MIT PDE/Analysis Seminar

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.