Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak's Mobius randomness conjecture, which deals with the disjointness of actions of (N,+) and (N,*). This talk is based on joint work with Vitaly Bergelson.

Abstract: The polynomial Szemerédi theorem of Bergelson and Leibman is a central result at the interface between ergodic theory and additive combinatorics, extending earlier results of Szemerédi and Furstenberg on arithmetic progressions. It states that each dense subset of integers contains certain polynomial configurations. The theorem follows from an ergodic theoretic result on the convergence of multiple ergodic averages with polynomial iterates. The limiting behaviour of such averages has been an object of intensive study by ergodic theorists and additive combinatorists alike. In this talk, I will discuss new results in this direction. Specifically, I will characterise the limits of multiple ergodic averages related to certain families of polynomial configurations for which little has been known previously. While doing so, I will highlight an interesting connection between the form that these limits take and the algebraic relations satisfied by the polynomial configurations. I will also discuss some consequences of this result, in particular a multiple recurrence result that extends the classical recurrence theorem of Khintchine.

Abstract: Translational tiling is a covering of a space using translated copies of a certain building block, called the “tile”, without any positive measure overlaps. Which are the possible ways that a space can be tiled? In the talk, we will survey the study of this question and report on recent progress, joint with Terence Tao. We will also discuss some applications and connections to other interesting problems.

Abstract: The lattice point counting problem for (2q+1)-dimensional Cygan-Koranyi balls is the problem of establishing error estimates for the number of integer lattice points lying inside Heisenberg dilates of the unit ball with respect to the Cygan-Koranyi norm. In this talk, I shall explain how this problem arises naturally in the context of the Heisenberg groups, and how it relates in the higher dimensional case to the (in)famous Gauss circle problem. I shall then survey some of the major results obtained to date for this lattice point counting problem, including some recently obtained results on the fluctuating nature of the error term in the 3-dimensional case.

Abstract: Divide by two if you can, otherwise multiply by 3 and add 1. Define in this way an integer map. The notorious Collatz Conjecture, which has been haunting the minds of many mathematicians since the 30’s, states that every orbit of this map eventually hits one. In this talk we will discuss generalisations of this conjecture, their connection with computation theory and recent results which link the problem with a 2-dimensional renewal process and the question of avoiding triangles.

Abstract: Let R1,R2 be the Riesz transforms on the complex plane. The singular integral R=R2+iR1 is called the complex Riesz transform. We establish the estimates on Lp, 1<p< \infty , for the integer powers Rk of R that are sharp simultaneously in k and p. This answers a question suggested in a 1996 work by Iwaniec and Martin. We present three different proofs of this result. We also conjecture the exact values of the Lp norms of Rk. This is a joint work with Andrea Carbonaro and Vjekoslav Kovač.

Abstract: In this talk we give an overview of some recent results and open questions in the area of multilinear singular integrals. We also discuss their connection with problems about existence of certain geometric point configurations in large subsets of the Euclidean space and quantitative norm convergence of some ergodic averages.

Abstract: Green and Tao famously showed that the von Mangoldt function is Gowers uniform and used this to give an asymptotic formula for the number of solutions to any linear system of equations (of finite complexity) in the primes. Their theorem however gave no quantitative rate of decay for the Gowers uniformity norms (of degree greater than 4). In this talk, I will discuss a quantification of their result and applications to e.g. Szemerédi's theorem with shifted prime differences. This is based on joint work with Terence Tao.

Abstract: Banach space valued Poincaré inequality and dimension free estimates of singular integrals on discrete cube are the topics of this talk. For Banach space valued functions Poincaré inequality is usually replaced by Pisier’s inequality. It is interesting to understand precisely for which Banach spaces X Pisier inequality on Hamming cube is dimension free. This has been done in a paper of Ivanisvili-Van Handel-Volberg (IVHV). This, in particular, gave a solution to Enflo’s conjecture. There is a whole scale of related inequalities filling the gap between Pisier’s inequality and singular integral inequalities on Hamming cube. For those inequalities the description of class of Banach spaces X that allows the dimension free estimates is not known, the reason is related to the following fact: we are used to the “fact” that singular integrals on X-valued functions have to be bounded in Lp(X) if X is UMD. But on Hamming cube this is not true anymore. However, we will show a wide class of spaces for which those inequalities hold. The proofs are the mixture of the formula of IVHV and quantum random variables technique á la Francoise Lust-Piquard.

Abstract: For a given topological dynamical system (X,T) we are interested in the distribution of orbits of an initial condition x\in X sampled at prime times. We will recall known results, main methods and discuss some recent progress. We will also mention some other non-standard ergodic averages related to Mobieus disjointness conjecture of Sarnak.

Abstract: A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge in the mean to the product of the integrals. We give necessary and sufficient conditions for joint ergodicity that are flexible enough to recover most of the known examples of jointly ergodic sequences and also allow us to answer some related open problems. An interesting feature of our arguments is that they avoid deep tools from ergodic theory that were previously used to establish similar results. Our approach is inspired by a technique developed by Peluse and Prendiville in order to give quantitative variants for the finitary version of the polynomial Szemer\'edi theorem.

Abstract: We present some progress on polynomial progression in finite fields and/or in real line. Many questions are still open in this field.