L6

Let us fix a prime $p$. The Erdős-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero.

In this talk, we discuss a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method, as well as some new combinatorial ideas.

A subset $D$ of a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ is called a "perfect difference set" if every nonzero element of $\mathbb{Z}/m\mathbb{Z}$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that $m=n^2+n+1$ for some nonnegative integer $n$. Singer constructed examples of perfect difference sets in $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ whenever $n$ is a prime power, and it is an old conjecture that these are the only such $n$ for which $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ contains a perfect difference set. In this talk, I will discuss a proof of an asymptotic version of this conjecture.

Witten-Reshetikhin-Turaev quantum invariants of links and 3 dimensional manifolds are obtained from quantum sl(2). There exist different versions of quantum sl(2) leading to other families of invariants. We will briefly overview the original construction and then discuss two variants. First one, so called unrolled quantum sl(2), allows construction of invariants of 3-manifolds involving C* flat connections. In simplest case it recovers Reidemeister torsion. The second one is the non restricted version at a root of unity. It enables construction of invariants of links equipped with a gauge class of SL(2,C) flat connection. This is based respectively on joined work with Costantino, Geer, Patureau and Geer, Patureau, Reshetikhin.

Given two metric spaces $X$ and $Y$, it is natural to ask how faithfully, from the point of view of the metric, one can embed $X$ into $Y$. One way of making this precise is asking whether there exists a coarse embedding of $X$ into $Y$. Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many seemingly elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. Highlights of the talk include a new algebraic dichotomy for connected unimodular Lie groups, and a method of calculating a lower bound on the conformal dimension of a compact Ahlfors-regular metric space.
 

The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arationaltrees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called `non-uniquely ergodic'. I will talk briefly about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.