Valley Geometry Seminar Fridays, LGRT 1634 University of Massachusetts Amherst, MA Tea: 3:45 PM

 

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Autumn 2002 Schedule:

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		6 September	Business Meeting
		3:30-4:30
		13 September	Eric Babson, University of Washington.
		4:00-5:00	A Space of Simplices
		20 September	David Cox, Amherst College
		4:00-5:00	Universal Rational Parametrizations and Toric Varieties
		27 September	Joint VGS --- Amherst College Colloquium
			Sarah Greenwald, Appalachian State University
		4:00-5:00	Geometric Properties of Spherical Orbifolds
		4 October	Ivan Soprounov, U Mass
		4:00-5:00	Tame Symbol and Product of Roots Formula
		11 October	Eduardo Cattani, U Mass    Room Change: LGRT 319
		4:00-5:00	Periodic Instantons
		18 October	Marcos Jardim, U Mass
		4:00-5:00	Periodic instantons
		25 October	Jenya Soprounova, U Mass
		4:00-5:00	Zeros of systems of exponential sums and trigonometric polynomials
		1 November	Jessica Sidman, UC Berkeley, MSRI, and Mt. Holyoke
		4:00-5:00	Bounding Castelnuovo-Mumford regularity using approximations
		8 November	Greg Smith, Banard College, Columbia University
		4:00-5:00	Orbifold Chow rings of stacky toric varieties
		15 November	No Seminar
		22 November	Sarah Witherspoon, Amherst College
		4:00-5:00	Algebraic deformations arising from orbifolds
		29 November	No Seminar: Thanksgving week
		6 December	Dan Abramovich, Boston University
		4:00-5:00	Title
		13 December	Kale Karu, Harvard University
		4:00-5:00	Title: TBA

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Supported in part by the

Abstract:
We study an incidence variety on complex projective space. The general points are simply configurations of n general points in projective n-1 space. The space is singular and we consider a geometrically defined resolution that preserves the various group actions and maps to Grassmannians. The case n=3 is classical, and was studied by Schubert, Semple, Roberts-Speiser, and Collino-Fulton. For n=4 we can show that the resulting resolution is smooth and compute various facts about this space. The combinatorics turns out to be fairly complicated and so for larger values of n we can say very little about the spaces.
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Abstract:
When parametrizing a given rational variety in geometric modeling, one problem is that many different parametrizations can give the same variety. However, if the variety is a sufficiently nice projection of an embedded toric variety, then one can get a precise description of all possible rational maps from affine space to the variety. The talk will consist of a series of examples to illustrate our results. This is joint work with R. Krasauskas and M. Mustat
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Abstract:
A Riemmanian orbifold is, roughly speaking, a metric space locally modeled on quotients of Riemannian manifolds by finite groups of isometries. Since orbifolds are the most tractable singular spaces, they furnish a natural starting point for the study of the geometry of singular spaces in general. Many results for Riemannian manifolds generalize easily to the orbifold setting. For example, results of local analysis still hold for orbifolds since one can simply work in the local finite cover, which is a smooth manifold. However, most global results in Riemannian geometry do not generalize to orbifolds easily: many of them no longer hold, while others take on a different form. After an introduction to geometric orbifolds, we will discuss examples and results related to the diameter and spectrum of spherical orbifolds.
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Abstract: The Vieta formula for the product of roots of a polynomial is a particular case of Weil's reciprocity for the tame symbol on a complex projective curve. For a wide class of sparse polynomial systems we have a similar situation: the generalized Vieta formula for the product of roots of a system (recently obtained by Khovanskii) has a simple explanation in the theory of tame symbols for toric varieties.
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Abstract: Periodic instantons are anti-self-dual connections on the Euclidean 4-dimensional space which are periodic in one, two or three directions. They are of great interest to many physicists and string theorists. This talk focuses on the geometrical aspects of such objects, like their asymptotic behaviour and moduli spaces, with special attention given to the case of doubly-periodic instantons. An interesting duality among invariant instantons called Nahm transform will also be described.
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Abstract:
Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial at the zeros of a system of n Laurent polynomials in (C\0)ⁿ with generically positioned Newton polytopes. A similar formula should hold in the case of exponential sums with real frequencies. We prove the existence-part of the conjectured formula not only in the complex situation but also in a very general real setting. In most cases the conjectured formula gives answer zero. We prove that this is indeed true.
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Abstract:
Due to rapid advances in the field of computational algebra over the past twenty years, the problem of determining the complexity of working algebraically has become increasingly important. The Castelnuovo-Mumford regularity of a finitely generated graded module M over a polynomial ring k[x₀, ... , x_n] is a key ingredient in estimating the computational resources needed in working with M.

In the course of general investigations into the behavior of the Castelnuovo-Mumford regularity of products of ideals, A. Conca and J. Herzog were recently able to compute the regularity of products of linear ideals. I will discuss joint work with H. Derksen which shows how the computation of Conca and Herzog can be generalized to give a method for bounding the regularity of a module with sufficiently nice ``approximations.'' In particular, we can bound the regularity of the ideal of an arrangement of linear subspaces in projective space as well as certain other modules whose descriptions are combinatorial in nature.
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Abstract:
How do we see the creprant resolutions of a space X in the geometry of X? In this talk, we examine this question when X is a simplicial toric variety. In particular, we show that the appropriate Chow ring of an associated Deligne-Mumford stack (a stacky toric variety) encodes the Chow ring of any creprant resolution. This is joint work with Lev Borisov and Linda Chen.
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Abstract:
An orbifold is essentially a quotient of a manifold X by an action of a finite group G. Such a quotient is usually singular, and one would like to resolve or deform the singularities away to obtain a smooth manifold again. Such geometric deformations are related to deformations of algebras (or sheaves of algebras) associated to the orbifold. In this talk, we will focus on these algebraic deformations, connections to Hochschild cohomology, and how to compute explicitly the Hochschild cohomology and deformations for certain types of algebras arising from orbifolds. The talk will be introductory, and students are welcome.
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