Title: Methods of nonlinear equations in the theory of algebraic geometry
Speaker: Nathan Hayford
Date/time: Wednesday, February 17th, 3:00pm-4:00pm
Abstract: The Riemann-Schottky problem, in modern language, asks for a characterization of Jacobian varieties of among abelian varieties. From Riemann’s point of view, the question was to address the discrepancy between the 3g-3-dimensional space of moduli for compact Riemann surfaces of genus g from the g(g+1)/2-dimensional space of Siegel matrices, in which the space of Riemann surfaces embeds, via the period matrix. The problem remained open for almost a millennium. The first hint of a solution to this problem was found (somewhat surprisingly) by I. Krichever (1977), when he showed (more or less) that the period matrix for a Riemann surface could be used to construct tau functions for the KP hierarchy. S. Novikov conjectured that the converse also held, thus answering the Riemann-Schottky problem a la methods of integrable systems. This turned out to indeed be the case, with a proof from T. Shiota (1986) shortly thereafter. In this talk, I will attempt to fill in some of the details of this story, and explain some of the consequences of this theorem, both in integrable systems and algebraic geometry.