The asymptotically locally Euclidean Ricci-flat self-dual 4-manifolds were classified and constructed by Kronheimer as hyperkahler quotients. Each belongs to a finite-dimensional family and a particularly interesting subfamily consists of manifolds with a circle action which can be identified with the minimal resolution of a quotient singularity C^2/G where G is a finite subgroup of SU(2). The resolved singularity is a configuration of rational curves and there is a distinguished one which is pointwise fixed by the circle action. The talk will give an explicit description of the induced metric on this central sphere, and involves twistor theory and the geometry of the lines on a cubic surface.
 

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.

I will discuss recently published examples of SCFTs in
two dimensions and their dual backgrounds. Aspects of the
integrability of these string backgrounds will be described in
correspondence with those of the dual SCFTs. The comparison with four and
six dimensional examples will be presented. It time allows, the case of
conformal quantum mechanics will also be addressed.