Bohnenblust--Hille (BH) inequalities are an extension of Littlewood's 4/3 inequality and have found many applications to harmonic analysis. A variant of BH inequalities for Boolean cubes has been proven with constants that are dimension-free and subexponential in degree. Such inequalities have found great applications in learning low-degree Boolean functions. Motivated by learning quantum observables, a quantum analog of BH inequality for Boolean cubes was recently conjectured and resolved unaware of the conjecture. In this talk, we give a simpler proof with better constants. As applications, we study learning problems for quantum observables of low degrees. Joint work with Alexander Volberg.

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.