Analytic series expansion for $\sigma$-functions: applications, examples and open questions

Julia Bernatska (National University of Kyiv-Mohyla Academy)

Tuesday 10th February, 2015 15:00-16:00 Maths 516

Abstract

This talk is devoted to the theory of multivariate $\sigma$-functions developed by V. Buchstaber, D. Leykin, V. Enolski and C. Eilbeck, M. England,  Onishi Yo. The theory is based on series expansions, and has the advantage to be effective and easy for computation. The first part of talk describes a construction of the series expansion of $\sigma$-function associated with a so called (n;s)-curve. As a by-product of the construction we obtain the basis of second kind differentials associated to the standard first kind differentials. The general scheme is illustrated by the examples of small genera. Further we discuss on some applications and open problems related so called polylinear relations, namely, bilinear Hirota relations, which can be alternatively obtained from Klein's bidifferential formula; and trilinear relations, which produce addition formulas. We focus on the problem of regularization of the second kind integrals, which appears nontrivial in non-hyperelliptic case.

Work in collaboration with Dmitry Leykin.