# Video Lectures

I will present results establishing cancellation in short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases, or more general nilsequence phases. These results imply the...

We will discuss multilinear variants of Weyl's inequality for the exponential sums arising in pointwise convergence problems related to the Furstenberg-Bergelson-Leibman conjecture. We will also illustrate how to use the multilinear Weyl inequality...

In this talk we present a natural generalization of a sumset conjecture of Erdos to higher orders, asserting that every subset of the integers with positive density contains a sumset $B_1+\ldots +B_k$ where $B_1, \ldots , B_k$ are infinite. Our...

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In unpublished lecture notes, William A. Veech considered the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