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.

Strong law of large numbers gives a method to estimate the
average of a function on the Boolean cube so that it is accurate with
high probability. But there is still a little risk that it is
inaccurate. I will present a polynomial time method to estimate the
averages of certain functions on Boolean cube without risk of being
inaccurate.

Maximal functions play a central role in the study of differentiation,
singular integrals and almost everywhere convergence. With Sergey Tikhonov
(ICREA, Barcelona) we recently proved some pointwise estimates for maximal
functions in terms of smoothness and rearrangements. I plan to discuss the recent
progress on these topics and some applications. In particular, I will discuss the
Fefferman-Stein inequality for the sharp maximal function for r.i. spaces which
are close to L∞.