Combinatorial solutions to integrable hierarchies

Sergei Lando (HSE University)

Tuesday 6th May 16:00-17:00
Maths 311B

Abstract

There is a large class of enumerative problems whose solutions are generating functions that solve integrable hierarchies of mathematical physics. The possibility of the appearance of solutions of integrable hierarchies (namely, the Korteweg–de Vries hierarchy) in such problems has been known for more than thirty years, but the then existing constructions required a non-trivial investigation of the singularities of the generating functions in question and passage to the so-called double-scaling limit. The first natural example was the generating function for the double Hurwitz numbers. R.Pandharipande conjectured in 2000 that this function is a solution to the Toda lattice hierarchy, and Okounkov proved the conjecture in the same year. In turn, interest in the study of Hurwitz numbers arose from interest in the geometry of moduli spaces of complex curves. This geometry serves as an alternative model of two-dimensional quantum gravity, and the appearance in it of solutions to the Korteweg–de Vries hierarchy was conjectured by Witten in the end of 1980ies, based on the double-scaling limit. After Okounkov’s work, natural examples of combinatorial solutions to integrable hierarchies began to increase in number. It turned out that non-linear partial differential equations can serve as efficient computational tools in enumerating various combinatorial objects, and this, in turn, provides efficient methods for computing various asymptotic characteristics of the corresponding sequences that were not known earlier.

No preliminary knowledge is expected from the audience.

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