Abstract: The Gowers uniformity k-norm on a finite abelian group measures the averages of complex functions on such groups over k-dimensional arithmetic cubes. The inverse question about these norms asks if a large norm implies correlation with a function of an algebraic origin. The analogue of the Gowers uniformity norms for measure-preserving abelian actions are the Host-Kra-Gowers seminorms which are intimately connected to the Host-Kra-Ziegler factors of such systems. The corresponding inverse question in the dynamical setting asks for a description of such factors in terms of systems of an algebraic origin. In this talk, we survey recent results about the inverse question in the dynamical and combinatorial settings, and in particular how an answer in the former setting can imply one in the latter. This talk is based on joint works with Asgar Jamneshan and Terence Tao.

Abstract: The classical edge-isoperimetric inequality on the hypercube states that $|\nabla A| \geq |A| \log_2 (1/|A|)$ for any set $A \subseteq \{0,1\}^d$, where $\nabla A$ is the set of edges between A and its complement. This is sharp, since the inequality saturates on any subcube. Extensions and variants of this inequality have been studied by several authors, but so far none of them has the property of saturating on all subcubes. In this talk, we will present such an inequality, as well as improved versions of existing estimates. We will also discuss some applications. This is joint work with Paata Ivanisvili and José Madrid.

Abstract: Following Birkhoff's proof of the Pointwise Ergodic Theorem, it has been studied whether convergence still holds along various subsequences. In 2020, Bergelson and Richter showed that under the additional assumption of unique ergodicity, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors on $n$ counted with multiplicity. In this talk, we will see that removing this assumption, a pointwise ergodic theorem does not hold along $\Omega(n)$. We will then study the interplay of the dynamics with certain number theoretic properties of $\Omega(n)$ to obtain further information on the asymptotic behavior of this sequence. 

Abstract: Recent years have witnessed important breakthroughs regarding arithmetic combinatorics in the finite field setting, such as Peluse’s power saving bounds in the linearly independent case of the polynomial Szemerédi theorem (2019).

We will promote a dynamical point of view for interpreting various combinatorial statements and use this perspective to give several new results about polynomial configurations in finite fields. In particular, we will see how adapting ergodic-theoretic considerations to a finitary context leads to a power saving bound for the Furstenberg—Sárközy theorem and provides new families of partition regular polynomial equations over finite fields. We hope to demonstrate the utility of infinitary mathematical tools and methods (coming primarily from ergodic theory, equidistribution, and measure theory on ultraproduct spaces) in a finitary setting.

This talk is based on joint work with Vitaly Bergelson.

Abstract: Multiple results in harmonic analysis involve integrals of functions over curves, such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates. In this set-up, boundedness depends strongly on the non-vanishing of the torsion of the associated curve.

Over the past years there has been considerable interest in extending these results to two settings: On one hand, a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. On another hand, the discrete analogues of these problems, where one is summing over integer points on the curve.

While quite different in nature, these two settings share one property: One can provide operator bounds that are independent of the curve, but that depend only on its complexity (such as polynomial degree, frequency bounds.)

In this talk, I will introduce and motivate the concept of affine arclength measure, show new decomposition theorems for polynomial curves over characteristic zero local fields, and show how to use them to get uniformity results in both continuous and discrete harmonic analysis. 

Abstract: Consider a Nilpotent Lie group $G$ and a discrete subgroup $\Gamma$ such that the topological quotient $G/\Gamma$ is compact. Certain problems in arithmetic combinatorics are concerned with an equidistribution theory on $G/\Gamma$. This theory studies the behavior of orbits $g^n\Gamma$ and classification of their limit sets in $G/\Gamma$. 

In 2012, Green and Tao proved a quantitative equidistribution theory on $G/\Gamma$, achieving polynomial bounds on the rate of equidistribution and with exponent single exponential in the dimension of $G$. In this talk, we go over a recent result, which improves the bounds to have exponent polynomial in the dimension of $G$. We also discuss implications of this result to arithmetic combinatorics. A key obstruction that the proof of this result overcomes is "induction on dimensions", which also seem to appear elsewhere in higher order Fourier analysis over $\mathbb{Z}/N\mathbb{Z}$.

Abstract: We discuss convergence (in $L^2$) results for multiple ergodic averages along sequences of polynomial growth evaluated at primes. Building on the work of Frantzikinakis, Host, and Kra who showed that polynomial ergodic averages along primes converge, we generalize their results to other sequences with polynomial growth. Combining our results with Furstenberg's correspondence principle, we derive several applications in combinatorics. The most interesting application is that positive density subsets of $\mathbb{N}$ contain arbitrarily long arithmetic progressions with common difference of the form $\lfloor{p^c}\rfloor$, where $c$ is a positive non-integer and $p$ is a prime number. The main tools in the proof are a recent result of Matom\"{a}ki, Shao, Tao, and Ter\"{a}v\"{a}inen on the uniformity of the von Mangoldt function in short intervals, a polynomial approximation of our sequences with good equidistribution properties, and a lifting trick that allows us to replace $\mathbb{Z}$-actions on a probability space by $\mathbb{R}$-actions on an extension of the original system.

Abstract:  The notion of `stabilizer’ of an approximate subgroup has been extensively studied in work of Croot—Sisask, Hrushovski, Sanders, Tao and Weil among others. For a subset $X$ of a group $G$, the stabilizer of $X$ is simply defined as the set of those elements of $G$ that do not change $X$ too much. They are used to relate subsets of small doubling with approximate subgroups and often have very good properties. 

In this talk, I will discuss how these stabilizers are related to return times of cross-sections of dynamical systems. I will illustrate this by explaining two results in aperiodic order (i.e. the study of discrete subsets of homogeneous spaces with long-range aperiodic order). First, I will explain an extension of a theorem of Lagarias about quasi-coherent sets; and, secondly, an extension of a theorem of Meyer about the structure of Meyer sets. If time permits, I will discuss questions and conjectures related to these topics

Abstract: How many skewed affine hyperplanes are required to cover the vertices of the n-dimensional hypercube?  Here skewed means that the coordinates of the normal vector of the affine hyperplane  are nonzero real numbers.   A related question is the following: can one recover the multilinear polynomial of degree less than n/99 through those points of {0,1}^n whose hamming weights are divisible by 99? The talk will be based on joint work with Ohad Klein and Roman Vershynin. 

Abstract: The problem of determining the sharp constant for the endpoint Stein-Tomas Fourier extension inequality on the circle has received a lot of attention over the last decade. The same question for the case of the two-dimensional sphere has been elegantly answered by D. Foschi about ten years ago. In this seminar, we discuss a sharp endpoint Stein-Tomas Fourier extension inequality on the circle for functions whose spectrum satisfies a certain arithmetic constraint. Such arithmetic constraint corresponds to a generalization of the notion of $B_3$-set. The seminar is based on joint work with Felipe Gon\c{c}alves.

Abstract: In this paper we introduce and discuss various notions of doubling for measure- preserving actions of a countable abelian group G. Our main result characterizes 2-doubling actions, and can be viewed as an ergodic-theoretical extension of some classical density theorems for sumsets by Kneser. All of our results are completely sharp and they are new already in the case when G = (Z, +). With M. Bj\"orklund, A. Fish.