Mathematics & Statistics research seminar
Pizza and soft drinks will be served after the talk.
SPEAKER: Dr. Julia Bernatska
ABSTRACT: This talk is devoted to the theory of multivariate sigma-functions developed by V. Buchstaber, D. Leykin, V. Enolski (see [1–4]). The sigma-function is considered as a solution of a system of linear partial differential equations. The theory gives the way how to construct the system. The unique solution arises as a series expansion, and has the advantage to be effective and easy for computation.
The first part of talk describes the construction of sigma-function associated with a so called (n, s)-curve. As a by-product we obtain the basis of second kind differentials associated to the standard first kind differentials. The general scheme is illustrated by the examples of small genera.
Further we discuss on some applications and open problems related to so called polylinear relations, namely, bilinear Hirota relations, which can be alternatively obtained from Klein’s bidifferential formula; and trilinear relations, which produce addition formulas.
 Buchstaber V. M., Enolskii V. Z., and Leykin D. V. Rational Analogs of Abelian Functions, Functional Analysis and Its Applications, Vol. 33, No. 2, pp. 83–94, 1999.
 Buchstaber V. M., and Leykin D. V. Polynomial Lie Algebras, Functional Analysis and Its Applications, Vol. 36, No. 4, pp. 267–280, 2002.
 Buchstaber V. M., and Leykin D. V. Heat Equations in a Nonholonomic Frame, Functional Analysis and Its Applications, Vol. 38, No. 2, pp. 88–101, 2004.
 Buchstaber V. M., and Leykin D. V. Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of (n, s)-Curves, Functional Analysis and Its Applications, Vol. 42, No. 4, pp. 268–278, 2008.