The last decade has witnessed a revolution in the circle of
problems concerned with proving sharp moment inequalities for
exponential sums on tori. This has in turn led to a better
understanding of pointwise estimates, but this topic remains...
The notion of strong stationarity was introduced by Furstenberg
and Katznelson in the early 90's in order to facilitate the proof
of the density Hales-Jewett theorem. It has recently surfaced that
this strong statistical property is shared by...
Incidence bound for points and spheres in higher dimensions
generally becomes trivial in higher dimensions due to the existence
of the Lenz example consisting of two orthogonal circles in ${\Bbb
R}^4$, and the corresponding construction in higher...
The "intersectivity lemma" states that if a ∈
(0,1) and A_n, n ∈ N, are measurable sets in a
probability space (X,m) satisfying m(A_n) ≥ a for
all n, then there exist a subsequence n_k, k ∈ N, which
has positive upper density and such that the...
It is an open question as to whether the prime numbers contain
the sum A+B of two infinite sets of natural numbers A, B (although
results of this type are known assuming the Hardy-Littlewood prime
tuples conjecture). Using the Maynard sieve and the...
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...
Horospherical group actions on homogeneous spaces are famously
known to be extremely rigid. In finite volume homogeneous spaces,
it is a special case of Ratner’s theorems that all horospherical
orbit closures are homogeneous. Rigidity further...
The inverse theorem for the Gowers U^{s+1}-norms has a central
place in modern additive combinatorics, but all known proofs of it
are difficult and most do not give effective bounds.
Over this seminar and the next, I will give an outline of a
proof...
Finding the smallest integer N=ES_d(n) such that in every
configuration of N points in R^d in general position there exist n
points in convex position is one of the most classical problems in
extremal combinatorics, known as the Erdős-Szekeres...
Let $f: \mathbf C^2 \rightarrow \mathbf \C^2$ be a polynomial
transformation. The dynamical degree of $f$ is defined as $\lim_n
(\text{deg} f^n)^{1/n}$, where $\text{deg} f^n$ is the degree of
the $n$-th iterate of $f$. In 2007, Favre and Jonsson...
