MIT PDE/Analysis Seminar

Abstract: I will discuss the structure theory of zero and singular sets of solutions to parabolic PDEs, drawing an analogy with the corresponding theory in the polynomial setting. As in the polynomial setting, their geometric and measure-theoretic structure is controlled by frequency, a quantity that plays the role of degree for polynomials. However, this analogy breaks down at the level of topology.

This talk is based in part on joint work with Max Hallgren and Zilu Ma, and on joint work with Javier Gómez-Serrano and Alexander Logunov.

Abstract: Suppose that a probability measure on a compact manifold $M$ is invariant under a collection of diffeomorphisms of $M$. I will explain some criteria on the measure and diffeomorphisms which ensure that the measure is a smooth volume form on $M$. The assumptions say, informally, that the measure is of Lebesgue class when conditioned along a random collection of (measurable) foliations by submanifolds.

The results, joint with Brown, Eskin, and Rodriguez Hertz, are applicable to measures for which we can deduce additional rigidity properties (joint with the same authors). I will explain the necessary background and motivation from dynamics.

Abstract: Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the $L^\infty$ norm of every Laplace eigenfunction on $X_n$ with bounded eigenvalue decays polynomially in $n$. Using similar methods, we also show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. Our proof relies on the Selberg pre-trace formula and a variant of the polynomial method.

This is joint work with Zhongkai Tao.

Abstract: We establish Gaussian scaling limit, for the (elliptic) Anderson model in $4$ dimensions, with small coupling constant. This is a simplest example of a critical model in SPDEs, a theme which has been a major challenge of the field. Our theorem provides the first example where infinitely many renormalization constants are present and controlled.

This is joint work with Hao Shen.

Abstract: In this talk, I will describe the spectrum of a non-self-adjoint version of the periodic Harper’s equation. One example of this is the so-called Scottish Flag Operator. A description of the spectrum relies on exact WKB methods to approximate solutions to analytic finite difference equations. I will motivate the study of such operators, describe the main ideas of the proof, and explain the exact WKB techniques that are used.

This is joint work with Frédéric Klopp.

Abstract: The passage from interacting particles to PDEs is a central problem in statistical mechanics. For a dilute gas, the Boltzmann equation describes the evolution of the one-particle distribution, while the Boltzmann hierarchy keeps track of joint particle statistics and their correlations. I will explain how these descriptions are related and discuss the approach to equilibrium for correlated gases. The main analytic challenge is to control an infinite system in which each equation depends on the next. I will present an energy method that combines the dissipation produced by transport and collisions with estimates that remain uniform across the hierarchy. For a class of solutions near equilibrium satisfying suitable constraints on conserved quantities, these estimates give exponential relaxation toward an equilibrium of independent particles with Maxwellian velocities, together with quantitative decay of pair correlations. The talk will emphasize the statistical meaning of the equations and the ideas behind the estimates. [https://arxiv.org/abs/2610.24089]

Abstract: The periodic Zakharov system couples Schrödinger and wave equations, leading to a special shell-type structure in frequency space. Motivated by this observation, I study bilinear decoupling estimates for the paraboloid when the Fourier supports are restricted to thin spherical shells of two different radii. I establish sharp bilinear shell-type decoupling estimates that improve upon applying linear shell decoupling separately to the two factors. As applications, I obtain bilinear Strichartz estimates on tori, with further improvements on the square torus using a refined circle method. These estimates are motivated by the study of nonlinear smoothing for the periodic Zakharov system.