Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

We find the sharp constant C in the inequality $\|\phi\|_{L^r} \leq C \|\phi\|_{L^p}^{p/r} \|\phi\|_{BMO}^{1-p/r}$, where $1\leq p\leq r<+\infty$. We use the Bellman function machinery to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. Based on joint works with D. Stolyarov, V. Vasyunin,  and  I. Zlotnikov. 

Translational tiling is a covering of a space using translated copies of some building blocks, called the “tiles”, without any positive measure overlaps.  Which are the possible ways that a space can be tiled?  In the talk, we will discuss the study of this question as well as its applications, and report on recent progress, joint with Terence Tao. 

Poincaré series are natural functions that arise in Riemannian geometry 
when one wants to count the number of geodesic arcs of length less than 
T between two given points on a compact manifold. I will begin with an 
introduction on this topic. Then I will discuss some recent results with 
N.V. Dang (Univ. Paris Sorbonne) showing that, in the case of negatively 
curved manifolds, these series have a meromorphic continuation to the 
whole complex plane. This can be shown by relating Poincaré series with 
the resolvent of the geodesic vector field and by exploiting recent 
results on this resolvent obtained through microlocal methods. If time 
permits, I will also explain how the genus of a surface can be recovered 
from the analysis of these series.
 

Two years ago, a colleague from economics asked me for the ”best” way to compute the average income over the last year. At first I didn’t understand but then he explained it to me: suppose you are given a real-valued function f(x) and want to compute a local average at a certain scale. What we usually do is to pick a nice probability measure u, centered at 0 and having standard deviation at the desired scale, and convolve f ∗ u. Classical candidates for u are the characteristic function or the Gaussian. This got me interested in finding the ”best” function u – this problem comes in two parts: (1) describing what one considers to be desirable properties of the convolution f ∗ u and (2) understanding which functions u satisfy these properties. I tried a basic notion for the first part, ”the convolution should be as smooth as the scale allows”, and ran into lots of really funky classical Fourier Analysis that seems to be new: (a) new uncertainty principles for the Fourier transform, (b) that potentially have the characteristic function as an extremizer, (c) which leads to strange new patterns in hypergeometric functions and (d) produces curious local stability inequalities. Noah Kravitz and I managed to solve two specific instances on the discrete lattice completely, this results in some sharp weighted estimates for polynomials on the unit interval – both the Dirichlet and the Fejer kernel make an appearance. The entire talk will be completely classical Harmonic Analysis, there are lots and lots of open problems and I will discuss several.