Abstract: Classical Brunn-Minkowski inequality states that nth-root of volume in $R^n$ is a concave function under Minkowski addition of bounded measurable sets.

In this talk, I will discuss Brunn-Minkowski type inequalities for nonlinear capacity, called A-capacity, associated to A-harmonic PDE which is a quasi-linear elliptic PDE whose structure is modelled on the p-Laplace equation. We show that A-capacity satisfies a Brunn-Minkowski type inequalities for convex sets and in the case of equality, we show that the sets should be homothetic. I will briefly talk about the proof and, if time permits, I will also discuss some problems related to this inequality.

Abstract: We consider the problem of restriction of the Fourier transform to surfaces in R^3 with nonvanishing gaussian curvature. The methods developed since the nineties (bilinear, multilinear or polynomial partitioning), produced new results for positive curvature surfaces and the particular case of the saddle. In joint work with Detlef Müller and Stefan Buschenhenke, we have obtained the desired result for the general negative curvature case.

Abstract: In this talk I will introduce the field of modern pointwise ergodic theory, dating back to the work of Bourgain in the late 1980s and early 1990s. I will then discuss recent work concerning pointwise convergence of bilinear ergodic averages along polynomial orbits, joint with M. Mirek (Rutgers University) and T. Tao (UCLA).

Abstract: A finite set $\{a_i\}$ of reals is convex if the sequence $b_i=a_{i+1}-a_i$ is strictly monotone. Convexity presents an obstruction to additive structure. One can iterate the definition, by postulating convexity of $\{b_i\}$, etc. In a series of recent papers, we have used some old ideas of Solymosi and Garaev to prove lower bounds on cardinality of iterated sumsets of sufficiently convex sets, as well as upper bounds on the corresponding energies. This has allowed for small improvements of the state-of-the-art convex sumset bounds, as well as breaking the ice on the question of the minimum number of distinct pairwise dot products, defined by a finite set of vectors in the plane.