Sheinman, Oleg Karlovich

Steklov Mathematical Institute, Department of Geometry and Topology (leading researcher)
and Independent University of Moscow (permanent professor)

Phone: (095) 938-3787 (office)

The area of main scientific interest: infinite-dimensional Lie algebras (Krichever-Novikov algebras, Lax operator algebras), representation theory, related problems of geometry of moduli spaces and mathematical physics, integrable systems.

Vita and Education
Learned Bodies
Principal Scientific Results
Teaching at the Independent University of Moscow
Selected talks

Vita and Education:

Born: June, 09, 1949, Moscow.
Father - Sheinman, Karl Mikhailovich  (1926-1996), an aircraft engineer, constructor of
Mother - Sheinman-Topstein, Cecile Yakovlevna (1923-1991), a phylologist, translator of
               classical phylosophical works (Plato, Kant, Descartes and others) to Russian.
1966-1971 -- Study in Moscow State University (MSU), Department of Mathematics and
1971 -- Student diploma from the Department of Mathematics and Mechanics of MSU.
             Thesis title: "Orbits of the real simplectic group", (Prof. A. A. Kirillov - adviser).
1974-1977 Aspirantura, the Central Institute for Economics and Mathematics of the Academy
                  of Sciense
USSR (Prof. S.S.Lebedev - adviser).
1982 Candidate of Science (=PhD) in Physics and Mathematics.
         Dissertation title: Duality and subadditive functions in integer linear programming.
2007 Doctor of Science in Physics and Mathematics.
         Dissertation title: Krichever-Novikov algebras, their representations and applications in geometry and mathematical physics.
1974 Married, 2 children: the daughter and the son.

1971-1974 A member of staff at the Central Institute for Economics and Mathematics (Moscow),
                   junior researcher
1977-1995 A member of staff at the Krzhizhanovski Power Engineering Institute (Moscow),
                   junior researcher, senior researcher
1995-2000 A member of staff at the Institute for Economics of Energetics (Moscow), senior
1993           A member of staff at the Independent University of Moscow
                   since 2004 a permanent professor of this university
2000           A member of staff at the Steklov Mathematical Institute, leading researcher

1996            RFBR project 96-01-00063 (Head of the project)
1996-98       joint RFBR and DFG project 96-01-00055G (co-Head of the project)
1999-2001   RFBR project 99-01-00198 (Head of the project)
    (RFBR=Russian Foundation for Basic Research,  DFG=Deutsche Forschungsgemeinschaft)
Since 2002-   RFBR projects 02-01-00803, 05-01-00170
Since 2002-  Mathematical methods of nonlinear dynamics. The project of the Russ. Acad. Sci.

Learned Bodies:
Moscow Mathematical Society

Principal Scientific Results

Lax operator algebras:

Introduced in 2006 in the joint work with I.Krichever. Certain basic properties were established there, such as almost graded structure, existense of the almost graded central extensions (the corresponding 2-cocycles are constructed explicitly).The classification of all almost graded central extensions is given in the joint work with M.Schlichenmaier (2007). In my subsequent works I developed certain applications to the theory of integrable systems: the existence and properties of integrable hierarchies.

Krichever-Novikov algebras and their representations:

Classification of  the coadjoint orbits and its relation to the 21-th problem of Hilbert. Hitchin-Tyurin invariants of Krichever-Novikov algebras. Construction of wedge representations of Krichever-Novikov algebras and their classification by holomorphic bundles on Riemann surfaces. Description of the second order casimirs for the Krichever-Novikov algebras and some more general operators (semi-casimirs). Relation between semi-casimirs, conformal blocks and tangent spaces to certain moduli spaces of Riemann surfaces with marked points and fixed jets of local coordinates. Analog of Weil-Kac formula for characters for the special class of irreducible representations.

2D Conformal Field Theory:

Constructing the Conformal Field Theory on moduli spaces of Riemann surfaces with puctures using Krichever-Novikov algebras as algebras of gauge and conformal symmetries. The generalization of the Knizhnik-Zamolodchikov equations on positive genus. Formules for infinitesimal deformation of regular Krichever-Novukov functions and vector fields under deformation of moduli  (joint results with M.Schlichenmaier).

Discrete minimization (maximization) problems (1977-82).

The duality theory for linear discrete programming. Analog of Kuhn-Tacker theorem for discrete nonlinear programming. Lagrange multipliers for problems in discrete arguments. The extremal properties of subadditive cutting planes.

Teaching at the Independent University of Moscow

1993/94 differential geometry (lectures, seminar; together with I.Krichever)
1995                       Riemann surfaces (lectures, seminar; together with O.Schwarzman and
1996-98                  Seminar on Lie algebras and their applications (together with I.Paramonova)
2002-03                 Basic representation theory
2003-04                  Calculus on manifolds
2004 -- Krichever-Novikov algebras and their representations
2008/09 -- Lax operator algebras
2000-present time:
Seminar on Riemann surfaces, Lie algebras and mathematical physics
                                (together with S.Natanzon and O.Schwarzman)

Principal publications: (see the full list of publications here)

Monographs, Textbooks

- Current algebras on Riemann surfaces. De Gruyter Expositions in Mathematics, 58, Walter de Gruyter GmbH & Co, Berlin–Boston, 2012, ISBN: 978-3-11-026452-4, 150 pp.

- Алгебры Кричевера-Новикова, их представления и приложения в геометрии и математической физике. Совр. пробл. матем., том 10. Москва, МИАН, 2007. 140 стр.

- Основы теории представлений. Москва, МЦНМО, 2004, 64 стр. (English translation: Basic representation theory. Moscow, MCCME, 2005).

- Задачи семинара "Алгебры Ли и их приложения". Москва, МЦНМО, 2004, 48 стр. (совместно с И.М.Парамоновой).

Scientific Articles

- Lax equations and Knizhnik-Zamolodchikov connection. math/1009.4706

- Lax operator algebras and Hamiltonian integrable hierarchies. math/0910.4173 and Uspekhi Mat. Nauk, 2011, no.1, 151-178 (Dedicated to I.Krichever on the occasion of his 60-th Birthday).

- Lax operator algebras and integrable hierarchies. Proceedings of the Steklov Math. Institute, v.263, p.216-226 (2008).
           In English       In Russian

- On certain current algebras related to finite-zone integration. Geometry, topology and mathematical physics. S.P.Novikov's seminar 2006-2007. Ed. by V.M.Buchstaber and I.M.Krichever. AMS Transl. Ser.2, v.224 (2008).

- Lax operator algebras. Funct. Anal. and Applications, v.41, no.4, p.46-59 (2007). math.RT/0701648 (joint work with I.M.Krichever)

- Projectively flat connections on the moduli space of Riemann surfaces and Knizhnik-Zamolodchikov equations. Proceedings of the Steklov Mathematical Institute, "Nonlinear dynamics", v. 251. ( postscript file in Russian).

- Krichever-Novikov algebras and their representations. Proceedings of the conference "Noncommutative geometry and representation theory in mathematical physics", Karlstad, Sweden, 4-11 July 2004. Contemp. Math., 391, p. 313--321. Amer. Math. Soc., Providence, RI, 2005.

- Knizhnik-Zamolodchikov equations for positive genera (joint work with M.Schlichenmaier). Uspekhi Mat.Nauk(=Rusian. Math. Surv.), 2004, n 4, p.147-180.(ArXiv: The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, II. math.AG/0312040)

- Affine Krichever-Novikov algebras, their representations and applications. In: Geometry, Topology and Mathematical Physics. S.P.Novikov's Seminar 2002-2003, V.M.Buchstaber, I.M.Krichever, eds. AMS Translations (2) 212, p.p. 297-316. Math.RT/0304020

- Second order casimirs for the Krichever-Novikov algebras $\widehat{\mathfrak{gl}}_{g,2}$ and $\widehat{\mathfrak{sl}}_{g,2}$. Moscow Mathematical Journal 1(4) 2001; math.RT/0109001

- The fermion model of representations of the affine Krichever-Novikov algebras. Funkt. Anal. Prilozh., 35 (3)2001, 60–72; math.RT/0204178.

- Krichever-Novikov algebras and self-duality equations on Riemann surfaces.  Uspekhi Mat.Nauk, v.56, No. 1 (2001) (postscript file in Russian .

- The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras. Russian. Math. Surveys, v.54, N 1, p. 213-249 (1999) (translation from: Uspekhi Mat.Nauk, v.54, No. 1, p. 213-250 (1999)) (joint work with M. Schlichenmaier); math.QA/9812083

- Orbits and representations of Krichever-Novikov affine-type algebras. Journal of Mathematical Sciences, 1996, 82, No. 6, 3834-3843

- Sugawara construction and Casimir operators for Krichever-Novikov algebras. Journal of Mathematical Sciences, 1998, 92, No. 2, 3807- 3834; also: Mannheimer Manuskripte Nr. 201 and q-alg/9512016 (joint work with M. Schlichenmaier)

- Highest weight modules for affine Lie algebras on Riemann surfaces. Funct. Anal. Appl. 29, No.1, 44-55 (1995) (translation from: Funkt. Anal. Prilozh., 29, No.1, p. 56 -71, (1995).

- Representations of Krichever-Novikov algebras. In: Topics in topology and mathematical physics (Novikov, S.P., ed.), Amer. Math. Soc. Translations, Ser.2, Vol.170, p.185 -198, R.I., U.S.A., 1995

- Krichever-Novikov algebras and CCC-groups. Russian Math. Surveys 50, No.5, 1097-1099 (1995) (translation from: Uspehi Mat. Nauk, 50, No.5 253 -254 (1995)).

- The orbits and representations of Krichever-Novikov algebras of affine type.  In: International Congress of Mathematicians, Abstracts, short communications, p.82. Zurich, 1994.

- Affine Lie algebras on Riemann surfaces. Funct. Anal. Appl. 27, No.4, 266-272 (1993) (translation from: Funkt. Anal. Prilozh. 27, No.4,54-62 (1993))

- Highest weight modules over certain quasigraded Lie algebras on elliptic curves.  Funct. Anal. Appl. 26, No.3, 203-208 (1992) (translation from: Funkt. Anal. Prilozh. 26, No.3, 65-71 (1992))

- Elliptic affine Lie algebras. Funct. Anal. Appl. 24, No.3, 210-219 (1990) (translation from: Funkt. Anal. Prilozh. 24, No.3, 51-61 (1990))
           In English       In Russian

- Hamiltonian string formalism and discrete groups. Funct. Anal. Appl. 23, No.2, 124-128 (1989) (translation from: Funkt. Anal. Prilozh. 23, No.2, 49-54 (1989)).

- Kernel of evolution operator in the space of sections of a vector bundle as integral over trajectories. Funct. Anal. Appl. 23, No.2, 124-128 (1989) (translation from: Funkt. Anal. Prilozh. 23, No.2, 49-54 (1989)).

- Dedekind \eta-function and indefinite quadratic forms. Funct. Anal. Appl. 19, No.3, 232-234 (1985) (translation from: Funkt. Anal. Prilozh. 19, No.3, 80-81 (1985)).

Selected talks :

Lax operator algebras: unexpected outcome, and a new tool of the theory of integrable systems. Southeast Lie theory workshop. College of Charleston, Charleston, SC, USA, December 16-18, 2012.