Abstracts for Extra talks

We will describe a joint work with P. Achar, A. Henderson and D. Juteau which gives a geometric construction relating two fundamental constructions in geometric representation theory, the Satake equivalence and the Springer correspondence. More precisely we will describe a functor from perverse sheaves on the "small" part of the affine Grassmannian of a reductive group G to perverse sheaves on the nilpotent cone of G which realizes geometrically the functor which sends a (small) module for the Langlands dual group to its weight zero subspace (a representation of its Weyl group).

 A crossing probability is the probability of finding, in a physical model,
a critical cluster that touches specified boundary arcs; its density
conditions on a point z being in a specified cluster.  We consider various
examples, for percolation and related models, on a rectangle.
Surprisingly, all known crossing formulas have modular properties, being
either modular forms, second-order modular forms or transforming like
Hermitian Jacobi modular functions.  This is unexpected because a rectangle

Given a finite group G, consider the locus M_g(G), in M_g, consisting of curves which
admit an effective action by G. We propose numerical invariants of the G-action to
distinguish irreducible components of M_g(G). These invariants take into account the
local monodromies at the branch points of the associated Galois cover and certain
classes in H_2(G,Z), and extend those already introduced to study the case of abelian
groups.
When G=D_n, the dihedral group of order 2n, we show that these invariants

Khovanov homology is a homology theory for links in the three-sphere. 
In the past 10 years, various mathematicians have given several different
constructions of this homology theory; some use algebraic geometry, some
use symplectic geometry, and some use representation theory.  This talk
will describe two more constructions of Khovanov homology.