Algebraic Geometry SeminarMondays 3:003:50 PM

Related seminars:
All Mathematics Seminars 
Date  Speaker  Title (click for abstract) 

1/14/08  Frank Sottile, TAMU  Galois groups of Schubert problems via homotopy computation 
1/28/08  Frank Sottile, TAMU  Betti number bounds for fewnomial hypersurfaces via stratified Morse theory 
2/4/08  No meeting  No meeting this week (special meeting of Algebra
& Combinatorics Seminar) Bill Schmitt, Milner 317, 34pm 
2/11/08  Colleen Robles, TAMU  Rigidity of projective homogeneous varieties 
2/18/08  Eric Katz, UT Austin  Tropical Curves and Monodromy 
2/25/08  No meeting  No meeting this week (special meeting of Algebra
& Combinatorics Seminar) Charles Conley, Milner 216, 34pm 
3/3/08  Zach Teitler, TAMU  Multiplier ideals of hyperplane arrangements 
3/10/08  Spring breakno meeting  
3/17/08  Bruce Reznick, UIUC Room change: Milner 317 
On Hilbert's construction of positive
polynomials which are not a sum of squares Room change: Milner 317 
3/24/08  We will feature two 20minute talks: Luke Oeding, TAMU Aaron Lauve, TAMU 
The abstracts are located at the AMS web site: The geometry of the relations among principal minors of symmetric matrices On matrix inversion using mixed inversion 
3/31/08  Susan Morey, Texas State University  Relations between Commutative Algebra, Combinatorics, and Algebraic Geometry 
4/7/08  Leonardo Mihalcea, Duke  Quantum Ktheory of Grassmannians 
4/11/084/13/08  TAGS  Texas
Algebraic Geometry Seminar Hosted by Rice University 
4/14/08  No meeting  
4/28/08  Clarence Wilkerson, Purdue  Equivariant Cohomology, Localization, and Fixed Points 
The Galois group of a problem in enumerative geometry encodes the structure of the set of solutions. This invariant was introduced by Jordan in 1870, and shown by Harris in 1979 to be a monodromy group of the total space of the problem. That is, it is the group of permutations of solutions obtained by varying the conditions.
Numerical homotopy continuation, a method to compute numerical solutions to systems of equations, was developed for applications of mathematics. With Anton Leykin, we apply it to the problem from pure mathematics of computing Galois groups of Schubert problems, a class of geometric problems including the problem of four lines.
In this talk, I will describe this work, also giving the
necessary background. In particular, I will explain how we show by
direct computation that the Galois group of the Schubert problem of
3planes in 8dimensional complex space meeting 15 fixed 5planes
nontrivially is the full symmetric group S_{6006}.
TOP
We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a hypersurface in R^{n}_{>} defined by a polynomial with n+l+1 terms.
This is joint work with Frédéric Bihan.
TOP
The problem of identifying homogeneous varieties from their
local differential geometry dates back to Monge, and has been
studied by Fubini, Griffiths and Harris, Hwang and Yamaguchi, and
others. I will describe recent work with J.M. Landsberg that
establishes a general recognition theorem. The key component is the
resolution of exterior differential systems by Lie algebra
cohomology.
TOP
Mircea Mustata computed the multiplier ideals of hyperplane arrangements using jet schemes. The result nicely reflects the combinatorics of the arrangement. I present a more elementary way to get the same result. When combined with the "wonderful models" introduced by De Concini and Procesi this approach allows one to simplify the result. I will also discuss some very recent progress by Nero Budur on the jumping numbers of these multiplier ideals.
All these results fail for arbitrary subspace arrangements, but
it is an open question whether similar results hold for certain
special subspace arrangements. Some questions along these lines may
be loosely related to Mark Haiman's conjectures.
TOP
In 1888, Hilbert described how to find real polynomials which
take only nonnegative values, but are not a sum of squares of
polynomials. His construction was so restrictive that no examples
appeared until the 1960s, under a variation of his original plan.
We revisit and generalize Hilbert's original construction and show
how the underlying mechanism can be simplified and
generalized.
Note room change to Milner 317.
TOP
As an algebraist, I am interested in studying properties of
squarefree monomial ideals in a polynomial ring. These ideals have
a combinatorial realization, as well as a geometric interpretation.
In this talk, I will discuss the interactions between these
viewpoints and give some examples of theorems which combine ideas
from the three areas. I will then focus on a favorite property of
algebraists, the CohenMacaulay property. I will give examples of
combinatorial criterion on the generators of a squarefree monomial
ideal I that imply R/I is CohenMacaulay.
TOP
If X is a Grassmannian (or an arbitrary homogeneous space) the 3point, genus 0, GromovWitten invariants count rational curves of degree $d$ satisfying certain incidence conditions  if this number is expected to be finite. If the number is infinite, Givental and Lee defined the Ktheoretic GromovWitten invariants, which compute the sheaf Euler characteristic of the space of rational curves in question, embedded in Kontsevich's moduli space of stable maps. The resulting quantum cohomology theory  the quantum Ktheory algebra  encodes the associativity relations satisfied by the Ktheoretic GromovWitten invariants.
In joint work with Anders Buch, we have shown that the
(equivariant) Ktheoretic GromovWitten invariants for
Grassmannians are equal to structure constants of the ordinary
(equivariant) Ktheory of certain twostep flag manifolds. We have
therefore extended  and also reproved  the "quantum=classical"
phenomenon earlier discovered by BuchKreschTamvakis in the case
of the usual GromovWitten invariants. Further, we have obtained a
Pieri and a Giambelli rule, which yield an effective algorithm to
multiply any two classes in the quantum K algebra.
TOP
I'll give a recipe or algorithm for how to recover the cohomology of the fixed points from the (Borel) equivariant cohomology of the group action, at least for certain groups G. I'll then make some remarks about the nonsingular toric variety case.
For more information, email Zach Teitler.