Cambridge

There is a conjectural relationship due to Yau-Tian-Donaldson between stability of projective manifolds and the existence of canonical Kahler metrics (e.g. Kahler-Einstein metrics). Embedding the projective manifold in a large projective space gives, on one hand, a Geometric Invariant Theory stability problem (by changing coordinates on the projective space) and, on the other, a notion of balanced metric which can be used to approximate the canonical Kahler metric in question. I shall discuss joint work with Richard Thomas that extends this framework to orbifolds with cyclic quotient singularities using embeddings in weighted projective space, and examples that show how several obstructions to constant scalar curvature orbifold metrics can be interpreted in terms of stability.

Let $A$ be a finite set in some ambient group. We say that $A$ is a $K$-approximate group if $A$ is symmetric and if the set $A.A$ (the set of all $xy$, where $x$, $y$ lie in $A$) is covered by $K$ translates of $A$. I will illustrate this notion by example, and will go on to discuss progress on the "rough classification" of approximate groups in various settings: abelian groups, nilpotent groups and matrix groups of fixed dimension. Joint work with E. Breuillard.

I will describe joint work with Mohammed Abouzaid, in which we complete the proof of homological mirror symmetry for the standard four-torus and consider various applications. A key tool is the recently-developed holomorphic quilt theory of Mau-Wehrheim-Woodward.