Tomohiro Sasamoto(Tokyo Institute of Technology)
Determinantal formulas for Kardar-Parisi-Zhang models
17:00--18:30, January 31 (Thu), 2019
Room B314, Faculty of Science
The Kardar-Parisi-Zhang (KPZ) equation, is a stochastic partial differential equation, introduced in 1986 to explain behaviors of surface growth. Large scale behaviors of its solution are universal and shared by various systems, constituting the KPZ universality class. A class of models in the KPZ class have turned out to be integrable and admit determinantal formulas, from which one can establish the limiting universal laws of fluctuations, typically given by the Tracy-Widom distributions. In this presentation, after a brief review on the basics about the KPZ equation, we introduce several models in the KPZ class and explain how the determinantal formulas are obtained for these models. In particular we will see that our approach of using Ramanujan’s bilateral sum and Frobenius determinant proposed in [1,2] can be applied to many models in a unified fashion. The talk is based on joint works with M. Mucciconi and T. Imamura. The talk will be in Japanese (slides will be prepared in English). References: [1]T. Imamura and T. Sasamoto, Fluctuations for stationary q-TASEP, arXiv:1701.05991, to appear in Prob. Th. Rel. Fields. [2]M. Mucciconi, T. Imamura and T. Sasamoto, Stationary Stochastic Higher Spin Six Vertex Model and q-Whittaker measure, in preparation.

Anton Zabrodin (National Research University Higher School of Economics )
Elliptic solutions to BKP equation and many-body systems
17:00--18:30, February 5 (Tue), 2019
Room B314, Faculty of Science
We derive equations of motion for poles of double-periodic (elliptic) solutions to the BKP equation. The basic tool is the auxiliary linear problem for the wave function. The result is a new many-body dynamical system with three-body interaction expressed through the Weierstrass elliptic function. This system does not admit Lax representation but, instead, it is equivalent to a sort of Manakov's triple equation with a spectral parameter. We also discuss integrals of motion which follow from the equation of the spectral curve and analyze analytic properties of the wave function on the spectral curve.

Akane Nakamura(Josai University)
Recovering linear from nonlinear
17:00--18:30, February 14 (Thu), 2019
Room B301, Faculty of Science
One of the important aspects of the integrable systems is that these nonlinear systems possess linear problems. However, it is not easy to find a linear problem (Lax equation) just by looking at the nonlinear equations. In this talk, we will explain a way to recover a linear problem from the nonlinear autonomous 4-dimensional Painlevé-type systems (the Hitchin systems). Our way is to compare generic degenerations of the families of curves arising from the nonlinear problem (i.e., the boundary divisors adjoined in the compactification of the Liouville tori) and curves appearing in the linear side (the spectral curves). We have proved that the Jacobian of generic curve of these systems has unique principal polarization, so that we can recover curves. This talk is based on joint work with Eric Rains.