Abstract: The Weakly Mixing PET (Polynomial Ergodic Theorem) established in Ergodic Theory Dynam. Systems 7 (1987), no. 3, 337–349, states, roughly, that for any weakly mixing transformation T and any nonconstant polynomials p_i(n) ∈ ℤ[n], i = 1,2,...,k, satisfying p_i – p_j ≠ const for any distinct i, j, the polynomial actions T^p_i(n) exhibit independent behavior. The purpose of this talk is to discuss some new PETs which involve commuting measure preserving transformations. (Joint work with Rigo Zelada).

Abstract: We discuss a proof of multiple recurrence for ergodic systems (and thereby of Szemerédi's theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs-de Leeuw-Glicksberg decomposition and properties of the Gowers-Host-Kra uniformity seminorms.

Abstract: We will present forthcoming work which proves a sharp L^7 square function estimate for the moment curve in R^3 using ideas from decoupling theory. Consider a function f with Fourier support in a small neighborhood of the moment curve. Partition the neighborhood into box-like subsets and form a square function in the Fourier projections of f onto these box-like regions. Bounding f in L^p by the square function in L^p is an important way to quantify the cancellation that f has from its specialized Fourier support. 

Abstract: In recent work Kra, Moreira, Richter and I showed that positive density sets always contain the sum of any finite number of infinite sets. Central to our approach was the existence of certain cubical dynamical configurations that I will describe in this talk. After outlining the connection with our combinatorial result, I will explain how ergodic theory can be used to guarantee existence of such cubes.

Abstract:  The scattering transform for the Dirac system of differential equations is commonly viewed as a non-linear version of the classical Fourier transform. This connection leads to natural questions on finding analogs of various  properties of the Fourier transform in non-linear settings. In my talk I will give  a short overview of that area and present a version of Carleson's theorem on pointwise convergence in the non-linear case.

Abstract:  Since Szemerédi's seminal work on graphs in the 70s, regularity lemmas have been of fundamental importance in many areas of discrete mathematics, with analogous results being proven in more analytic settings. This talk will survey recent work on regularity decompositions of subsets of finite groups under additional assumptions such as stability or bounded VC-dimension, which turn out to have particularly desirable properties. In the second half of the talk, we will describe very recent joint work with Caroline Terry (Ohio State University) which extends these ideas to the realm of higher-order Fourier analysis.

Abstract: In 1920s, M. Riesz and E.C. Titchmarsh proved that the (continuous) Hilbert transform is bounded on $L^p$, and that its various discretizations are bounded $\ell^p$. The exact value of the operator norm of the Hilbert transform was only found half a century later by Pichorides and Cole, but already Riesz and Titchmarsh asked whether the corresponding norms of the continuous and discrete Hilbert transforms are equal. Together with Rodrigo Bañuelos from Purdue University, we were able to provide an affirmative answer to this question for one variant of the discrete Hilbert transform (the one with kernel 1/n), and a partial result for another one (the Riesz–Titchmarsh transform, with kernel 1/(n + 1/2)). During my talk, I will briefly discuss the history of the problem, present our results, and show the key ideas of the proof.

Abstract: Examine a subset S of the real line, or, more generally of a Euclidean space. Consider the Hilbert space of square integrable functions on S. Is it possible to find a Riesz basis for this space consisting only of complex exponentials (the frequencies may be any real numbers)? This tantalising question is known for very few S. We will discuss the first construction of a set that does not have any such basis. All terms will be explained in the talk. Joint work with Alexander Olevskii and Shahaf Nitzan.

Abstract: A quasi-crystallline point process is a translation-invariant probability measure on the space of uniformly discrete configurations in a Euclidean space whose difference sets are uniformly discrete approximate subgroups.  We investigate the growth rates of the number variances for such processes via their diffraction measures and show  that they depend in a subtle way on diophantine properties of the processes. On the way we indicate a few unanswered  questions.

Abstract: It is quite natural to expect that minimization of  pairwise interaction energies leads to uniform distributions, at  least for "nice" kernels. However, the opposite  effect occurs in many interesting examples, especially for attractive-repulsive energies with very weak  repulsion: minimizers tend to cluster, i.e. minimizing measures are discrete (or at least are very non-uniform, e.g. supported on "thin" or lower-dimensional sets). At the same time it appears that weak repulsion at small scales is not the only reason causing this effect.  We shall discuss some results related to this curious phenomenon and its relation to problems and conjectures in analysis, signal processing, discrete geometry etc. 

Abstract: I will discuss tilings of the real line by translates of a function $f$, which means systems $\{f(x - \lambda), \lambda \in \Lambda\}$ of translates of $f$ that form a partition of unity. Which functions $f$ can tile by translations, and what can be the structure of the translation set $\Lambda$? I will survey the subject and present some recent results.

Abstract:  In this talk, I will discuss a proof of a quantitative version of the inverse theorem for Gowers uniformity norms U^5 and U^6 in F_2^n. The proof starts from an earlier partial result of Gowers and myself which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The most of the argument is a study of the relationship between the natural actions of the symmetric groups Sym_4 and Sym_5 on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. One of the outcomes of the proof is a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in 5 variables, which is known to be false in the case of 4 variables. Finally, I will discuss the possible generalization of the argument for U^k norms.