Algebraic Geometry SeminarMondays 3:003:50 PM

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Date  Speaker  Title (click for abstract) 

9/1/08  Frank Sottile, TAMU  Toric polar Cremona transformations 
9/8/08  Zach Teitler, TAMU  Computing asymptotic multiplier ideals 
9/15/08  Chris Hillar, TAMU  Positive semidefinite matrix word equations 
9/22/08  Jarek Buczynski, TAMU  Maps between toric varieties in terms of Cox coordinates 
9/29/08  Paulo LimaFilho, TAMU  Invariants for real curves 
10/6/08  Javier Elizondo, UNAM  Equivariant cohomology of real toric varieties 
10/13/08  Christine Berkesch, Purdue  The rank of a hypergeometric system 
10/20/08  David Jorgensen, UTArlington  Dualizing complexes old and new 
10/27/08  Jeremy Martin, U Kansas  Counting simplicial and cubical spanning trees 
11/3/08  Zhenhua Qu, UT Austin  Tropical Compactifications 
11/10/08  Mara Neusel, Texas Tech  Degree bounds in invariant theory 
11/17/08  Ashraf Ibrahim, TAMU  Roots of Polynomials over Local Fields 
11/24/08  Sarah Kitchen, U. Utah  Representation Theory and the Geometry of Flag Varieties 
12/1/08  Frank Sottile, TAMU  Frontiers of Reality in Schubert Calculus 
Garcia and Sottile showed that problem of classifying toric patches that posses linear precision is equivalent to a classifying homogeneous polynomials whose toric derivatives (which generate its toric polar linear system) define a Cremona transformation (a birational isomorphism).
In this talk, I will explain the motivation from geometric
modelling and then outline the classification of forms F in three
variables whose toric derivatives define a Cremona transformation,
which solves an open question in geometric modeling. This is joint
work with Kristian Ranestad and HansChristian Graf von
Bothmer.
TOP
Most of the celebrated applications of multiplier ideals in algebraic geometry, such as the invariance of plurigenera and existence of flips, have actually involved asymptotic multiplier ideals. However virtually no examples of asymptotic multiplier ideals have actually been computed. In contrast to the case of ordinary multiplier ideals, whose definition is an explicitbut very difficultalgorithm for computing them, there is no apparent algorithm for computing asymptotic multiplier ideals.
I will give an expository introduction to asymptotic multiplier
ideals, including the applications that have made them important
tools in algebraic geometry, and the relation to the analytic
approach (involving approximation of plurisubharmonic forms). I
will describe some cases in which asymptotic multiplier ideals have
now been computed. This is work in progress. I will mention some
open questions.
TOP
Matrix word equations arise naturally in many contexts. In the
simplest incarnation, one is given a word W(X,B), matrices B and P,
and a solution matrix X is desired such that W(X,B) = P. In many
applications, people are interested in equations in which B, P, and
X are resetricted to be positive semidefinite matrices (of the same
size). For instance, one might desire positive semidefinite
solutions X to the ubiquitous Riccati equation XBX = P given fixed
B and P. In this talk, we will give an overview of what is known
about such equations in real algebraic geometry. In particular, we
will discuss the very general result (of which there are now two
proofs) that positive semidefinite word equations W(X,B) = P always
have (generically finite) positive semidefinite solutions when W is
palindromic (an unavoidable restriction). This result was motivated
by the longstanding BMV conjecture in statistical physics. (partly
joint with C. R. Johnson and separately, S. Armstrong).
TOP
Toric varieties are algebraic varieties, which admit an action
of a torus (C^*)^n, such that this action has an open orbit. Cox
observed that by analogy to affine and projective spaces, any toric
variety can be understood in terms of its homogeneous coordinate
ring, which is always a polynomial ring. This is a very convenient
tool for computational purposes, especially if you are interested
in subvarieties of toric varieties. We propose an elementary way of
describing any map between any two toric varieties in terms of Cox
coordinates, which fills in a gap left by Cox. (This is joint work
with Gavin Brown.)
TOP
This is a very elementary talk in which we present explicit
computations of certain bigraded equivariant cohomology rings of
real algebraic curves. We will first give a gentle introduction to
the theory and present the computations. At the end, we will
explain how these invariants relate to other invariants, from
classical objects such as Brauer groups to recent versions of
Deligne cohomology for real curves. This culminates with a new
proof of Weichold's classical description of the Picard group of a
real curve.
TOP
In this talk we overview a work in progress on the equivariant
cohomology of real toric structures (RTS). We will introduce RTS
and give a couple of examples of them, and we also introduce the
Borel equivariant cohomology. At the end we want to show the
computations we have in equivariant cohomology for RTS.
TOP
An Ahypergeometric system is a parametric system of PDE arising
from a toric ideal. The dimension of its solution space, called its
rank, is constant for generic parameters. I will discuss the
combinatorial nature of its rank at nongeneric parameters. No
background is necessary; it may be helpful to have a basic
understanding of semigroup rings.
TOP
In this talk we will give a brief history of dualizing
complexes, and their more concrete manifestations, dualizing
modules. We will then introduce semidualizing modules, and discuss
existence questions for them in the CohenMacaulay context.
TOP
The classical matrixtree theorem expresses the number of
spanning trees of a graph G in terms of the eigenvalues of its
Laplacian matrix; one wellknown special case is Cayley's formula
n^{n2} in the case that G is the complete graph on n vertices.
Based on work of Gil Kalai, we extend the definition of spanning
tree from graphs to CWcomplexes in such a way that the matrixtree
theorem remains valid. As an application, we enumerate the
simplicial spanning trees of any shifted simplicial complex X by
their facetvertex degree sequences, by finding a combinatorial
interpretation for the eigenvalues of a certain weighted Laplacian.
We also have obtained some results on cubical complexes that
generalize the formula for the number of spanning trees of a
hypercube graph.
TOP
Tropical compactification is defined and studied by Tevelev
originally in an effort of compactifying moduli spaces of del Pezzo
surfaces. I will talk about this method and some new results on
tropical compactification and sketch how it is applied to the study
of compactification of moduli spaces of line arrangements in
projective plane. As we shall see, this involves algebraic geometry
on one side and combinatorics on the other. (joint with Mark
Luxton)
TOP
In this talk I want to give a survey on invariant theory of
finite groups. For that I have picked the particular problem of
degree bounds: what are they, why are they relevant, how do we find
them, and what are current results and open problems? are the
things I want to cover.
TOP
Let K be a padic field and f be a univariate polynomial with
coefficients in K and nonvanishing discriminant. In this talk, we
are going to present an algorithmic method for counting the number
of roots of f in K. This method is based on a result connecting the
number of roots of f with the number of roots of its reduction
modulo some ideal. The connection with so called lower binomials
will also be discussed.
TOP
The BorelWeilBott theorem identifies finite dimensional
representations for a complex semisimple algebraic group with line
bundles on a projective variety associated to that group. I will
discuss generalizations of this principle of associating
representations to sheaves. In particular, the structure of a
representation will give the associated sheaf the structure of a
Dmodule. We aim to understand representations by relating
Dmodules on various partial flag varieties associated to our
group. I will begin with a brief overview of Dmodules and
equivariant sheaves, highlighting some advantageous differences
from Omodules on projective varieties, then explain how to extend
known results relating representations and sheaves on the full flag
variety to improve our understanding of the geometric
picture.
TOP
For more information, email Zach Teitler.