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 (University of 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 emphasize a further step towards
understanding focusing problems and extending the applicability to other integrable
models. As a concrete application, we examine the perturbation of the one-dimensional
defocusing 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.
November 3, 2025, 4:00–5:00 pm, East Hall 1866
Speaker: Shi-Zhuo Looi (Caltech)
Title: Late-time tails and nonlinear Price's law for semilinear wave equations in 3-D
Abstract: I will present sharp late-time pointwise asymptotics for semilinear wave equations with power nonlinearities on stationary, asymptotically flat spacetimes (including black hole exteriors). Under standard spectral and local energy decay hypotheses for perturbations of black-hole backgrounds, we show a clean dichotomy: cubic nonlinearities generate a nonlinear t^{-2} tail with an explicit coefficient, while for powers p \ge 4 the linear Price's law t^{-3} decay holds with a modified coefficient; in both regimes we identify the leading term throughout the forward causal domain via a blend of radiation-field methods and low-energy resolvent analysis. Joint work with Haoren Xiong.
November 11, 2025, 9:00–10:00 am, zoom
Speaker: Alex Little (ENS de Lyon)
Title: The partition function of β-ensembles with complex potentials
Abstract: The asymptotic behaviour of the partition function is one of the central questions of statistical mechanics. In our work we consider this problem when the external potential is complex valued and for a particular statistical-mechanical model, a β-ensemble. We prove a full 1/N expansion of the logarithm of the partition function, the so-called free energy. Our method can be regarded as an infinite dimensional version of the method of steepest descent for contour integrals. This is joint work with A. Guionnet and K. Kozlowski.
November 17, 2025, 4:00–5:00 pm, East Hall 1866
Speaker: Antonios Zitridis (University of Michigan)
Title: From entropic propagation of chaos to concentration bounds for stochastic particle systems
Abstract: We shall discuss about weakly interacting stochastic particle systems with possibly singular pairwise interactions. In this setting, we observe a connection between entropic propagation of chaos (proved by Jabin and Wang, 2018) and exponential concentration bounds for the empirical measure of the system. In particular, we will show how to establish a variational upper bound for the probability of a certain rare event, and then use this upper bound to show that ''controlled" entropic propagation of chaos implies an exponential concentration bound for the empirical measure.
December 1, 2025, 4:00–5:00 pm, East Hall 1866
Speaker: Andres Contreras Hip (University of Chicago)
Title: Gaussian fluctuations for the open one-dimensional KPZ equation
Abstract: In this talk we consider the open one-dimensional KPZ equation on the interval $[0,L]$ with Neumann boundary conditions. For $L \sim t^{\alpha}$ and stationary initial conditions, we obtain matching upper and lower bounds on the variance of the height function for $\alpha \in [0,\frac23]$ for different choices of the boundary parameters. Additionally, for fixed $L$ and an arbitrary probability measure as initial conditions, we show Gaussian fluctuations for the height function as $t\to \infty$. Joint work with Sayan Das and Antonios Zitridis.
December 8, 2025, 4:00–5:00 pm, East Hall 1866
Speaker: Andrei Prokhorov (University of Cincinnati)
Title: TBD
Abstract: TBD