Category Archives: Fall 2025

Random tensors and fractional NLS

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The HADES seminar on Tuesday, September 30th, will be at 3:30pm in Room 740.

Speaker: Rui Liang

Abstract:In this talk, we will consider the Schrödinger equation with cubic nonlinearity on the circle, with initial data distributed according to the Gibbs measure.  We will discuss the challenges and strategies involved in establishing the Poincaré recurrence property with respect to the Gibbs measure in the full dispersive range. This work, using the theory of the random averaging operator developed by Deng-Nahmod-Yue ’19, addresses an open question proposed by Sun-Tzvetkov ’21. We will also explain why the Gibbs dynamics for the full dispersive range is sharp in some sense. Finally, we will see how the theory of random tensors works for extending this work to multi-dimensional settings.

Integral formulas for under/overdetermined differential operators

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The HADES seminar on Tuesday, September 23rd, will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (IHÉS) that introduces a new versatile approach to constructing integral solution operators (i.e., right-inverses up to finite rank operators) for a broad class of underdetermined operators, including the divergence operator, linearized scalar curvature operator, and the linearized Einstein constraint operator. They are optimally regularizing and, more interestingly, have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). My goal is to (1) describe our approach, (2) demonstrate how it generalizes the well-known construction of Bogovskii, which has proved very useful in fluid dynamics, and (3) explain how it connects underdetermined PDEs with the rich literature on the dual problem on overdetermined differential operators.

Instability, chaos, and nonlinear energy transfer

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The HADES seminar on Wednesday, September 17th, will be at 4:00pm in Room 732.

Speaker: Jacob Bedrossian

Abstract: In this talk we survey several recent results regarding nonlinear dynamics of stochastic differential equations. First, we discuss joint results with Alex Blumenthal, Keagan Callis, and Kyle Liss regarding the existence of stationary measures to SDEs with degenerate damping. This requires the nonlinearity to consistently pump energy from the forced modes to the damped modes. We determine sufficient conditions on the nonlinearity for this and then prove that “generic” examples of fluid-like SDEs satisfy these conditions. Second, we discuss joint results with Alex Blumenthal and Sam Punshon-Smith regarding estimating lower bounds of Lyapunov exponents and using this to prove non-uniqueness of stationary measures for SDEs with almost-surely invariant subspaces. In particular, we prove for L96 with every 3rd mode stochastically forced that for strong forcing there is exactly 2 stationary measures — the trivial one supported only on the forced modes and a second mode which is absolutely continuous with respect to Lebesgue measure and so determines the long term dynamics of almost every initial condition.

Nonuniqueness of solutions to the Euler equations with integrable vorticity

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The HADES seminar on Thursday, September 11th, will be at 3:30pm in Room 736.

Speaker: Anuj Kumar

Abstract: Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in $L^p (p > 1)$.  A celebrated open question is whether the uniqueness result can be generalized to solutions with $L^p$ vorticity. In this talk, we resolve this question in negative for some $p > 1$. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bruè and Maria Colombo.