Ringel-Hall algebras of cyclic quivers

These are notes from my mini-course at ICRA13, Sao Paulo, 200

A Prehistory of n-Categorical Physics

This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open-closed topological string theory, the categorification of quantum groups, Khovanov homology, and Lurie's work on the classification of topological quantum field theories.Comment: 129 pages, 8 eps figure

Interactions between Algebraic Geometry and Noncommutative Algebra

The workshop presented the current developments in the field of noncommutative algebra geometry and its interactions with algebraic geometry and representation theory

Lectures on canonical and crystal bases of Hall algebras

These are the notes for a series of lectures given on the theory of canonical and crystal bases for Hall algebras (for a summer school in Grenoble in 2008). It may be viewed as a follow-up to arXiv:math/0611617. It covers the construction, due to Lusztig, of the canonical bases for the Hall algebra of a quiver Q in terms of a certain category of perverse sheaves over the moduli space of representations of Q. It also contains an exposition of Kashiwara and Saito's geometric construction of the crystal graph in terms of irreducible components of Lusztig's lagrangian in the cotangent bundle to the above moduli spaces. The last section deals with the Hall algebras of curves. It contains a few new results and conjectures. Apart from these, the text is purely expositional.Comment: 101 pages, Latex; minor revision