In this talk, I will discuss various bounds for the $L^p$ distance of polynomials on discrete hypercubes from Walsh tail spaces, extending some of Oleszkiewicz’s results (2017) for Rademacher sums. This is based on joint work with Alexandros Eskenazis (CNRS, Sorbonne University).

For planar billiard tables, the marked length spectrum encodes the lengths
of action (minus the length) minimizing orbits of a given rational rotation
number. For strictly convex tables, a renormalization of these lengths extends
to a continuous function called Mather’s beta function or the mean minimal
action. We show that using the algebraic structure of its Taylor coefficients,
one can prove C infinity compactness of marked length isospectral sets. This
gives a dynamical counterpart to the Laplace spectral results of Melrose,
Osgood, Phillips and Sarnak.

Abstract: In this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review a direct approach to the construction of asymptotic Bergman projections, developed by Deleporte--Hitrik--Sjöstrand in the case of real analytic weights, and Hitrik--Stone in the case of smooth weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.

We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé approximant of a random power series $f(z) = \sum_{k=0}^\infty a_k z^k$, for $a_k$ independent, with finite first moment, and anti-concentrated. Under the same assumptions we show that almost surely $f$ has infinitely many zeros in the unit disc, with the unit circle serving as a (strong) natural boundary for $f$. For $R_m$ the radius of the largest disc containing at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our setting the poles of the $[m,n]$-Padé approximant almost surely cluster uniformly at the circle of radius $R_m$ when $n\to \infty$ and $m$ stays fixed, and we provide almost sure rates of convergence of these $R_m$'s to $1$. We also show that our results on the clustering of the zeros hold for log-concave vectors $(a_k)$ with not necessarily independent coordinates. This is joint work with S. Dostoglou (University of Missouri).