L3

The theory of Special Lagrangian (SL) submanifolds is the natural point of intersection between various classical (Lagrangian and volume-minimizing submanifolds) and contemporary (Mirror Symmetry and invariants of Calabi-Yau manifolds) topics. The key problem is how to characterize the compactified moduli space of SLs. Equivalently, to understand which SL singularities admits desingularizations. Our aim is to present some explicit examples, topological results and simple observations which shed some light on the nature and complexity of this problem, and which we expect will be a useful foundation for future progress in the field. This is joint work with M. Haskins (Imperial College), cfr. arXiv:math/0609352.

Category theory is used to study structures in various branches of

mathematics, and higher-dimensional category theory is being developed to

study higher-dimensional versions of those structures. Examples include

higher homotopy theory, higher stacks and gerbes, extended TQFTs,

concurrency, type theory, and higher-dimensional representation theory. In

this talk we will present two general methods for "categorifying" things,

that is, for adding extra dimensions: enrichment and internalisation. We

will show how these have been applied to the definition and study of

2-vector spaces, with 2-representation theory in mind. This talk will be

introductory; in particular it should not be necessary to be familiar with

any category theory other than the basic idea of categories and functors.

I will discuss how one can construct nice cellular

algebras using the cohomology of Springer fibres associated with two

block nilpotent matrices (and the convolution product). Their

quasi-hereditary covers can be described via categories of highest

weight modules for the Lie algebra sl(n). The combinatorics of torus

fixed points in the Springer fibre describes decomposition

multiplicities for the corresponding highest weight categories. As a

result one gets a natural subcategory of coherent sheaves on a

resolution of the slice to the corresponding nilpotent orbit.

There are many triangulated categories that arise in the study

of group cohomology: the derived, stable or homotopy categories, for

example. In this talk I shall describe the relative cohomological

versions and the relationship with ordinary cohomology. I will explain

what we know (and what we would like to know) about these categories, and

how the representation type of certain subgroups makes a fundamental

difference.