Continuing my series of talks from last fall, I will define the Chow ring of cycles on a nonsingular variety and the Chern classes of vector bundles.  I will discuss and use (without proof) the Kunneth formula, the projection formula, and the generalized Riemann-Roch formulas.  The examples will come from curves, surfaces and projective spaces.

We shall apply the techniques of intersection theory to prove the classical Brill-Noether existence theorem on the existence of special linear series, and its generalization in dimensions 1 and 2 to linear series with a ramification point. In case the expected dimension is zero, we can actually count the expected number of such points.

"Search for Primary-type intersection representations of ideals in non-noetherian rings"

This is a survey talk on the above-mentioned topic in the ideal theory in commutative rings.  It starts with Lasker's primary representation of polynomial ideals, and continues with the four representations by Emmy Noether and two others in the literature for noetherian rings.  The study of the general case begins with Krull's principal and isolated components.  Different definitions of associated primes (in the sense of Krull, Bourbaki, Nagata, or Zariski-Samuel, etc.) will be introduced.  The various intersection representations of ideals in non-noetherian rings will be compared.  Applications to rings with notherian spectra and to arithmetical rings will also be mentioned.

"TBA"

After proving the existence theorems, we will begin the background for the Finiteness and Non-Existence theorems by discussing the moduli space of curves and the moduli space of pointed curves.

"TBA"

"The Semigroup of Right Ideals of a Semigroup"

Let S be a semigroup, and let R(S) be the collection of right ideals of S. The set R(S) is a semigroup under right ideal multiplication. We discuss the relation between the structure of S and the structure of R(S). We concentrate mostly on semigroups for which every right ideal H satisfies . Such semigroups are called right weakly regular. This talk is similar to previous results on the semigroup of right ideals of a ring.

" QI- modules "

An R-module MR is quasi-injective if each homomorphism of any submodule N into M can be extended to a homomophism of M into M.  A ring R is called a right QI-ring if every quasi-injective right R-module is injective.

"TBA"

"Multiplier ideals of hyperplane arrangements"

Multiplier ideals have emerged in the last decade as a fundamental tool in algebraic geometry. They have played key roles in the solution of a number of outstanding problems and they are interesting objects in their own right. But they are quite difficult to calculate, and in fact very few instances have been computed. There are many associated problems: to characterize the jumping numbers of an ideal; to characterize the ``relevant'' exceptional divisors (or valuations) that impose conditions on the multiplier ideals; to characterize which ideals have the same multiplier ideal.

"Brill-Noether Theorems with a Movable Ramification Point"

Completing our seminar series on Brill-Noether theory, we shall bound the dimension of the special linear systems with a ramification point, and prove an exact theorem (if and only if) in the case of maps to the plane.  It time permits, I shall conclude by discussing the obstructions to extending the theorem to higher dimensions, non-general curves, and multiple ramification points, and why I believe these three open problems are closely related.

Tài Huy Hà, Tulane University

"Cycles in graphs via commutative algebra"

We will introduce an algebraic approach to study the cycle structure in graphs. In particularly, we will discuss how to use computational and commutative algebra to tackle an open problem in graph theory - determining the complexity of finding odd cycles in a graph.

"Root Systems and Del Pezzo Surfaces"

It is a classical fact that the Picard group of a Del Pezzo surface of degree less than seven contains a simply-laced root system.  It is natural to wonder how this root system and the symmetries arising from the associated Weyl group affect the geometry of a Del Pezzo surface.  We will talk about how the symmetries can be used to give a symmetric triangulation of the nef cone of a Del Pezzo surface.  This allows for the computation of the volume of the nef cone truncated in an appropriate manner, and this volume plays a role in Manin’s conjecture on the distribution of rational points on a Del Pezzo surface.  Time permitting, we will also talk about potential applications to analysis of the homogeneous coordinate ring of a Del Pezzo surface.

"Solving the Likelihood Equations "