Yuri Manin in nLab

Yuri Ivanovich Manin (1937-2023, Russian: Юрий Иванович Манин) was a Russian-born mathematician of polymath broadness, with main works in number theory and arithmetic geometry, noncommutative geometry, algebraic geometry and mathematical physics.

• Wikipedia entry

• MPI obituary

His diverse work includes a classification theorem in the theory of commutative formal group, early study of monoidal transformations and exposition on motives in 1960-s, a fundamental starting work in quantum information theory, proposals on quantum logics, an approach to quantum groups, ADHM construction in soliton theory, work with Maxim Kontsevich on Gromov-Witten invariants, work on Frobenius manifolds (and introduced more general “F-manifolds” with Claus Hertling). He published a number of influential monographs including on noncommutative geometry, quantum groups, complex geometry and gauge theories, introduction to schemes, Frobenius manifolds, mathematical logics…

Manin’s students include:

• A. Beilinson,

• V. Drinfel'd

• Vera Serganova

• Ivan Penkov

• M. Kapranov

• Yuri Tschinkel

• Ralph Kaufmann

• …

Selected writings

Introducing what came to be called the Gauss-Manin connection:

• Yuri Manin, Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR Ser. Mat. 22 6 (1958) 737-756 [[mathnet:izv3998, pdf]] (in Russian)

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]

Introducing the notion of quantum computation:

• Yuri I. Manin, Introduction to: Computable and Uncomputable, Sov. Radio (1980) [Russian original: pdf], English translation on p. 69-77 of Mathematics as Metaphor: Selected essays of Yuri I. Manin, Collected Works 20, AMS (2007) [ISBN:978-0-8218-4331-4]

  Perhaps, for a better understanding of [molecular biology], we need a mathematical theory of quantum automata.

and review of Shor's algorithm:

• Yuri I. Manin, Classical computing, quantum computing, and Shor’s factoring algorithm, Astérisque, 266 Séminaire Bourbaki 862 (2000) 375-404 [arXiv:quant-ph/9903008, numdam:SB_1998-1999__41__375_0]

Early discussion of mathematical supergeometry:

• Yuri Manin, New Dimensions in Geometry, in: Arbeitstagung Bonn 1984, Lecture Notes in Mathematics 1111, Springer (1985) 59–101 [doi:10.1007/BFb0084585), pdf]

On the Penrose-Ward transform relating twistor space to Minkowski spacetime, on its generalization to superalgebra and supergeometry, and on applications to super Yang-Mills theory and supergravity:

• Yuri Manin, Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]

Introducing quantum linear groups as universal co-acting bialgebras (and their quotient Hopf algebras):

• Yuri I. Manin, Quantum groups and non-commutative geometry, CRM Short COurses, Springer (1988) [doi:10.1007/978-3-319-97987-8]

On quantum cohomology and Gromov-Witten invariants

• Maxim Kontsevich, Yuri Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Physics 164 (1994) 525-562 doi arXiv:hep-th/9402147

• Yuri Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloqium Publications 47, 1999

• Maxim Kontsevich, Yuri Manin, Ralph Kaufmann, Quantum cohomology of a product, Invent. Math. 124 (1996) 313-339 doi arXiv:q-alg/9502009

On homological algebra and homotopical algebra (via triangulated categories and including the model structure on dgc-algebras for rational homotopy theory):

• Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (doi:10.1007/978-3-662-12492-5)

On Frobenius manifolds and quantum cohomology:

• Yuri Manin: Frobenius manifolds, quantum cohomology, and moduli spaces, Colloqium Publications 47, Amer. Math. Soc. (1999) [ams:]

and generalized to Frobenius supermanifolds in supergeometry:

• Yuri I. Manin, Sergei A. Merkulov: Semisimple Frobenius (super)manifolds and quantum cohomology of P rP^r, Topol. Methods in Nonlinear Analysis 9 (1997) 107-161 [arXiv:alg-geom/9702014, doi:10.12775/TMNA.1997.006]

On relations of AdS3/CFT2 to hyperbolic geometry and Arakelov geometry of algebraic curves:

• Yuri Manin, Matilde Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5 (2002) 617-650 (arXiv:hep-th/0201036

On doubly monoidal categories and quadratic operads:

• Yuri Ivanovich Manin, Bruno Vallette, Monoidal structures on the categories of quadratic data, Documenta Mathematica 25 (2020) 1727-1786 [arXiv:1902.03778]

Quotes

What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live “nowhere” and “nowhen”; only poetry and mathematics are capable of speaking meaningfully about such things

Mathematics as Metaphor: Selected Essays of Yuri I. Manin (ed. 2007) (libquotes)