Fall 2025
September 9, 2025, 9:00–10:00 am, zoom
Speaker: Yuancheng Xie (Shenzhen MSU-BIT University)
Title: Pfaffians as τ -functions of the BKP hierarchy: a constructive parametrization of complex pure spinors de E. Cartan
Abstract: It is well known that τ -functions of KP hierarchy are parameterized by points in Sato’s Universal Grassmannian manifold (UGM). These τ -functions have Schur expansions with coefficients satisfying Plücker relations. In this talk we will show that all τ -functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. These τ -functions are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). They have natural Schur-Q expansions with coefficients satisfying Cartan-Plücker relations. As a byproduct this parameterization gives a constructive description for complex pure spinors de E. Cartan. As an application, we reprove a theorem due to A. Alexandrov which states that τ -functions of KdV satisfy BKP up to rescaling of the time parameters by 2. We prove this by showing that the KdV hierarchy can be viewed as 4-reduction of the BKP hierarchy. This interpretation gives complete characterization for the KdV orbits nside the BKP hierarchy. This talk is based on preprint arXiv:2210.03307.
September 22, 2025, 4:00–5:00 pm, East Hall 1866
Speaker: Yelena Mandelshtam (University of Michigan)
Title: KP solitons from algebraic curves and the positive Grassmannian
Abstract: The Kadomtsev–Petviashvili (KP) equation is an important nonlinear PDE in the theory of integrable systems, with rich families of solutions arising both from algebraic geometry and from combinatorics. On one hand, Krichever showed how to build solutions from algebraic curves using Riemann theta functions. On the other, Kodama and Williams connected soliton solutions to the geometry of the positive Grassmannian. In this talk I will describe recent and ongoing work with various collaborators, where we study what happens when algebraic curves degenerate tropically. In this limit, theta-function solutions collapse to soliton solutions, and we can track how the geometry of the tropical curve manifests in the combinatorial structure of the soliton. This provides a new bridge between the algebro-geometric and combinatorial approaches to KP solutions.
September 30, 2025, 9:00–10:00 am, zoom
Speaker: Xi Chen (University of Basel)
Title: Gérard-type explicit formulas and their applications
Abstract: This talk will focus on the explicit formulas initially introduced by Patrick Gérard (2022), a powerful tool that provides more direct access to analyzing the solutions of integrable equations such as the Benjamin-Ono equation and the Calogero-Moser DNLS equation. We will also discuss three applications: the analysis of the zero dispersion limit, the characterization of long-time behavior, and the construction of new numerical schemes.
October 7, 2025, 9:00–10:00 am, zoom
Speaker: Jiaqi Liu (Chinese Academy of Sciences)
Title: On perturbation of completely integrable PDEs
Abstract: We revisit the perturbative theory of infinite dimensional integrable systems developed
by P. Deift and X. Zhou, aiming to provide new and simpler proofs of some key
L∞ bounds and Lᵖ a priori estimates. Our proofs emphasizes a further step towards
understanding focussing problems and extends the applicability to other integrable
models. As a concrete application, we examine the perturbation of the one-dimensional
defocussing cubic nonlinear Schrödinger equation and modified KdV equations.
October 21, 2025, 9:00–10:00 am, zoom
Speaker: Anton Dzhamay (BIMSA)
Title: On the geometric approach to the Painlevé equivalence problem
Abstract: We show how the techniques from the Okamoto-Sakai geometric theory of Painlevé equations can be used to solve the Painlevé equivalence, i.e., how to recognize an equation as a Painlevé equation and find an explicit change of variables transforming it into some canonical form. We illustrate the geometric approach by considering two examples recently obtained by M. van der Put and J. Top in their study of a certain ansatz of isomonodromic deformations of linear ODEs. We provide explicit coordinate transformations identifying these examples with standard form of some Painlevé equations and also explicitly identify their Hamiltonians.