Algebraic Geometry Seminar Mondays 3:00--3:50 PM Texas A&M University Milner 216

Related seminars: Algebra & Combinatorics Seminar Geometry Seminar Number Theory Seminar Several Complex Variables Seminar All Mathematics Seminars

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Spring 2009 Schedule:

Abstracts:

The Waring problem for polynomials asks how to write a homogeneous polynomial of degree d as a sum of dth powers of linear polynomials. The rank of a polynomial is the least number of terms in such an expression. The border rank is the least rank of an approximation of the polynomial. The problem of finding the rank and border rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to signal processing and computational complexity.

In 1916, Macaulay gave a lower bound for rank and border rank in terms of catalecticant matrices. In the almost 100 years since, there has been relatively little progress on the problem of determining or bounding rank (although related questions have proved very fruitful). I will describe new upper and lower bounds, with especially nice results for some examples including monomials and cubic polynomials. This is joint work with J.M. Landsberg.
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Let R be a commutative noetherian local ring. In the 1980s, work by Buchweitz, Greuel, and Schreyer, Knörrer, Yoshino and others established remarkable connections between the module theory of R and the character of its singularity. I will report on recent progress in this area.
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A division algebra over a field is decomposable if it is isomorphic to the tensor product of two subalgebras of positive degree. In this talk I will discuss the existence of indecomposable Brauer classes over various fields, including new examples over Q_p(t) with exponent strictly less than index.
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The Littlewood-Richardson rule appears in many areas of mathematics, and has many proofs. We will look at the first step of a program designed to give a commutative algebra proof of the geometric form of this rule. We will sketch the main points in proving the Pieri formula in this setting.
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Given a rational homogeneous variety G/P (where G is complex, simple and of ADE type), its tangent bundle T_G/P is simple, i.e. its only endomorphsims are scalar multiples of the identity. If G/P is Hermitian symmetric, then this is a consequence of a result of Ramanan from the 60s on stability of irreducible bundles. In this talk I will show how both simplicity and stability hold in the general case. My main tool will be the equivalence of categories between homogeneous vector bundles on G/P and finite dimensional representations of a given quiver with relations, that I will describe in detail.
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A powerful tool in birational geometry is the cone of effective divisors of a variety. We examine this cone when the given variety is the moduli space of curves with one marked point, in some low genus cases. In particular, we identify divisors that lie on extremal rays of the cone - most notably, pointed analogues of Brill-Noether divisors that have been studied previously by Adam Logan.
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We define some intersection numbers between tropical hypersurfaces. These numbers are sums of certain mixed volumes and behave like corresponding intersection numbers for complex hypersurfaces. This is based on a joint work with Benoit Bertrand.
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Let K be a field with a non-archimedian discrete valuation and let F be a square system of polynomial equations over K (n variables and n equations). In this talk I will present a result that allows us to determine the exact number of solutions of F in (K*)^n under some assumptions on F. With more relaxed hypothesis, we do not have an exact formula for the number of solutions, but we can show the upper bound O((tq)^n) where t is the number of non-zero terms of F and q is the cardinality of the residue field of K. The constant implied in the big-O notation is universal. This is part of a joint work with A. Ibrahim and J.M. Rojas that has not been published yet.
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Toric varieties are fundamental objects of combinatorial algebraic geometry. The reason for this is that they are canonically associated to a fan in a lattice and may be viewed either as varieties or as objects in geometric combinatorics. They are also characterized as normal varieties that have an action of a diagonal (split) torus with a dense orbit.

An arithmetic toric variety is a normal variety over a field k equipped with the action of a (not necessarily split) torus having a dense orbit. Extending scalars to the algebraic closure, they become a usual toric variety. Their classification is via (nonabelian) Galois cohomology which mixes the combinatorics with the symmetry of a discrete group action.

In this talk, which represents joint work with Javier Elizondo, Paulo Lima-Filho, and Zach Teitler, I will introduce you to the topics of toric varieties and Galois cohomology, stating our classification theorem, present some examples, and discuss future work in this area. It is the foundation for our work with Clarence Wilkerson on equivariant cohomology for real toric varieties, which is the subject of a future talk.
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For more information, email Zach Teitler.