A longstanding folklore conjecture in combinatorial number theory is the following: given an additive set $S$ not containing the identity, $S$ can be ordered as $s_1, \ldots, s_k$ so that the partial sums $s_1+\cdots+s_j$ are distinct for each $j\in[k]$. We discuss a recent resolution of this conjecture in the finite field model (where the ambient group is $\mathbb{F}_2^n$, or more generally, any bounded exponent abelian group). This is joint work with B. Bedert, M. Bucic, N. Kravitz, and R. Montgomery.

I will report on ongoing joint work with Sam Fisher on showing that the mapping class group has a finite index subgroup whose group ring embeds in a division ring. Our methods involve p-adic analytic groups, but no prior knowledge of this will be assumed and much of the talk will be devoted to explaining some of the underlying theory. Time permitting, I will also discuss some consequences for the profinite topology for the mapping class group and potential extensions to Out(RAAG).

Basic Turán theory asks how many edges a graph can have, given certain restrictions such as not having a large clique. A more generalized Turán question asks how many copies of a fixed subgraph $H$ the graph can have, given certain restrictions. There has been a great deal of recent interest in the case where the restriction is planarity. In this talk, we will discuss some of the general results in the field, primarily the asymptotic value of ${\bf N}_{\mathcal P}(n,H)$, which denotes the maximum number of copies of $H$ in an $n$-vertex planar graph. In particular, we will focus on the case where $H$ is a cycle.

It was determined that ${\bf N}_{\mathcal P}(n,C_{2m})=(n/m)^m+o(n^m)$ for small values of $m$ by Cox and Martin and resolved for all $m$ by Lv, Győri, He, Salia, Tompkins, and Zhu.

The case of $H=C_{2m+1}$ is more difficult and it is conjectured that ${\bf N}_{\mathcal P}(n,C_{2m+1})=2m(n/m)^m+o(n^m)$. 

We will discuss recent progress on this problem, including verification of the conjecture in the case where $m=3$ and $m=4$ and a lemma which reduces the solution of this problem for any $m$ to a so-called "maximum likelihood" problem. The maximum likelihood problem is, in and of itself, an interesting question in random graph theory.