School of Mathematics & Statistics

Tropical KP theory

Yelena Mandelshtam (University of Michigan)

Tuesday 17th March 16:00-17:00
Maths 311B

Abstract

The Kadomtsev–Petviashvili (KP) equation is a central example of an integrable nonlinear PDE with deep connections to algebraic geometry. A classical construction of Krichever produces quasi-periodic solutions from algebraic curves together with divisor data via Riemann theta functions. At the same time, soliton solutions have a rich combinatorial structure: work of Kodama-Williams and others relates them to the geometry and combinatorics of the positive Grassmannian.

In this talk I will describe recent and ongoing work with several collaborators that develops a “tropical KP theory’’ connecting these two viewpoints. When an algebraic curve degenerates to a tropical curve, the associated theta-function solutions collapse to soliton solutions. We show that the algebro-geometric data in the Krichever construction admits a direct tropical/combinatorial description that determines the resulting soliton solution. In particular, one can translate the geometric data of the degeneration into purely combinatorial objects that encode the soliton structure. This perspective provides a concrete way to pass from algebraic curves to soliton solutions and reveals a new combinatorial layer underlying the classical algebro-geometric theory of KP.

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