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 2008 Schedule:

Abstracts:

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 3-planes in 8-dimensional complex space meeting 15 fixed 5-planes non-trivially is the full symmetric group S₆₀₀₆.
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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 Rⁿ_> defined by a polynomial with n+l+1 terms.

This is joint work with Frédéric Bihan.
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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.
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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.
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In 1888, Hilbert described how to find real polynomials which take only non-negative 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.
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As an algebraist, I am interested in studying properties of square-free 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 Cohen-Macaulay property. I will give examples of combinatorial criterion on the generators of a square-free monomial ideal I that imply R/I is Cohen-Macaulay.
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If X is a Grassmannian (or an arbitrary homogeneous space) the 3-point, genus 0, Gromov-Witten 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 K-theoretic Gromov-Witten 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 K-theory algebra - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants.

In joint work with Anders Buch, we have shown that the (equivariant) K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary (equivariant) K-theory of certain two-step flag manifolds. We have therefore extended - and also reproved - the "quantum=classical" phenomenon earlier discovered by Buch-Kresch-Tamvakis in the case of the usual Gromov-Witten 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.
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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.

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For more information, email Zach Teitler.