Previous Special Year Seminar

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...

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...

In the study of some dynamical systems, the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for...

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...

In model theory Fraisse limits are certain highly homogeneous countable structures -- examples include the rational numbers as the unique dense linear order without endpoints, and the Rado graph as the "unique infinite random graph".  I will discuss...

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 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...

In these two talks, I will discuss the structure of certain varieties that depend functorially on the choice of a finite-dimensional vector space. Examples include the variety of d-way tensors of "slice rank" at most k and the variety of degree-d...

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...