Special Year Research Seminar

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...

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...

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. This...

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...

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...

Given a topological dynamical system (X,T) a bounded sequence (an) and f∈C(X) we are interested in the asymptotic behavior of

1∑n≤N|an|∑n≤Nanf(Tnx)

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.