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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Fall 2008 Schedule:

Abstracts:

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 Hans-Christian Graf von Bothmer.
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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 explicit---but very difficult---algorithm 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.
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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 long-standing BMV conjecture in statistical physics. (partly joint with C. R. Johnson and separately, S. Armstrong).
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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.)
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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.
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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.
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An A-hypergeometric 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 non-generic parameters. No background is necessary; it may be helpful to have a basic understanding of semigroup rings.
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In this talk we will give a brief history of dualizing complexes, and their more concrete manifestations, dualizing modules. We will then introduce semi-dualizing modules, and discuss existence questions for them in the Cohen-Macaulay context.
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The classical matrix-tree theorem expresses the number of spanning trees of a graph G in terms of the eigenvalues of its Laplacian matrix; one well-known special case is Cayley's formula n^{n-2} 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 CW-complexes in such a way that the matrix-tree theorem remains valid. As an application, we enumerate the simplicial spanning trees of any shifted simplicial complex X by their facet-vertex 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.
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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)
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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.
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Let K be a p-adic field and f be a univariate polynomial with coefficients in K and non-vanishing 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.
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The Borel-Weil-Bott theorem identifies finite dimensional representations for a complex semi-simple 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 D-module. We aim to understand representations by relating D-modules on various partial flag varieties associated to our group. I will begin with a brief overview of D-modules and equivariant sheaves, highlighting some advantageous differences from O-modules 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.
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For more information, email Zach Teitler.