Richard Melrose - November 2, 2009
I will spend the first part of the lecture outlining the
definition
and properties of (radial) blow up of a submanifold. Most of the remainder
of the time will be devoted to a description of various applications, in
particular resolutions of various types. At the end I will point to some
unsolved problems.
David Vogan - November 16, 2009
Suppose K \subset G are compact connected Lie groups. The sets of
irreducible representations of K and of G can be parametrized very
concretely and explicitly: each is the set of lattice points in a certain
rational cone. If you restrict an irreducible representation of G to K,
you therefore get a finite set of lattice points (each with a positive
integer multiplicity). The subject of this talk is how to understand this
finite set geometrically. A theorem of Kirwan says that it's
approximately the set of lattice points in a compact convex polytope.
I'll talk about the (few) things that are known about this polytope, some
examples, and the (many) things that somebody ought to be able to prove.
Roman Bezrukavnikov - October 5, 2009
An important pattern of connecting algebra to geometry
has been established by Beilinson and Bernstein in the celebrated
localization theorem, which says roughly that a complicated algebraic object
-- a representation of a semisimple Lie algebra---can be translated into a
geometric family of algebraic objects of much simpler nature.
I will describe how a variation of this pattern leads to new ways to apply
algebraic geometry to representation theory and vice versa.
The talk should be easier to enjoy for someone who had seen definitions of a
Lie algebra and of a vector bundle on an algebraic variety, but that's about
all the prerequisites.