# Video Lectures

In the 1970’s Erdos asked several questions about what kind of infinite structures can be found in every set of natural numbers with positive density. In recent joint work with Kra, Richter and Robertson we proved that every such set A can be...

Given a fractal set E in $R^n$ and a set F in $Gr(k,n)$, can we find k-plane S in F such that the orthogonal projection of E onto S is large?

We will survey some classical and recent projection theorems and discuss their applications. This...

Given $B \subset N$, we consider the corresponding set $FB$ of $B$-free integers, i.e. $n \in FB i_ no b \in B$ divides $n$. We $de_{ne} X \eta_}$ the B-free subshift _ as the smallest subshift containing $\eta := 1FB \in {0, 1}Z$. Such systems are...

A central question in additive combinatorics is to determine what class of structured functions is enough to determine multilinear averages such as

$\mathbb{E}_{x,a} f_1(x) f_2(x+a) f_3(x+2a) f_4(x+3a)$.

In ergodic theory the...

In unpublished lecture notes, William A. Veech considered the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 to the...

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 the Cesàro...

I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of S^3 along T(2,q) torus knots to deduce information about the dynamics of symplectomorphisms of the genus (q-1)/2 pages...

### On Random Polynomials and Counting Number Fields: Fourier Analysis Meets Arithmetic Statistics

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 wide variety...

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 Teravainen.

...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...