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∞.

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. We discuss a general method based on the multi-dimensional polynomial interpolation identity related to the so called Combinatorial Nullstellensatz of Alon that is powerful enough to establish many such identities, both known before and new, in a simple manner. Based on joint works with R. Karasev, G. Karolyi, Z. Nagy and V. Volkov.

Let a_ij be a finite collection of real numbers, and let s_i and s_j be spins (they take only two values +1 or -1). The goal is to choose the spins s_i and s_j so  that to maximize the double sum a_ij s_i s_j, where the summation runs over all indecies from 1 to n such that i does not qual to j.  If a_ij =1 then the sum is maximized if all spins have the same sign which gives the result to be of order n^2. However, if a_ij=-1 then the sum is maximized when half of the spins take value 1 and the other half -1. When a_ij are arbitrary real numbers this becomes a nontrivial problem (also known as Dean's problem). The problem is well understood  in average when a_ij are i.i.d symmetric  +1 or -1 random vairables  (the Sherrington--Kirkpatrick model).  We will speak about possible  lower bounds of  the maximum in terms of arbitrary real numbers a_ij,  its extensions to d-spin case, some conjectures in this area, and their applications in quantum computing.