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)
Email:  sheinman@mi.ras.ru

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:

Born: June, 09, 1949, Moscow.
Father - Sheinman, Karl Mikhailovich  (1926-1996), an aircraft engineer, constructor of
nozzles.
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
Mechanics,
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
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.

Employment:
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
researcher
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

Projects
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:
The
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
A.Kulakov)
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.

- Àëãåáðû Êðè÷åâåðà-Íîâèêîâà, èõ ïðåäñòàâëåíèÿ è ïðèëîæåíèÿ â ãåîìåòðèè è ìàòåìàòè÷åñêîé ôèçèêå. Ñîâð. ïðîáë. ìàòåì., òîì 10. Ìîñêâà, ÌÈÀÍ, 2007. 140 ñòð.

- Îñíîâû òåîðèè ïðåäñòàâëåíèé. Ìîñêâà, ÌÖÍÌÎ, 2004, 64 ñòð. (English translation: Basic representation theory. Moscow, MCCME, 2005).

- Çàäà÷è ñåìèíàðà "Àëãåáðû Ëè è èõ ïðèëîæåíèÿ". Ìîñêâà, ÌÖÍÌÎ, 2004, 48 ñòð. (ñîâìåñòíî ñ È.Ì.Ïàðàìîíîâîé).

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.