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. 

Let K be a convex body with volume one and barycentre at the origin.
How small is the volume of the intersection of K and -K? I shall
discuss such lower bounds and present applications to the Hadwidger
covering/illumination conjecture. Based on joint work with H. Huang,
B. Slomka and B. Vritsiou.