Travis Schedler (Imperial): Poisson and noncommutative geometry Abstract: A geometric space is typically encoded by its commutative algebra of functions. I will recall how noncommutative perturbations, first arising in quantum mechanics, can be profitably viewed as the functions on a noncommutative space. Such deformations give rise to Poisson structures on the original space, which generalize symplectic structures. I will explain some aspects of the intricate relationship between Poisson and noncommutative geometry. In the remaining time, I will briefly give applications to representation theory (e.g., semisimple Lie algebras, finite groups acting on vector spaces, and quivers) and discuss open questions related to mirror symmetry / symplectic duality. Iain Gordon (Edinburgh): Why be noncommutative with algebraic combinatorics? Abstract: I’ll explain a little about a problem about combinatorics around the invariant theory of reflection groups: how noncommutativity helped to solve a problem in this a few years ago, what this led to in representation theory and algebraic geometry, and what people would like to know now!” Toby Stafford (Manchester): Noncommutative Projective Algebraic Geometry Abstract: In recent years a surprising number of insights and results in noncommutative algebra have been obtained by using the global techniques of projective algebraic geometry. Many of the most striking results arise by mimicking the commutative approach: classify curves, then surfaces, and we will use this approach here. As we will discuss, the noncommutative analogues of (commutative) curves are well understood while the study of noncommutative surfaces is on-going. In the study of these objects a number of intriguing examples and significant techniques have been developed that are very useful elsewhere. In this talk I will to discuss several of them. This talk will be aimed at postgraduate students, so the technicalities will be kept to a minimum. |