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
Spring 2009 Schedule:
Date | Speaker | Title (click for abstract) |
---|---|---|
Jan. 26, 2009 | No meeting | |
Feb. 2, 2009 | Zach Teitler | Ranks and border ranks of polynomials |
Feb. 9, 2009 | Lars Christensen, Texas Tech | Simple hypersurface singularities via totally reflexive modules |
Feb. 16, 2009 | Kelly McKinnie, Rice | Indecomposable division algebras in the Brauer group of Q_p(t) |
Feb. 23, 2009 | Nick Hein, TAMU | A Commutative Algebra Approach to Proving the Pieri Formula |
Mar. 2, 2009 | Ada Boralevi, TAMU | Simplicity and stability of tangent bundles on rational homogeneous varieties |
Mar. 9, 2009 | David Jensen, UT Austin | The Birational Geometry of Moduli Spaces of Curves with One Marked Point |
Mar. 16, 2009 | Spring break---no meeting | |
Mar. 23, 2009 | no meeting | |
Mar. 30, 2009 | Mounir Nisse, Institut de Mathématiques de Jussieu | Coamoebas |
Apr. 6, 2009 | Frédéric Bihan, Université de Savoie | Intersection numbers in tropical geometry |
Apr. 13, 2009 | Martin Avendaño, TAMU | Multivariate ultrametric root counting |
Apr. 20, 2009 | Frank Sottile, TAMU | Arithmetic Toric Varieties |
Apr. 27, 2009 | ||
May 1-3, 2009 | TAGS | Texas Algebraic Geometry Seminar |
May 4, 2009 |
Abstracts:
February 2, 2009
Zach Teitler, TAMU
Ranks and border ranks of polynomials
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.
• • • • •
February 9, 2009
Lars Christensen, Texas Tech
Simple hypersurface singularities via totally reflexive
modules 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.
• • • • •
February 16, 2009
Kelly McKinnie, Rice
Indecomposable division algebras in the Brauer group of
Q_p(t)
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.
• • • • •
February 23, 2009
Nick Hein, TAMU
A Commutative Algebra Approach to Proving the Pieri
Formula
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.
• • • • •
March 2, 2009
Ada Boralevi, TAMU
Simplicity and stability of tangent bundles on rational
homogeneous varieties
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.
• • • • •
March 9, 2009
David Jensen, UT Austin
The Birational Geometry of Moduli Spaces of Curves with One
Marked Point
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.
• • • • •
March 30, 2009
Mounir Nisse, Institut de Mathématiques de
Jussieu
Coamoebas
PDF
abstract
• • • • •
April 6, 2009
Frédéric Bihan, Université de
Savoie
Intersection numbers in tropical geometry
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.
• • • • •
April 13, 2009
Martin Avendaño, TAMU
Multivariate ultrametric root counting
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.
• • • • •
April 20, 2009
Frank Sottile, TAMU
Arithmetic Toric Varieties
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.
Previous semesters:
Fall 2008 • Spring 2008 • Fall 2007
Last modified: 5 February 2009 by Zach Teitler