Positively folded galleries arise as images of retractions of buildings onto a fixed apartment and play a role in many areas of maths (such as in the study of affine Hecke algebras, Macdonald polynomials, MV-polytopes, and affine Deligne-Lusztig varieties). In this talk, we will define positively folded galleries, and then look at how these can be used to study affine flag varieties. We will also look at a new recursive description of the set of end alcoves of folded galleries with respect to alcove-induced orientations, which gives us a combinatorial description of certain double coset intersections in these affine flag varieties. This talk is based on joint work with Elizabeth Milićević, Petra Schwer and Anne Thomas.

For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.