Special Year 2022-23: Dynamics, Additive Number Theory and Algebraic Geometry

Abstract: In unpublished lecture notes, William A. Veech considered the following potential property of the Möbius function:
"In any Furstenberg system of the Möbius function, the zero-coordinate is orthogonal to any function measurable with respect...

Abstract: We construct examples showing that the correlation in the Mobius disjointness conjecture can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of μ(n) one can put any bounded sequence such that...

Abstract: Behaviors of objects of algebraic interest, such as polynomials, elliptic curves and number fields -- many of which are still unknown -- fall into the field of arithmetic statistics..  By bringing in Fourier analysis to interplay with a...

Abstract: Low-lying horocycles are known to equidistribute on the modular curve. Here we consider the joint distribution of two low-lying horocycles of different speeds in the product of two modular curves and show equidistribution under certain...

Abstract: We discuss some consequences of the existence of a Siegel zero for various questions relating to the distribution of the prime numbers, and in particular to conjectures of Hardy-Littlewood and Chowla type.  This is joint work with Joni...

Abstract: A subset of a group is said to be product free if it does not contain the product of two elements in it. We consider how large can a product free subset of the alternating group $A_n$ be? 

In the talk we will completely solve the problem...

Abstract: Approximate lattices in locally compact groups are approximate subgroups that are discrete and have finite co-volume. They provide natural examples of objects at the intersection of algebraic groups, ergodic theory and additive...

Abstract: Work of Mark Shusterman and myself has proven an analogue of Chowla's conjecture for polynomial rings over finite fields, which controls k-points correlations of the Möbius function for k bounded by a certain function of the finite field...

Abstract: We present recent work, joint with J. Wolf, in which we define a natural ternary analogue of the order property, called the functional order properly, and show that subsets of $F_p^n$ without the functional order property admit especially...

Abstract: In this talk, I will discuss a proof of a quantitative version of the inverse theorem for Gowers uniformity norms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$. The proof starts from an earlier partial result of Gowers and myself...