Seminars Sorted by Series

I'll introduce o-minimality from a user's perspective assuming zero background. I'll talk about some of the main examples of o-minimal structures: as a user of o-minimality your first goal is to find out whether your favorite set lives in one of...

I'll focus specifically on point counting results in o-minimal structures. I'll start with the classical theorem of Pila and Wilkie and move on to improved versions that only hold in the "sharp" variant of o-minimality.

A Hodge structure is a certain linear algebraic datum.  Importantly, the cohomology groups of any smooth projective algebraic variety come equipped with Hodge structures which encode the integrals of algebraic differential forms over topological...

Differential Galois groups are algebraic groups that describe symmetries of some systems of differential equations. The solutions considered can live in any differential field and thus a natural framework to consider such symmetries is the setting...

A Hodge structure is a certain linear algebraic datum.  Importantly, the cohomology groups of any smooth projective algebraic variety come equipped with Hodge structures which encode the integrals of algebraic differential forms over topological...

The goal of these lectures is to present the fundamentals of Simpson’s correspondence, generalizing classical Hodge theory, between complex local systems and semistable Higgs bundles with vanishing Chern classes on smooth projective varieties.

Ben Bakker explained to us how to construct moduli spaces of polarized Hodge structures, and then produced period maps associated to families of smooth projective varieties. However, in practice one often encounters a family of smooth varieties...

During the 2026-27 academic year the School will have a special program on Conformally Symplectic Dynamics and Geometry. Michael Hutchings, University of California, Berkeley will be the Distinguished Visiting Professor.

The purpose of this special...

This will be a survey talk about recent progress on pointwise convergence problems for multiple ergodic averages along polynomial orbits and their relations with the Furstenberg-Bergelson-Leibman conjecture.

Given a topological dynamical system $(X,T)$ a bounded sequence $(a_n)$ and $f\in C(X)$ we are interested in the asymptotic behavior of $$\frac{1}{\sum_{n\leq N}|a_n|}\sum_{n\leq N}a_nf(T^nx)$$

The restriction conjecture, one of the most central problems in harmonic analysis, studies the Fourier transform of functions defined on curved surfaces; specifically, it claims that the level sets of such Fourier transforms are relatively small...

A number is called y-smooth if all of its prime factors are bounded above by y. The set of y-smooth numbers below x forms a sparse subset of the integers below x as soon as x is sufficiently large in terms of y. If f_1, …, f_r \in Z[x_1,…,x_s] is a...

A word on d letters is an element of the free group of rank d, say, with basis x_1,…,x_d. Given a word w=w(x_1,…,x_d) on d letters, for every group G, there is a word map w:G^d—> G given by substituting the x_i’s with elements of G. We say that a...

A famous conjecture of Littlewood states that the Fourier transform of every set of N integers has $l^1$ norm at least log(N), up to a constant multiplicative factor. This was proved independently by McGehee-Pigno-Smith and Konyagin in the 1980s...

The talk will consists of a long historical introduction to  the topic of deviation
of ergodic averages for locally Hamiltonian flows on compact surafces  as well as
some current results obtained in collaboration with Corinna Ulcigrai  and Minsung...

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| \gg \min \{ |A|^{1+c}, |G| \}. In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset...

A discrete subgroup of PSL(2,C) is called a Kleinian group. I will present  a criterion on  when a discrete faithful representation of a Kleinian group into PSL(2,C) is a conjugation, or equivalently  a criterion on when an equivariant embedding of...

If a set of integers is syndetic (finitely many translates cover the integers), must it contain two integers whose ratio is a square?  No one knows.  In the broader context of the disjointness between additive and multiplicative configurations and...

A conjecture of Erdős states that for every large enough prime q, every reduced residue class modulo q is the product of two primes less than q. I will discuss my on-going work with Kaisa Matomäki establishing among other things a ternary variant of...

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...

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 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...

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...

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...

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...

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 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...

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...

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...

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...

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...

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...

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...