Previous Special Year Seminar

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...

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...

The Chowla conjecture from 1965 predicts that all autocorrelations of the Liouville function vanish. In fact, after an adaptation, the Chowla conjecture was expected to hold for all aperiodic multiplicative functions with values in the unit disc (cf...

We discuss some recent progress on the model-theoretic problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include Castle’s recent solution of the Restricted Trichotomy...

The Bowen-Ruelle conjecture predicts that geodesic flows on negatively curved manifolds are exponentially mixing with respect to all their equilibrium states. In a breakthrough in '98, Dolgopyat pioneered a method rooted in the thermodynamic...