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...

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School of Mathematics

A powerful technique developed and extended in the past decade in communication complexity is the so-called "lifting theorems." The idea is to translate the hardness results from "easier" models, e.g., query complexity, polynomial degrees, to the...

In this talk, we show a strong XOR lemma for bounded-round two-player randomized communication. For a function f:X×Y→{0,1}, the n-fold XOR function f^⊕n:X^n×Y^n→{0,1} maps n input pairs (x_1,...,x_n), (y_1,...,y_n) to the XOR of the n output bits f...

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the q^vol measure, corresponding to random cylindric partitions, the height...

Given a set of integers, we wish to know how many primes there are in the set. Modern tools allow us to obtain an asymptotic for the number of primes, or at least a lower bound of the expected order, assuming certain strength Type-I information...

Let G be a simply-connected complex semisimple algebraic group and let C be a smooth projective curve of any genus. Then, the moduli space of semistable G-bundles on C admits so called determinant line bundles. E. Verlinde conjectured a remarkable...

Several equivalent definitions of rank for matrices yield non-equivalent definitions of rank when generalized to higher order tensors. Understanding the interplay between these different definitions is related to important questions in additive...

In this talk I will first review some basics about communication complexity. Secondly I will survey some (old and new) equivalences between central problems in communication complexity and related questions in additive combinatorics. In particular...

ECH capacities have found many applications to symplectic embedding problems, most of which in the toric setting. I will discuss a new application of ECH to studying optimal embeddings for non-toric rational surfaces. The key convex geometric...

Projective modules over rings are the algebraic analogs of vector bundles; more precisely, they are direct summands of free modules. Some rings have non-free projective modules. For instance, the ideals of a number ring are projective, and for some...