiv.org/abs/hep-th/9808016), [spire:474214](http://inspirehep.net/record/474214))

On [[Yang-Mills instantons]] as [[Dp-D(p+4) brane bound states]]:

* [[Michael Douglas]], _Branes within branes_, In: Baulieu L., Di Francesco P., Douglas M., Kazakov V., Picco M., Windey P. (eds.) _[Strings, Branes and Dualities](https://link.springer.com/book/10.1007/978-94-011-4730-9)_ NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 520. Springer, Dordrecht ([arxiv:hep-th/9512077](https://arxiv.org/abs/hep-th/9512077), [doi:10.1007/978-94-011-4730-9_10](https://doi.org/10.1007/978-94-011-4730-9_10))

* [[Michael Douglas]], _Gauge fields and D-branes_, J. Geom. Phys. 28 (1998) 255-262 ([arXiv:hep-th/9604198](https://arxiv.org/abs/hep-th/9604198))

On [[2d QCD]]:

* [[Michael Douglas]], Keke Li, Matthias Staudacher, _Generalized two-dimensional QCD_ ([arXiv:hep-th/9401062](https://arxiv.org/abs/hep-th/9401062))

On [[chiral fermions]] in [[intersecting D-brane models]]:

* {#BerkoozDouglasLeigh96} [[Micha Berkooz]], [[Michael Douglas]], [[Robert Leigh]], _Branes intersecting at angles_, Nucl. Phys. B480:265-278, 1996 ([arXiv:hep-th/9606139](https://arxiv.org/abs/hep-th/9606139))

On the [[mass gap]]-problem in [[Yang-Mills theory]]:

* {#Douglas04} [[Michael Douglas]], _Report on the Status of the Yang-Mills Millennium Prize Problem_ (2004) [[[Douglas-StatusOfYMMillenniumProb.pdf:file]]]

On [[F-theory on K3]]:

* {#DouglasParkSchnell14} [[Michael Douglas]], Daniel S. Park, Christian Schnell, _The Cremmer-Scherk Mmechanism in F-theory compactifications on K3 manifolds_, JHEP05 (2014) 135 ([arXiv:1403.1595](https://arxiv.org/abs/1403.1595))

On [[machine learning|large language models]]:

* Michael R. Douglas, _Large language models_, [arXiv:2307.05782](https://arxiv.org/abs/2307.05782)

On [[KK-compactification]] in [[string theory]]:

* [[Michael R. Douglas]], [[Liam McAllister]], *Compactification of Superstring Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2310.20118](https://arxiv.org/abs/2310.20118)]

category: people

[[!redirects Michael R. Douglas]]


Michael Duff

\linebreak

Michael Duff is professor of theoretical physics at Imperial College London. $\;\;$

He made foundational contributions to [[string theory]] and [[M-theory]]. $\;\;$

* [personal webpage](http://michaelduff.weebly.com)

* [institute webpage](http://www.imperial.ac.uk/AP/faces/pages/read/Home.jsp?person=m.duff&_adf.ctrl-state=6bq6bx4cv_3&_afrRedirect=2848555508000)

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

Autobiographical notes around encountering [[Chris Isham]]:

* [[Michael Duff]], *Chris Isham: mentor, colleague, friend* ([arXiv:2112.13722](https://arxiv.org/abs/2112.13722))

See also:

* [[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)]

\linebreak

## Students

Among Duff's former students is [[Hisham Sati]].

\linebreak

## Intervies

* Interview by
Graham Fermelo, _The universe speaks in numbers -- Interview 14_ ([web](https://grahamfarmelo.com/the-universe-speaks-in-numbers-interview-14/))

* Interview by [[Hisham Sati]], [[Urs Schreiber]], at _[[M-Theory and Mathematics]]_ 2020:

  see: _[[Duff interview at M-Theory-Mathematics 2020]]_

## Quotes
 {#Quotes}

On the open problem of formulating [[M-theory]]:

\linebreak

[Duff 96, totality of Section 6](#Duff96):

> The overriding problem in superunification in the coming years will be to take the Mystery out of M-theory, while keeping the Magic and the Membranes.

\linebreak

[Duff 98a, last paragraph (p. 6)](#Duff98):

> Despite all these successes, physicists are glimpsing only small corners of M-theory; the big picture is still lacking.

> $[...]$

> Indeed future historians may judge the late 20th century as a time when theorists were like  children  playing  on  the  seashore, diverting themselves with the smoother pebbles or prettier shells of superstrings while the great ocean of M-theory lay undiscovered before them.

\linebreak

[Duff 98b, p. 6](#Duff98b):

> we are only just beginning to scratch the surface of the ultimate meaning of M-theory, and for the time being therefore, M stands for Magic and Mystery too.

\linebreak

[Duff 99, p.330](#Duff99):

> future historians may judge the period 1984-95 as a time when theorists were like boys playing by the sea shore, and diverting themselves with the smoother pebbles or prettier shells of perturbative ten-dimensiorial superstrings while the great ocean of non-perturbative eleven-dimensional M-theory lay all undiscovered before them.

\linebreak

Interview with [[Mike Duff]] by Graham Fermelo, _The universe speaks in numbers -- Interview 14_ ([web](https://grahamfarmelo.com/the-universe-speaks-in-numbers-interview-14/)):

> {#WeDontKnowWhatItIs} (17:04) The problem we face is that we have a patchwork understanding of M-theory, like a quilt. We understand this corner and that corner, but what's lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don't know what it is.

> In a certain sense, and this is not a popular statement, I think it's premature to be asking: "What are the empirical consequences", because it's not yet in a mature enough state, where we can sensibly make falsifiable prediction.

\linebreak

{#Duff20} [[Duff interview at M-Theory-Mathematics 2020]]:

> (06:59) But the matrix model itself was not all of M-theory; it was a corner of M-theory, and it told us certain interesting things, but there were interesting things about M-theory that it didn't tell us.

> (07:13) I think we are still looking, in fact, for what M-theory really is.

> (07:19) We have a patchwork picture. We understand various corners. But the overarching big picture of M-theory is still waiting to be discovered, in my view.

> (08:12) M-theory in 1995 was very promising, and it's taught us a lot about the fundamental interactions; but the final theory is still not with us.

> (12:46) I wouldn't like to predict what the ultimate picture of M-theory will be; I imagine it will be something quite different from what we can imagine now.

> (16:36) That's why I think M-theory is not yet in a mature enough stage for us to make falsifiable predictions.

> (16:44) We don't understand the theory sufficiently well yet to do so.

## Selected writing

On [[KK-compactification]] of [[11-dimensional supergravity]] on a [[squashed sphere|squashed]] [[7-sphere]] with [[G2-holonomy]] ([[M-theory on G2-manifolds]]):

* {#AwadaDuffPope83} M. A. Awada, [[Mike Duff]], [[Christopher Pope]], _$N=8$ Supergravity Breaks Down to $N=1$_, Phys. Rev. Lett. 50, 294 (1983) ([doi:10.1103/PhysRevLett.50.294](https://doi.org/10.1103/PhysRevLett.50.294))

* [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere_, Phys. Rev. Lett. 50, 2043 (1983) ([doi:10.1103/PhysRevLett.50.2043](https://doi.org/10.1103/PhysRevLett.50.2043), [erratum](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.846))

On [[double dimensional reduction]] of the [[super-membrane]] (the [[M2-brane]]) to the [[Green-Schwarz superstring]]:

* {#DuffHoweInamiStelle87} [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], _Superstrings in $D=10$ from Supermembranes in $D=11$_, Phys. Lett. B191 (1987) 70 and in [[Michael Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 205-206 (1987) ([spire](http://inspirehep.net/record/245249))

On the emergence of [[conformal field theory]] in the perturbations of the [[super-membrane]] around a classical solution stretched along the [[asymptotic boundary]] of [[anti de Sitter spacetime]]

* {#DuffSutton88} [[Mike Duff]], C. Sutton, _The Membrane at the End of the Universe_, New Sci. 118 (1988) 67-71 ([spire:268230](http://inspirehep.net/record/268230?ln=en))

predating the modern formulation of the [[AdS-CFT correspondence]], amplified in

* {#Duff98} [[Mike Duff]], _Anti-de Sitter space, branes, singletons, superconformal field theories and all that_, Int. J. Mod. Phys. A14:815-844, 1999 ([arXiv:hep-th/9808100](https://arxiv.org/abs/hep-th/9808100))

On the corresponding [[Green-Schwarz sigma-model|brane scan]] of [[super-conformal algebra|super-conformal]] branes:

* [[Miles P. Blencowe]], [[Mike Duff]], _Supersingletons_, Physics letters B, **203** 3(1988) 229-236 ([cds:184143](http://cds.cern.ch/record/184143), doi:10.1016/0370-2693(88)90544-8)

* [[Michael Duff]], _Near-horizon brane-scan revived_,  Nucl. Phys. B 810:193-209 2009 ([arXiv:0804.3675](http://arxiv.org/abs/0804.3675))

* {#Duff21} [[Michael Duff]], *The conformal brane-scan: an update* ([arXiv:2112.13784](https://arxiv.org/abs/2112.13784))

On [[quantization]] of the [[M2-brane]] [[Green-Schwarz sigma-model]]:

* {#DuffInamiPopeSezginStelle88} [[Mike Duff]], T. Inami, [[Christopher Pope]], [[Ergin Sezgin]],  [[Kellogg Stelle]], _Semiclassical Quantization of the Supermembrane_, Nucl.Phys. B297 (1988) 515-538 ([spire:247064](http://inspirehep.net/record/247064))

On [[Kaluza-Klein compactification]]

* {#Duff94} [[Michael Duff]], *Kaluza-Klein Theory in Perspective*, Proc. of the Symposium: *The Oskar Klein Centenary*, World Scientific, Singapore (1994) [[arXiv:hep-th/9410046](http://arxiv.org/abs/hep-th/9410046), [doi:10.1142/2816](https://doi.org/10.1142/2816)]

and specifically in [[supergravity]]:

* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Kaluza-Klein supergravity_, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 ([spire:229417](https://inspirehep.net/record/229417), doi:10.1016/0370-1573(86)90163-8)

On [[black brane|black]] [[M2-brane]]-solutions to [[D=11 supergravity]]:

* {#DuffStelle91} [[Mike Duff]], [[Kellogg Stelle]], _Multi-membrane solutions of $D = 11$ supergravity_, Phys. Lett. B 253, 113 (1991) ([spire:299386](http://inspirehep.net/record/299386), doi:10.1016/0370-2693(91)91371-2)

On [[black branes]] and [[solitonic branes]]:

* [[Michael Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String and Fivebrane Solitons: Singular or Non-singular?*, Nucl.Phys. B **377** (1992) 281-294 [[arXiv:hep-th/9112023](https://arxiv.org/abs/hep-th/9112023), doi:10.1016/0550-3213(92)90025-7]

* [[Michael Duff]], [[Jian Xin Lu]], *Black and super $p$-branes in diverse dimensions*, Nucl. Phys. B **416** (1994) 301-334 [[arXiv:hep-th/9306052](http://arxiv.org/abs/hep-th/9306052)]

* {#DuffKhuriLu94} [[Mike Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String Solitons*, Phys. Rept. **259** (1995) 213-326 [[arXiv:hep-th/9412184](https://arxiv.org/abs/hep-th/9412184), doi:10.1016/0370-1573(95)00002-X]

On [[M-theory]]:

* {#Duff96} [[Michael Duff]], _M-Theory (the Theory Formerly Known as Strings)_,  Int. J. Mod. Phys. A11 (1996) 5623-5642 ([arXiv:hep-th/9608117](http://arxiv.org/abs/hep-th/9608117))

* {#Duff98} [[Michael Duff]], _The Theory Formerly Known as Strings_, Scientific American 1998 ([pdf](https://www.nikhef.nl/pub/services/biblio/bib_KR/sciam14395569.pdf), [[DuffFormerlyStrings98.pdf:file]])

* {#Duff98b} [[Michael Duff]], _A Layman's Guide to M-theory_, 	Abdus Salam Memorial Meeting, Trieste, Italy, 19 - 22 Nov 1997, pp.184-213 ([arXiv:hep-th/9805177](https://arxiv.org/abs/hep-th/9805177), [cds:355721](https://cds.cern.ch/record/355721))

* {#Duff99} [[Michael Duff]], _[[The World in Eleven Dimensions]]_ (1999)

On the [[brane scan]] which classifies super $p$-[[brane]] [[sigma-models]] given by [[Green-Schwarz action functionals]]:

* {#Duff87} [[Michael Duff]], _Supermembranes: the first fifteen weeks_, Class. Quant. Grav. 5 (1988) 189 ([doi:10.1088/0264-9381/5/1/023](https://doi.org/10.1088/0264-9381/5/1/023), [spire:248034](https://inspirehep.net/record/248034))

On [[string-string duality]]:

* {#DuffKhuri93} [[Michael Duff]], [[Ramzi Khuri]], _Four-Dimensional String/String Duality_, Nucl.Phys. B411 (1994) 473-486 ([arXiv:hep-th/9305142](https://arxiv.org/abs/hep-th/9305142))

* [[Michael Duff]], [[Ruben Minasian]], _Putting string/string duality to the test_, Nucl.Phys. B436 (1995) 507-528 ([arXiv:hep-th/9406198](https://arxiv.org/abs/hep-th/9406198))

and specifically on the [[I8]]-term:

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

On [[supergravity]]:

* {#Duff04} [[Michael Duff]], *The status of local supersymmetry*, in: *From Quarks to Black Holes: Progress in understanding the logic of Nature*, World Scientific (2005) 60-116 [[arXiv:hep-th/0403160](http://arxiv.org/abs/hep-th/0403160), [doi:10.1142/9789812701794_0004](https://doi.org/10.1142/9789812701794_0004)]

On [[Freudenthal magic square|magic squares]] of [[magic supergravities]]  and [[U-duality]] via the [[classical double copy]]:

* {#BorstenDuffHughesNagy13} [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic square from Yang-Mills squared*, Phys. Rev. Lett. **112** (2014) 131601  [[arXiv:1301.4176](http://arxiv.org/abs/1301.4176)]

* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]

category: people

[[!redirects Mike Duff]]

[[!redirects Mike J. Duff]]


Michael Dummett

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

* [IEP entry](http://www.iep.utm.edu/dummett/)

## Related entries

* [[metaphysics]]

category: people


Michael Dütsch

* [website](http://www.theorie.physik.uni-goettingen.de/forschung/qft/research/index.en.html)

## Selected writings

On [[perturbative quantum field theory]], rifgorously via [[causal perturbation theory]], with emphasis on [[master Ward identities]]:

* [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, Progress in Mathematical Physics, Birkhaeuser 2019 ([publisher page](https://www.springer.com/de/book/9783030047375))

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

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

## Related $n$Lab entries

* [[perturbative quantum field theory]]

* [[master Ward identity]]

* [[AQFT]]

* [[perturbative AQFT]], [[causal perturbation theory]]

* [[AQFT on curved spacetimes]]

category: people

[[!redirects Michael Duetsch]]


Michael Eastwood

* [Institute page](https://maths-people.anu.edu.au/~eastwood/)

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

## Selected writings

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

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

On [[twistors]]:

* {#BastonEastwood89} [[Robert J. Baston]], [[Michael G. Eastwood]], _The Penrose transform its interaction with representation theory_ Oxford Science Publications, Clarendon Press, 1989 (ISBN:9780198535652, [spire:290342](https://inspirehep.net/literature/290342), [pdf](http://www.math.utah.edu/~ptrapa/math-library/baston-eastwood/baston-eastwood.pdf))

category: people

[[!redirects Michael G. Eastwood]]


Michael Faraday

Michael Faraday (1791-1867)

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

* [[Raymond Brock]], *Faraday's Lines of Force -- Modern Physics Begins*, chapter 11 in: *Quarks, Spacetime, and the Big Bang*, lecture notes [chapter: [pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/man_faraday_up.pdf), full text:[pdf](https://web.pa.msu.edu/people/brock/file_sharing/QSandBB/QS&BB_manuscript/QS&BB.pdf)]

## Selected writings

Early discussion of [[electromagnetism]] with emphais on the magnetic [[flux]] lines ("lines of force"):

* {#Faraday1852} [[Michael Faraday]], *Delienation of Lines of Magnetic Force by iron filings*, §37 in: *Experimental Researches in Electricity.--Twenty-Ninth Series*, Philosophical Transactions of the Royal Society of London **142** (1852) 137-159 [[doi:10.1098/rstl.1852.0012](https://doi.org/10.1098/rstl.1852.0012), [jstor:108540](https://www.jstor.org/stable/108540)]

* {#Martin2009} Thomas Martin (ed), *Faraday's diary of experimental investigation* 1820-1862, HR Direct (2009)  [[webpage](http://faradaysdiary.com/), preview:[pdf](http://faradaysdiary.com/ws3/faraday.pdf)]

  vol IV on Wikimedia:[pdf](https://upload.wikimedia.org/wikipedia/commons/8/89/Faraday%27s_diary_-_being_the_various_philosophical_notes_of_experimental_investigation_V.4_%28IA_faradaysdiarybei00fara_2%29.pdf)

## Related entries

* [[electromagnetism]], [[magnetic field]]

* [[flux]], [[flux tube]]

category: people

[[!redirects Faraday]]


Michael Farber

[[!redirects M. Farber]]
Michael Farber is Professor of Mathematics at the School of Mathematical Sciences, Queen Mary University of London. His research interests focus on applied and computational topology, topological robotics, applications of topology to statistics and computer science.

* [webpage](https://www.qmul.ac.uk/maths/profiles/farberm.html)

## Selected writings

Definition and basic properties of [[topological complexity]]:

* {#Farber01} [[Michael Farber]], _Topological complexity of motion planning_ (2001), [arXiv:math/0111197](https://arxiv.org/abs/math/0111197);

On [[topological complexity]] of [[real projective space]] and connection with their [[immersion]] into [[cartesian space]]:

* {#FarberTabachnikovYuzvinsky02} [[Michael Farber]], [[Serge Tabachnikov]], [[Sergey Yuzvinsky]], _Topological robotics: motion planning in projective spaces_ (2002), [arXiv:math/0210018](https://arxiv.org/abs/math/0210018);

category : people


Michael Faux

* [webpage](http://employees.oneonta.edu/fauxmg/)

## related $n$Lab entries

* [[adinkra]]

category: people

[[!redirects Mike Faux]]


Michael Finkelberg

* [webpage](https://www.hse.ru/en/org/persons/2806441)

## 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


Michael Flohr

* [webpage](https://www.itp.uni-hannover.de/en/institute/persons/michael-flohr/)

## Selected writings

On [[2d CFT|2d]] [[logarithmic CFT]]:

* [[Michael Flohr]], *Bits and pieces in logarithmic conformal field theory*, International Journal of Modern Physics A, **18** 25 (2003) 4497-4591 $[$[doi:10.1142/S0217751X03016859](https://doi.org/10.1142/S0217751X03016859), [arXiv:hep-th/0111228](https://arxiv.org/abs/hep-th/0111228)$]$

application to generalized [[Laughlin wavefunctions]]:

* {#GurarieFlohrNayak97} [[Victor Gurarie]], [[Michael Flohr]], [[Chetan Nayak]], *The Haldane-Rezayi Quantum Hall State and Conformal Field Theory*, Nucl. Phys. B **498** (1997) 513-538 $[$[arXiv:cond-mat/9701212](https://arxiv.org/abs/cond-mat/9701212), [doi:10.1016/S0550-3213%2897%2900351-9](https://doi.org/10.1016/S0550-3213%2897%2900351-9)$]$

category: people


Michael Forger

* [webpage](http://www.ime.usp.br/~forger/publdeut.html)

category: people


Michael Fourman

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

## related $n$Lab entries

* [[topos theory]]

* [[real number object]]

* [[smooth structure on a topos]]

category: people

[[!redirects Mike Fourman]]


Michael Freedman

* [webpage](http://stationq.cnsi.ucsb.edu/~freedman/)

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

## Selected writings

On [[topological quantum computation]] with a [[Chern-Simons theory]] for [[anyons]]:

* {#Freedman98} [[Michael Freedman]], *P/NP, and the quantum field computer*, Proc. Nat. Acad. Sci. **95** 1 (1998) 98-101 [[doi:10.1073/pnas.95.1.9](https://doi.org/10.1073/pnas.95.1.9)]

* [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], _Topological quantum computation_,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

On [[quantum error correcting codes]] associated with planar bulk/boundary systems (precursor to [[holographic tensor networks]]):

* [[Michael Freedman]], David A. Meyer, *Projective Plane and Planar Quantum Codes*, Found. Comput. Math. 1, 325–332 (2001) ([arXiv:quant-ph/9810055](https://arxiv.org/abs/quant-ph/9810055), [doi:10.1007/s102080010013](https://doi.org/10.1007/s102080010013))

On [[braiding]] of [[su(2)-anyons]] providing universal [[quantum gates]] for [[topological quantum computation]]:

* {#FreedmanLarsenWang00} [[Michael Freedman]], [[Michael Larsen]], [[Zhenghan Wang]], _A modular functor which is universal for quantum computation_, Communications in Mathematical Physics **227** (2002) 605–622 [[arXiv:quant-ph/0001108](http://arxiv.org/abs/quant-ph/0001108), [doi:10.1007/s002200200645](https://doi.org/10.1007/s002200200645)]

Approximating all [[topological quantum computation|topological quantum gates]] by just the "weaves" among all [[braids]]:

* [[Steven H. Simon]], [[Nick E. Bonesteel]], [[Michael H. Freedman]], N. Petrovic, [[Layla Hormozi]], *Topological Quantum Computing with Only One Mobile Quasiparticle*, Phys. Rev. Lett. 96 (2006) 070503 ([arXiv:quant-ph/0509175](https://arxiv.org/abs/quant-ph/0509175), [doi:10.1103/PhysRevLett.96.070503](https://doi.org/10.1103/PhysRevLett.96.070503))

On [[topological quantum computation]] with [[anyons]]/[[quantum Hall effect]]:

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. 80, 1083, 2008 ([arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083))

On [[topological quantum computation]] with [[anyons]] in [[topological superconductors]] ([[Majorana zero modes]]):

* [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], _Majorana Zero Modes and Topological Quantum Computation_, npj Quantum Information 1, 15001 (2015) ([nature:npjqi20151](https://www.nature.com/articles/npjqi20151))

On [[holographic entanglement entropy]] via [[bit threads]]:

* {#FreedmanHeadrick16} [[Michael Freedman]], [[Matthew Headrick]], *Bit threads and holographic entanglement*, Comm. Math. Phys. 352, 407 (2017) ([arXiv:1604.00354](https://arxiv.org/abs/1604.00354))

## Related entries

* [[quantum computation]]

category: people

[[!redirects Michael H. Freedman]]


Michael Friedman

_Not to be confused with the historian of mathematics, [[Michael Friedman (historian)]]._

Michael Friedman is an American philosopher. He has a developed a neo-Kantian philosophy which looks

> to understand the concrete historical process by which mathematical structures, physical theories of space, time, and motion, and mechanical constitutive principles organically evolve together so as to issue,  successively, in increasingly sophisticated mathematical representations of experience. ([Friedman 2010, p. 698](#Fried10))

This he does in particular through detailed historical studies of the response of [[Kant]] to [[Newtonian]] physics and that of the [[Vienna Circle]] to Einstein's [[general relativity]].

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

## Related entries

* [[philosophy of science]]

* [[David Corfield]], _[[davidcorfield:Friedman's Dynamics of Reason]]_, [slides](https://ncatlab.org/davidcorfield/files/WIP2.pdf).

## Selected writings

* _Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science_ (1983), Princeton University Press.

* _Kant and the Exact Sciences_ (1992), Harvard University Press.

* _Reconsidering Logical Positivism_ (1999), Cambridge University Press

* _A Parting of the Ways:  Carnap, Cassirer, and Heidegger_ (2000), Open Court.

* _Dynamics of Reason: The 1999 Kant Lectures at Stanford University_ (2001), CSLI/University of Chicago Press.

* {#Fried10} _Synthetic History Reconsidered_ (2010) in Mary Domski and Michael Dickson (eds.), Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science, pp. 571–813.

* _Extending the Dynamics of Reason_ (2011), Erkenntnis 75(3): 431-444.

* _Kant’s Construction of Nature_ (2013), Cambridge University Press.

category: people


Michael Friedman (historian)

 * [website](https://www.matters-of-activity.de/en/members/139/dr-michael-friedman)

_Not to be confused with the American philosopher of science [[Michael Friedman]]._

### Selected writings

On the early history of the discussion of the [[braid group]]:

* [[Michael Friedman (historian)]], *Mathematical formalization and diagrammatic reasoning: the case study of the braid group between 1925 and 1950*, British Journal for the History of Mathematics **34** 1 (2019) 43-59 [[doi:10.1080/17498430.2018.1533298](https://doi.org/10.1080/17498430.2018.1533298)]

category: people


Michael G. G. Laidlaw

## Selected writings

Early use of the [[configuration space of points]] in discussion of [[spin-statistics theorem|particle statistics]] (see also at *[[braid group statistics]]*):

* [[Michael G. G. Laidlaw]], [[Cécile Morette DeWitt]], *Feynman Functional Integrals for Systems of Indistinguishable Particles*, Phys. Rev. D **3** (1971) 1375 $[$[doi:10.1103/PhysRevD.3.1375](https://doi.org/10.1103/PhysRevD.3.1375)$]$

category: people

[[!redirects Michael Laidlaw]]


Michael Green

Michael Green is professor for theoretical [[physics]] at Cambridge. Together with [[John Schwarz]] he is one of the founding fathers of [[string theory]].

* [webpage](http://www.damtp.cam.ac.uk/people/m.b.green/)

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

## Quotes

On the computation of the [[anomaly cancellation]] now known as the [[Green-Schwarz mechanism]] in [[heterotic string theory]] (also known as [The First Superstring Revolution](http://theory.caltech.edu/people/jhs/strings/str133.html)):

> {#WeMustBeGettingPrettyClose} MICHAEL GREEN: I do remember a particular moment, when [[John Schwarz]] and I were talking at the blackboard and working out these numbers which had to fit, and they just had to match exactly. I remember joking with John Schwarz at that moment, because there was thunder and lightning -- there was a big mountain storm in Aspen at that moment -- and I remember saying something like, you know, "We must be getting pretty close, because the gods are trying to prevent us completing this calculation." And, indeed, they did match.

(Taken from [NOVA - The Elegant Universe - Transcript](https://www.scribd.com/document/185276/NOVA-The-Elegant-Universe-Transcript))

## Selected writings

Early argument that the [[RR-field]] [[flux]] [[differential form|density]]-expressions for [[D-brane charge]] are of the form of [[Chern characters]] on [[topological K-theory]], leading to the [[K-theory classification of D-brane charge]]:

* {#GreenHarveyMoore97} [[Michael Green]], [[Jeffrey A. Harvey]], [[Gregory Moore]], *I-Brane Inflow and Anomalous Couplings on D-Branes*, Class. Quant. Grav. **14** (1997) 47-52 [[arXiv:hep-th/9605033](https://arxiv.org/abs/hep-th/9605033), [doi:10.1088/0264-9381/14/1/008](https://doi.org/10.1088/0264-9381/14/1/008)]

Popular exposition of [[string theory]]:

* {#Green86} [[Michael Green]], _Superstrings_, Scientific American
Scientific American Vol. 255, No. 3 (September 1986), pp. 48-63 ([jstor:24976036](https://www.jstor.org/stable/24976036))

On the [[Green-Schwarz superstring]]:

* {#GreenSchwarz81} [[Michael Green]], [[John Schwarz]], _Supersymmetrical Dual String Theory_, Nucl. Phys. B 181 (1981) 502;

* {#GreenSchwarz82} [[Michael Green]], [[John Schwarz]], _Supersymmetrical Dual String Theory. 2. Vertices and Trees_, Nucl. Phys. B 198 (1982) 252.

* {#GreenSchwarz84} [[Michael Green]], [[John Schwarz]], _Covariant description of superstrings_, Phys. Lett. B136 (1984), 367–370 ([spire:193596](http://inspirehep.net/record/193596), arXiv;10.1016/0370-2693(84)92021-5)

* {#GreenSchwarz84b} [[Michael Green]], [[John Schwarz]], _Properties of the Covariant Formulation of Superstring Theories_, Nucl. Phys. B 243 (1984) 285 ([spire:196623](http://inspirehep.net/record/196623), doi:10.1016/0550-3213(84)90030-0)

On the [[Green-Schwarz mechanism]] of [[anomaly cancellation]] in [[heterotic string theory]]:

* [[Michael Green]], [[John Schwarz]], _Anomaly Cancellation in Supersymmetric $D=10$ Gauge Theory and Superstring Theory_, Phys. Lett. B 149 (1984) 117-122 ([spire:15583](https://inspirehep.net/literature/15583), doi:10.1016/0370-2693(84)91565-X)

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

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

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

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

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

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

On [[type I' string theory]] and [[geometric engineering of QFT]] on [[D3-D5 brane intersections]] subject to the [[s-rule]]:

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

and related to the [[Pauli exclusion principle]]:

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

On [[higher curvature corrections]] to the (abelian) [[DBI-action]] for [[D-branes]]:

* [[Constantin Bachas]], P. Bain, [[Michael Green]], _Curvature terms in D-brane actions and their M-theory origin_, JHEP 9905:011, 1999 ([arXiv:hep-th/9903210](https://arxiv.org/abs/hep-th/9903210))

## Related $n$Lab entries

* [[string theory]]

* [[superstring]]

* [[Green-Schwarz sigma-model]]

* [[Green-Schwarz mechanism]]


Michael Groechenig

* [personal page](http://individual.utoronto.ca/groechenig/)

## Selected writings

On [[algebraic stacks]]:

* [[Michael Groechenig]], *Algebraic Stacks*, Lecture notes (2014) [[web](http://individual.utoronto.ca/groechenig/stacks.html), [pdf](http://individual.utoronto.ca/groechenig/stacks.pdf), [[Groechenig-AlgebraicStacks.pdf:file]]]

category: people


Michael Hardy

I have a Ph.D. in statistics with a minor in mathematics. I've been pretty active in editing Wikipedia articles since 2002.

[[trigonometric identities and the irrationality of pi]]

category: people


Michael Hicks

* [personal page](https://mhicks.me/)

## Selected writings

Introducing [[polymonads]]:

* Nataliya Guts, [[Michael Hicks]], Nikhil Swamy, Daan Leijen, [[Gavin Bierman]], _Polymonads_, Extended version of POPL'13 submission [[pdf](https://www.cs.umd.edu/~mwh/papers/polymonadsTR.pdf), [[GHSLB-Polymonads.pdf:file]]]

* [[Michael Hicks]], [[Gavin Bierman]], Nataliya Guts, Daan Leijen, Nikhil Swamy, *Polymonadic Programming*, EPTCS **153** 79-99 (2014) [[arXiv:1406.2060](https://arxiv.org/abs/1406.2060), [doi:10.4204/EPTCS.153.7](https://doi.org/10.4204/EPTCS.153.7)]

On [[quantum circuit]] [[certified programming|certification]] with the [[quantum programming language]] *[[SQIR]]*:

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Xiaodi Wu, [[Michael Hicks]], *A verified optimizer for Quantum circuits*, Proceedings of the ACM on Programming Languages **5** Issue POPL 37 (2021) 1–29 [[doi:10.1145/3434318](https://doi.org/10.1145/3434318)]

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Liyi Li, [[Michael Hicks]], *Proving Quantum Programs Correct*, in *12th International Conference on Interactive Theorem Proving (ITP 2021)*, Leibniz International Proceedings in Informatics (LIPIcs) **193** (2021) [[arXiv:2010.01240](https://arxiv.org/abs/2010.01240)]

category: people


Michael Hill

* [webpage](http://people.virginia.edu/~mah7cd/home.html)

## Selected writings

On [[string bordism]] of the [[classifying space]] of [[E8|$E_8$]]:

* [[Michael Hill]], *The $String$  bordism of $B E_8$ and $B E_8 \times B E_8$ through dimension 14*, Ill. J. Math. **53** 1 (2009) 183-196 [[doi:10.1215/ijm/1264170845](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-53/issue-1/The-string-bordism-of-BE_8-and-BE_8times-BE_8-through-dimension/10.1215/ijm/1264170845.full)]

On [[tmf]]:

* [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]], _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

with [[level structure on an elliptic curve|level structure]]:

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], _Topological modular forms with level structure_, Inventiones mathematicae volume 203, pages 359–416 (2016) ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394), [doi:10.1007/s00222-015-0589-5](https://doi.org/10.1007/s00222-015-0589-5))

Solving the [[Arf-Kervaire invariant problem]] with methods of [[equivariant stable homotopy theory]]:

* {#HillHopkinsRavenel21} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem*, New Mathematical Monographs, Cambridge University Press (2021) [[doi:10.1017/9781108917278](https://doi.org/10.1017/9781108917278)]

* {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *On the non-existence of elements of Kervaire invariant one*, Annals of Mathematics **184** 1 (2016)[[doi:10.4007/annals.2016.184.1.1](https://doi.org/10.4007/annals.2016.184.1.1), [arXiv:0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf)]

* {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_, Current Developments in Mathematics, 2010: 1-44 (2011) ([[HHRKervaire.pdf:file]], [doi:10.4310/CDM.2010.v2010.n1.a1](https://dx.doi.org/10.4310/CDM.2010.v2010.n1.a1))

* {#HillHopkinsRavenelIntroduction} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire invariant problem in algebraic topology: introduction_ (2016) [[pdf](http://math.ucla.edu/~mikehill/Research/CDMHistory.pdf)]

category: people

[[!redirects Michael A. Hill]]


Michael Hopkins

Michael Jerome Hopkins is a mathematician at Harvard University.
He got his PhD from Northwestern University in 1984, advised by [[Mark Mahowald]].

* [website](http://www.harvardscience.harvard.edu/directory/researchers/michael-j-hopkins)

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

Hopkins is a world leading researcher in [[algebraic topology]] and ([[stable homotopy theory|stable]]-)[[homotopy theory]].

Among his notable achievements are his work on the [[Ravenel conjectures]], the introduction and discussion of the [[generalized cohomology theory]] _[[tmf]]_  and its [[string orientation of tmf|string orientation]], a formalization and construction of [[differential cohomology]], the proof of the [[Kervaire invariant]] problem.  More recently via [[Jacob Lurie]]'s work on the [[cobordism hypothesis]] Hopkins participates in work related to the foundations of [[quantum field theory]].

## Selected writings

On [[stable homotopy theory]]:

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

Introducing generalized [[differential cohomology]] motivated by the [[M5-brane]] partition function:

* _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_

On [[topological quantum field theory]]:

* _[[Topological Quantum Field Theories from Compact Lie Groups]]_

On [[ambidextrous adjunctions]] in [[stable homotopy theory]]

* _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

Introducing the [[nilpotence theorem]] in [[stable homotopy theory]]:

* {#DevinatzHopkinsSmith88} [[Ethan Devinatz]], [[Michael Hopkins]], [[Jeffrey Smith]], _Nilpotence and Stable Homotopy Theory I_, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 ([jstor:1971440](https://www.jstor.org/stable/1971440))

* {#DevinatzHopkinsSmith98} [[Ethan Devinatz]], [[Michael Hopkins]], [[Jeffrey Smith]], _Nilpotence and Stable Homotopy Theory II_, Annals of Mathematics Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 1-49 ([jstor:120991](https://www.jstor.org/stable/120991))

On the [[Conner-Floyd isomorphism]] for the [[Atiyah-Bott-Shapiro orientation]] of [[KU]] and [[KO]] ([[cobordism theory determining homology theory]]):

* {#HopkinsHovey92} [[Michael Hopkins]], [[Mark Hovey]], _Spin cobordism determines real K-theory_, Mathematische Zeitschrift volume 210, pages 181–196 (1992) ([doi:10.1007/BF02571790](https://doi.org/10.1007/BF02571790), [[HopkinsHoveyCobordismK.pdf:file]])

On [[elliptic genera]], the [[Witten genus]] and the [[string orientation of tmf]]:

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850 ([doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf))

* {#AndoHopkinsStrickland02} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _The sigma orientation is an H-infinity map_, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 ([arXiv:math/0204053](http://arxiv.org/abs/math/0204053), [doi:10.1353/ajm.2004.0008](https://doi.org/10.1353/ajm.2004.0008))

The construction of [[tmf]] was originally announced, as joint work with [[Mark Mahowald]] and [[Haynes Miller]], in

* {#Hopkins94} [[Michael Hopkins]], section 9 of _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Zürich 1994 ([[Hopkins_TopModFormsAtICM.pdf:file]], [doi:10.1007/978-3-0348-9078-6_49](https://doi.org/10.1007/978-3-0348-9078-6_49))

(There the spectrum was still called "$eo_2$" instead of "$tmf$".) The details of the definition then appeared in

* {#Hopkins02} [[Michael Hopkins]], section 4 of _Algebraic topology and modular forms_, Proceedings of the ICM, Beijing 2002, vol. 1, 283--309 ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

On [[stacks]] and [[complex oriented cohomology theory]]:

* {#Hopkins} [[Mike Hopkins]] (talk notes by [[Michael Hill]]), _Stacks and complex oriented cohomology theories_ ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter08/MikesTalk1.pdf))

On [[twisted equivariant K-theory]] with an eye towards [[twisted ad-equivariant K-theory]]:

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1
([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:

* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _[[Loop Groups and Twisted K-Theory]] I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. _[[Loop Groups and Twisted K-Theory]] II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _[[Loop Groups and Twisted K-Theory]] III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

On [[∞-groups of units]], [[Thom spectra]] and [[twisted generalized cohomology]]:

* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory_, Journal of Topology, Volume7, Issue 4, December 2014 ([arXiv:1403.4320](https://arxiv.org/abs/1403.4320), [arXiv:10.1112/jtopol/jtu009](https://doi.org/10.1112/jtopol/jtu009))

* {#ABGHR14} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology_, Journal of Topology Volume 7, Issue 3 2014 Pages 869–893  ([arXiv:1403.4325](http://arxiv.org/abs/1403.4325), [doi:10.1112/jtopol/jtt035](https://doi.org/10.1112/jtopol/jtt035))

Solving the [[Arf-Kervaire invariant problem]] with methods of [[equivariant stable homotopy theory]]:

* {#HillHopkinsRavenel21} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem*, New Mathematical Monographs, Cambridge University Press (2021) [[doi:10.1017/9781108917278](https://doi.org/10.1017/9781108917278)]

* {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *On the non-existence of elements of Kervaire invariant one*, Annals of Mathematics **184** 1 (2016)[[doi:10.4007/annals.2016.184.1.1](https://doi.org/10.4007/annals.2016.184.1.1), [arXiv:0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf)]

* {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_, Current Developments in Mathematics, 2010: 1-44 (2011) ([[HHRKervaire.pdf:file]], [doi:10.4310/CDM.2010.v2010.n1.a1](https://dx.doi.org/10.4310/CDM.2010.v2010.n1.a1))

* {#HillHopkinsRavenelIntroduction} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire invariant problem in algebraic topology: introduction_ (2016) [[pdf](http://math.ucla.edu/~mikehill/Research/CDMHistory.pdf)]

Introducing [[Hodge-filtered differential cohomology]] and its specialization to Hodge-filtered [[complex cobordism theory]]:

* {#HopkinsQuick14} [[Michael J. Hopkins]], [[Gereon Quick]], §5 in: *Hodge filtered complex bordism*, Journal of Topology **8** 1 (2014) 147-183 [[arXiv:1212.2173](https://arxiv.org/abs/1212.2173), [doi:10.1112/jtopol/jtu021](https://doi.org/10.1112/jtopol/jtu021)]

On classification of [[invertible TQFTs]] via [[reflection positivity]]:

* {#FreedHopkins21} [[Daniel Freed]], [[Michael Hopkins]], *Reflection positivity and invertible topological phases*, Geometry & Topology **25** (2021) 1165–1330  [[arXiv:1604.06527](https://arxiv.org/abs/1604.06527), [doi:10.2140/gt.2021.25.1165](https://doi.org/10.2140/gt.2021.25.1165)]

## Related entries

* [[A Survey of Elliptic Cohomology]]

* [[Hopkins-Miller theorem]]

category: people

[[!redirects Michael Hopkins]]
[[!redirects Michael J. Hopkins]]
[[!redirects Michael Jerome Hopkins]]
[[!redirects M. Hopkins]]
[[!redirects Hopkins]]
[[!redirects Mike Hopkins]]


Michael Hutchings

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

category: people


Michael J. Beeson

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

* [institute page](http://www.cs.sjsu.edu/faculty/beeson/)

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

## Selected writings

On [[constructive mathematics]]:

* [[Michael J. Beeson]], *Foundations of Constructive Mathematics*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **3** 6, Springer 1985 ([doi:10.1007/978-3-642-68952-9](https://link.springer.com/book/10.1007/978-3-642-68952-9))

## Related $n$Lab entries

* [[taboo]]

category: people

[[!redirects Michael Beeson]]


Michael Joachim

* [website](http://wwwmath.uni-muenster.de/u/joachim/)

## Selected writings

On [[KK-theory]] via [[homotopical structure on C*-algebras]] (and using the notion of *[[l.m.c.-C*-algebra]]*):

* {#JoachimJohnson07} [[Michael Joachim]], [[Mark Johnson]], _Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category_ ([arXiv:0705.1971](http://arxiv.org/abs/0705.1971))

On 3-[[twisted equivariant K-theory]] via the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]:

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

On [[bivariant cohomology]]:

* [[I. Barnea]], [[M. Joachim]], [[Snigdhayan Mahanta|S. Mahanta]], (2015). _Model structure on projective systems of C∗-algebras and bivariant homology theories_, [arXiv:1508.04283](http://arxiv.org/abs/1508.04283)

[[!redirects M. Joachim]]
category: people