Algebraic Solutions of the Multicomponent KP Hierarchy

KP 族的顶点算子与奇异代数曲线

Vertex Operators of the KP hierarchy and Singular Algebraic Curves

本文利用 Sato Grassmannian 证明 KP 族顶点算子作用产生的解对应奇异代数曲线上的无挠秩一 sheaf，并说明其可视为拟周期背景上的孤子。

Addition Formulae for Non-Abelian Theta Functions and Applications

This paper generalizes for non-abelian theta functions a number of formulae valid for theta functions of Jacobian varieties. The addition formula, the relation with the Szego kernel and with the multicomponent KP hierarchy and the behavior under cyclic coverings are given.

On Algebraic Theta Divisors and Rational Solutions of the KP Equation

In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Algebraic perspectives on signomial optimization

Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of ``degree'' that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the conc…

Tau functions and KP solutions on families of algebraic curves

Using abelian differentials and periods of the universal Mumford curve, we study the universal expression and asymptotic behavior of tau functions defined for stably degenerating families of algebraic curves with additional data. Furthermore, we apply this study to constructing solutions to the KP hierarchy which are expressed by nonarchimedean theta functions and by mixtures of quasi-periodic solutions with solitons containing real solutions defined on families of M-curves.

Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter …