Javid Validashti (Kansas)                  9h30 - 10h30  

Title: .A numerical condition for equisingularity

Abstract: . Multiplicity-based criteria for integral dependence play a
significant role in equisingularity theory, where one would like to use
numerical invariants to distinguish between members of a given a family
of singularities. These criteria are based on the Hilbert-Samuel or
Buchsbaum-Rim multiplicity and their variations, which rely on some kind
of finiteness conditions. To explore this problem, we introduce a few
notions of multiplicity without any finiteness assumption and we show
that these invariants can be used in detecting integral dependence of
modules and characterizing equisingularity conditions numerically. Parts
of this talk are based on joint works with Bernd Ulrich and Steven L.
Kleiman.

Joe Lipman (Purdue)                          11h - 12h

Title: Residues, Duality, and the Fundamental Class of a scheme-map

Abstract: .The duality theory of coherent sheaves on algebraic varieties
goes back to Roch's half of the Riemann-Roch theorem for Riemann
surfaces (1870s). In the 1950s, it grew into Serre duality on normal
projective varieties; and shortly thereafter, into Grothendieck duality
for arbitrary varieties and more generally, maps of noetherian schemes.
This theory has found many applications in geometry and commutative algebra.

We will sketch the theory in the reasonably accessible context of a
variety V over a perfect field k, emphasizing the role of differential
forms, as expressed locally via residues and globally via the
fundamental class of V/k. (These notions will be explained.)

As time permits, we will indicate some connections with Hochschild
homology, and generalizations to maps of noetherian (formal) schemes.
Even 50 years after the inception of Grothendieck's theory, some of
these generalizations remain to be worked out.

Evgeny Mukhin  (IUPUI)                     14h - 15h 

Title: Title: Schubert Calculus and Representation Theory.

Abstract: Schubert Caculus computes index of intersection of Schubert varieties in 
the Grassmannian in terms of Littlewood-Richardson coefficients. The same 
number gives a dimension of an appropriate multiplicity space in the 
representation theory. We present a natural construction which
identifies the scheme-theoretic intersection of Schubert varieties with a 
natural maximal commutative algebra of linear operators in the 
multiplicity space. This algebra is well-known in the theory of the 
Gaudin model. Our construction allows us to establish a number of long 
standing conjectures both in Algebraic Geometry and Integrable Systems.

We discuss possible generalizations of the construction.

This is a review of a joint project with V. Tarasov(IUPUI) and A. 
Varchenko (UNC at Chapel Hill).


 Coffee and snacks 15h-16h

Dave Anderson  (Washington)            16h - 17h

Title: Okounkov bodies, toric degenerations, and polytopes

Abstract: Given a projective variety X of dimension d, a "flag" of subvarieties 
Y_i, and a big divisor D, Okounkov showed how to construct a convex 
body in R^d, and in the last few years, this construction has been 
developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata.  
In general, the Okounkov body is quite hard to understand, but when X 
is a toric variety, it is just the polytope associated to D via the 
standard yoga of toric geometry.  I'll describe a more general 
situation where the Okounkov body is still a polytope, and show that 
in this case X admits a flat degeneration to the corresponding toric 
variety.  As an application, I'll describe some toric degenerations of 
flag varieties and Schubert varieties, and explain how the Okounkov 
bodies arising generalize the Gelfand-Tsetlin polytopes.

Food

There are many restaurants  within walking distance to the campus (including Indian, Chinese, Irish, Middle-eastern, Thai, Japanese, Korean, Vietnamese, Mexican etc.).  There are also several coffee-houses in and around campus as well as across the river in the town of Lafayette. See here for dining options in the Lafayette-West Lafayette area.

We will go for dinner on Sat evening at 6.30 to the Nine Irish Brothers.

Accommodation