### Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day,
informal meeting of mathematicians from the
University of Illinois, Purdue University and nearby universities, with interests in algebra,
geometry and combinatorics (widely interpreted).

Further details will be posted here as they become available. Or you may
contact the University of Illinois organizers

Hal Schenck and

Alexander Yong, or the Purdue organizers

Uli Walther
and

Saugata Basu .

__Next event__ : Spring, 2011

**Date: **March 26 (Sat), 2011

###
** Location: ** Department of Mathematics, Purdue University.

The talks will held at MATH 175.

__Speakers and schedule__:

### Cofee and snacks ** 8h30 - 9h30**

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.

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.

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**

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**

We have reserved a block of rooms at the Union Club Hotel which is conveniently located on campus and a two-minutes walk to the Mathematics department. **Parking**Parking is free on Saturdays on campus. The most convenient parking garage is on N. University street adjacent to the Math building.