Erdős-style geometry is concerned with combinatorial questions
about simple geometric objects, such as counting incidences between
finite sets of points, lines, etc. These questions can be typically
viewed as asking for the possible number of...
Given $n\in\mathbb{N}$ and $\xi\in\mathbb{R}$, let
$\tau(n;\xi)=\sum_{d|n}d^{i\xi}$. Hall and Tenenbaum asked in their
book \textit{Divisors} what is the value of $\max_{\xi\in[1,2]}
|\tau(n;\xi)|$ for a ``typical'' integer $n$. I will present
work...
Sets of recurrence were introduced by Furstenberg in the context
of ergodic theory and have an equivalent combinatorial
characterization as intersective sets, an observation
which has led to interesting connections between these areas.
Complex dynamics explores the evolution of points under
iteration of functions of complex variables. In this talk I will
introduce into the context of complex dynamics, a new approximation
tool allowing us to construct new examples of entire...
The Mackey-Zimmer representation theorem is a key structural
result from ergodic theory: Every compact extension between ergodic
measure-preserving systems can be written as a skew-product by a
homogeneous space of a compact group. This is used, e.g...
We will discuss a version of the Green--Tao arithmetic
regularity lemma and counting lemma which works in the generality
of all linear forms. In this talk we will focus on the qualitative
and algebraic aspects of the result.
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...
We show that for every positive integer k there are positive
constants C and c such that if A is a subset of {1, 2, ..., n} of
size at least C n^{1/k}, then, for some d \leq k-1, the set of
subset sums of A contains a homogeneous d-dimensional...
Ever since Furstenberg proved his multiple recurrence theorem,
the limiting behaviour of multiple ergodic averages along various
sequences has been an important area of investigation in ergodic
theory. In this talk, I will discuss averages along...
A density theorem for L-functions is quantitative measure of the
possible failure of the Riemann Hypothesis. In his 1990 ICM talk,
Sarnak introduced the notion of density theorems for families of
automorphic forms, measuring the possible failure of...
