Special Year 2022-23: Dynamics, Additive Number Theory and Algebraic Geometry

Abstract: A version of the polynomial Szemer´edi theorem was shown to hold in finite fields in [BLM05]. In particular, one has patterns

${x, x + P1(n), . . . , x + Pk(n)}$ (1)

for polynomials with zero constant term in large subsets of finite fields...

Abstract: The works of Furstenberg and Bergelson-Leibman on the Szemeredi theorem and its polynomial extension motivated the study of the limiting behavior of multiple ergodic averages of commuting transformations with polynomial iterates. Following...

Abstract: I will present results establishing cancellation in short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases, or more general nilsequence phases. These results imply...

Abstract: Abstract: We will discuss multilinear variants of Weyl's inequality for the exponential sums arising in pointwise convergence problems related to the Furstenberg-Bergelson-Leibman conjecture. We will also illustrate how to use the...

Abstract: In this talk we present a natural generalization of a sumset conjecture of Erdos to higher orders, asserting that every subset of the integers with positive density contains a sumset $B_1+\ldots +B_k$ where $B_1, \ldots , B_k$ are infinite...

ABSTRACT: In the 1970’s Erdos asked several questions about what kind of infinite structures can be found in every set of natural numbers with positive density. In recent joint work with Kra, Richter and Robertson we proved that every such set A can...

Abstract: Given a fractal set E in $R^n$ and a set F in $Gr(k,n)$, can we find k-plane S in F such that the orthogonal projection of E onto S is large? 

We will survey some classical and recent projection theorems and discuss their applications. ...

Abstract: For a topological dynamical systems (T, X) and a fixed $x \in X$ we are interested in the distribution of prime and semi-prime orbits, i.e. ${T px}p='$ and ${T p1p2 x}p1,p2='$. We are interested in systems for which the related sequences of...

Abstract: Given $B \subset N$, we consider the corresponding set $FB$ of $B$-free integers, i.e. $n \in FB i_ no b \in B$ divides $n$. We $de_{ne} X \eta_}$ the B-free subshift _ as the smallest subshift containing $\eta := 1FB \in {0, 1}Z$. Such...

Abstract: A central question in additive combinatorics is to determine what class of structured functions is enough to determine multilinear averages such as

$\mathbb{E}_{x,a} f_1(x) f_2(x+a) f_3(x+2a) f_4(x+3a)$.

In ergodic theory the analogous...