L3

Bridgeland's notion of stability condition allows us to associate a complex manifold, the space of stability conditions, to a triangulated category $D$. Each stability condition has a heart - an abelian subcategory of $D$ - and we can decompose the space of stability conditions into subsets where the heart is fixed. I will explain how (under some quite strong assumpions on $D$) the tilting theory of $D$ governs the geometry and combinatorics of the way in which these subsets fit together. The results will be illustrated by two simple examples: coherent sheaves on the projective line and constructible sheaves on the projective line stratified by a point and its complement.

Let X be a Gorenstein orbifold and Y a crepant resolution of

X. Suppose that the quantum cohomology algebra of Y is semisimple. We describe joint work with Iritani which shows that in this situation the genus-zero crepant resolution conjecture implies a higher-genus version of the crepant resolution conjecture. We expect that the higher-genus version in fact holds without the semisimplicity hypothesis.

There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.