"Schubert Polynomials and the Cohomology of Homogeneous Spaces"

We will survey the cohomology ring of homogeneous spaces and focus on the Schubert polynomials, which are representatives of the classical Schubert varieties in a certain presentation of the cohomology ring.  For example, in the special case of the Grassmannian, the Schubert polynomials are known as Schur polynomials and many beautiful facts in symmetric function theory have a geometric interpretation.  We will discuss what is known and what remains unsolved about Schubert polynomials in general.  This talk is aimed at providing motivation for the Monday learning seminar in algebraic geometry that will take place this semester.

"Gelfand pairs and parking functions"

Let be a subgroup. The pair (G,H) is called a Gelfand pair, if the convolution algebra of H-invariant functions on the symmetric space G/H is commutative. Equivalently, the induced trivial representation 1GH  is multiplicity-free.

In this talk, we:

1. first analyze (for the heck of it) the well known Gelfand pair: (Bn,Sn) the hyperoctahedral group and its symmetric subgroup Sn, then we evaluate its spherical functions.

2. study a Gelfand pair related to the parking functions. We show that the number of irreducible constituents of the induced representation is the Catalan numbers.
This is joint work with Kursat Aker (Feza Gursey Institute, Turkey).

"Gelfand pairs and parking functions, Part II "

Let be a subgroup. The pair (G,H) is called a Gelfand pair, if the convolution algebra of H-invariant functions on the symmetric space G/H is commutative. Equivalently, the induced trivial representation 1GH  is multiplicity-free.

In this talk, we:

1. first analyze (for the heck of it) the well known Gelfand pair: (Bn,Sn) the hyperoctahedral group and its symmetric subgroup Sn, then we evaluate its spherical functions.

2. study a Gelfand pair related to the parking functions. We show that the number of irreducible constituents of the induced representation is the Catalan numbers.
This is joint work with Kursat Aker (Feza Gursey Institute, Turkey).

"Local splitting of topological groups"

An example of a compact connected abelian group that is well known to continuum topologists is the solenoid. It is much less known outside the community of specialists; so a small nontrechnical introduction is in order, and I shall give it. If one writes the solenoid as a union of two closed subsets, one of them has to be the whole thing: It is indecomposable. However we find a compact neighborhood of the identity which a direcct products of a Cantor set and an interval; the former can be chosen to be a subgroup and the second one to be a local subgroup. Let us call such a represention of a neighborhood of the group identity a local splitting. It give us a pretty good idea how this globally complicated object looks locally. I shall give an overview of what I know about the local splitting of topological groups in the domain of locally compact groups and pro-Lie groups, perhaps, time permitting, leading to some recent insights. I shall honestly attempt to be nontechnical in these matters.
Literature:
[1] K. H. Hofmann and S .A. Morris: The Lie Theory of Connected Pro-Lie Groups, EMS Publishing House, Z¨urich, 2007, xvii+678pp.
[2] George Michael, A. A.: Local splitting of almost connected pro- Lie groups. E-mail correspondence.

"TBA"

"Regularity of powers of ideals"

Let $R = k[x_1, \dots, x_n]$ be a polynomial ring, and let $I$ be a homogeneous ideal in $R$. A celebrated result in the study of Castelnuovo-Mumford regularity states that $\text{reg}(I^q)$ is asymptotically a linear function, i.e., for $q \gg 0$, $\text{reg}(I^q) = aq + b$. While the linear constant $a$ is well understood from reduction theory, the free constant $b$ continues to elude researcher's efforts. In this talk, I will discuss how we can relate $b$ to a set of ``local'' data associated to fibers of certain projection map

"SEMIGROUPS OF IDEALS OF RIGHT WEAKLY REGULAR
RINGS"

This is joint work with Henry E. Heatherly and Karl A. Kosler. Let R be a right weakly regular (r.w.r.) ring; i.e., every right ideal of R is idempotent. Characterizations are given for r.w.r. rings in terms of a strong semilattice decomposition for the semigroup R(R) of all right ideals of the ring. The index set semilattice is shown to be isomorphic to the semigroup I(R) of all ideals of R. Connections are given between such a decomposition and the minimal and maximal right ideals of R.

"Cox rings of Generalized Del Pezzo Surfaces"

A construction of David Cox generalized the construction of the homogeneous coordinate ring of projective space to all toric varieties.  Under suitable assumptions on the success of the Minimal Model Program, Cox's construction generalizes to give a homogeneous coordinate ring for a much larger class of varieties, including all Fano varieties.  We will establish the finite generation of Cox rings of generalized Del Pezzo surfaces and relate the generators to certain root systems which encode information about the singularities of the surface obtained from the anticaonical linear series.

"Lecture Series on From Sums of Squares To Secant Varieties: Evolution of an Idea"

In these talks I would like to explain how some famous theorems of elementary number theory, which have analogues for polynomials in several variables, have given rise to interesting questions and investigations in commutative algebra and algebraic geometry.

Moreover, these purely geometric results have had practical applica tions in such diverse areas as: computational complexity (in particular,finding effcient algorithms for matrix multiplication) mathematical biology (in particular, on the genome project) and algebraic statistics.

"Lecture Series on From Sums of Squares To Secant Varieties: Evolution of an Idea"

In these talks I would like to explain how some famous theorems of elementary number theory, which have analogues for polynomials in several variables, have given rise to interesting questions and investigations in commutative algebra and algebraic geometry.

Moreover, these purely geometric results have had practical applica tions in such diverse areas as: computational complexity (in particular,finding effcient algorithms for matrix multiplication) mathematical biology (in particular, on the genome project) and algebraic statistics.

"Vector Bundles of Rank 2 on the grassmannian of lines in P3"

First, I will discuss some facts, results, and perspectives concerning vector bundles of rank 2 on the grassmannian. Then I will state a conjecture. The work I have done on this conjecture leads to the question of whether a very specific, very singular sub-scheme of the grassmannian exists. This is related to an important result in commutative algebra.

"Vector Bundles of Rank 2 on the grassmannian of lines in P3 - PART 2"

I will continue my presentation from last week: First, I will restate and explain the rationale for a conjecture concerning rank 2 vector bundles on the grassmannian. Then I will discuss the progress I have made on a special case of the conjecture. This involves a very specific, very singular sub-scheme of the grassmannian and an important result in commutative algebra.

"TBA"