It is unknown whether a bound on axion field ranges exists within quantum gravity. We study axion field ranges using extended supersymmetry, in particular allowing an analysis within strongly coupled regions of moduli space. We apply this strategy to Calabi-Yau compactifications with one and two Kähler moduli. We relate the maximally allowable decay constant to geometric properties of the underlying Calabi-Yau geometry. In all examples we find a maximal field range close to the reduced Planck mass (with the largest field range being 3.25 $M_P$). On this perspective, field ranges relate to the intersection and instanton numbers of the underlying Calabi-Yau geometry.

Work of Schoen--Yau in the 70's/80's shows that area-minimizing (actually stable) two-sided surfaces in three-manifolds of non-negative scalar curvature are of a special topological type: a sphere, torus, plane or cylinder. The torus and cylinder cases are "borderline" for this estimate. It was shown by Cai--Galloway in the late 80's that the torus can only occur in a very special ambient three manifold. We complete the story by showing that a similar result holds for the cylinder. The talk should be accessible to those with a basic knowledge of curvature in Riemannian geometry.

I will describe 5 definitions of Euler characteristic for a space with perfect obstruction theory (i.e. a well-behaved moduli space), and their inter-relations. This is joint work with Yunfeng Jiang. Then I will describe work of Yuuji Tanaka on how to this can be used to give two possible definitions of Vafa-Witten invariants of projective surfaces in the stable=semistable case.

For the programme see 

https://people.maths.ox.ac.uk/tillmann/OAC-manifolds.html

I will present several new  3d N=2 dualities with super-potentials involving monopole operators. Some of the theories that I will discuss describe systems of D3 branes ending on pq-webs. In these cases  3d mirror symmetry is a consequence of S-duality.