Aarhus University Seal / Aarhus Universitets segl

Combinatorial solutions to integrable hierachies

Sergei Lando (Higher School of Economics, Moscow)
Tirsdag, 27 januar, 2015, at 16:15-17:15, in Aud. D1 (1531-113)
Integrable hierarchies of partial differential equations appeared as a tool to describe the behavior of waves of special kind. It happened, however, that their solutions include very interesting formal ones, whose coefficients give answers to natural enumerative problems. According to Sato construction (1980), these solutions can be expressed in terms of Young diagrams and Schur polynomials. A spectacular example of such solution is the Witten–Kontsevich potential, which is the generating function for certain geometric parameters of moduli spaces of complex structures on curves. For such solutions, the equations of the hierarchies can be  interpreted as recurrence relations allowing one to efficiently compute the coefficients of the corresponding formal power series.
It will be explained how to construct solutions to the Kadomtsev–Petviashvili integrable hierarchy by means of Schur polynomials, and examples will be given, including those found recently, of enumerative problems leading to such solutions.
Organiseret af: QGM
Kontaktperson: Jørgen Ellegaard Andersen