Video Lectures

Infinite Sumsets in Sets with Positive Density

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...

Projection Theorems and Applications

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...

Thermodynamic Formalism for B-free Dynamical Systems

Joana Kulaga-Przymus

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...

The Complexity of Multilinear Averages

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...

On a Conjecture of Veech About Möbius Orthogonality

Thierry de la Rue

In unpublished lecture notes, William A. Veech considered the following potential property of the Möbius function: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...

Arbitrarily Slow Decay in the Mobius Disjointness Conjecture

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...

From Embedded Contact Homology to Surface Dynamics

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

Theresa Anderson

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...

Hardy-Littlewood and Chowla Type Conjectures in the Presence of a Siegel Zero

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.

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The Unipotent Mixing Conjecture

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...