Algebraic Geometry Seminar Fridays, Milner 317 Texas A&M University 2:00 - 2:50 PM

This Week's Seminar

3 December  Frank Sottile, Texas A&M 2:00-2:50   Equivariant Cohomology of the Quot Scheme

		24 September	E. Soprunova, University of Massachusetts, Amherst
		2:00-2:50	Lower bounds for some sparse polynomial systems
		1 October	Kevin Purbhoo, Fields Institute, Toronto
		2:00-2:50	The generalized Horn recursion
		8 October	Paulo Lima-Filho, Texas A&M University
		2:00-2:50	The Bredon cohomology ring of real quadrics
		15 October	J. Maurice Rojas, Texas A&M University
		2:00-2:50	A Complexity Threshold for Real Fewnomials
		22 October	Hal Schenck, Texas A&M University
		2:00-2:50	Syzygies of toric varieties
		29 October	Tara Holm, Berkeley
		2:00-2:50	Surjectivity techniques in symplectic geometry
		5 November	Jim Ruffo, Texas A&M
		2:00-2:50	Experimentation and Conjectures in the Real Schubert Calculus for Flag Manifolds
		12 November	Luis Garcia, MSRI, Texas A&M University
		2:00-2:50	Solving the Likelihood Equations of Small Phylogenetic Trees
		19 November
		26 November
		3 December	Frank Sottile, Texas A&M
		2:00-2:50	Equivariant Cohomology of the Quot Scheme

Abstract:
We present families of sparse polynomial systems having a lower bound on their number of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov on the degree of the Wronski map. This is joint work with Frank Sottile
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Abstract:
The classical statement of Horn's conjecture gives a recursive set of conditions on possible eigenvalues of triples of Hermitian matrices (A,B,C) satisfying A+B+C=0. Reformulated, it says that there is a recursive nature to the set of non-vanishing Littlewood-Richardson numbers. I'll discuss a generalisation which recursively characterises the non-vanishing Schubert intersection numbers for all minuscule flag varieties. The main focus will be on the example of the even dimensional quadric hypersurface (a type-D minuscule flag variety). The cohomology of these spaces is relatively simple, but even here, the recursion has interesting things to say.
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Abstract:
In this talk we will briefly introduce equivariant Bredon cohomology and explain its relevance in the study of real varieties. This stems from the fact that this cohomology is a natural recipient for cycle maps from motivic (hence from classical Chow groups) cohomology, and these cycle maps are compatible with characteristic classes. We then compute the equivariant Bredon cohomology of real quadric hypersurfaces and exhibit relations to their Chow groups (as real varieties) and classical computations in both algebraic geometry and topology.
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Abstract:
    Let f be a polynomial in n variables with integer coefficients and let RFEAS denote the following problem:

Abstract:
A toric variety is a rational variety, and is defined by a fan of polyhedral cones. Toric varieties give the simplest examples of varieties defined intrinsically (i.e. in terms of local information). If a toric variety is projective, then choosing a divisor yields a map to projective space. This map is determined by the polytope consisting of the global sections of the line bundle associated to the divisor. In this talk, we'll investigate the interplay between properties of the polytope and properties of the embedded variety (defining equations, free resolution, etc.). Some portions of the talk are joint work with Greg Smith (Queens).
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Abstract:
I will talk about the topology of symplectic (and other) quotients. I will briefly review Kirwan's techniques for proving that the restriction map from the equivariant cohomology of the originial space to the ordinary cohomology of the symplectic reduction is a surjection. I will show how this result can be used to understand various aspects of the topology of quotients, touching on such themes as real loci, orbifolds and orbifold cohomology.
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Abstract:
The classical Schubert calculus solves the problem of enumerating linear subspaces of an ambient vector space satifying conditions imposed by fixed flags of linear subspaces, when the vector spaces are complex. Over the real numbers, the answers depend very subtly on the fixed subspaces, and the situation is further complicated if we consider the more general case of enumerating flags. The Shapiro conjecture states that all solutions are real when the fixed flags are chosen in a certain way. This conjecture fails for flags, but extensive computation has lead to a very interesting refinement. We will discuss this project, its implications and some new results it has inspired. This is joint work with Y. Sivan, E. Soprunova, and F. Sottile.
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Abstract:
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. The algebraic degree of the critical equations of this optimization problem was studied by Catanse, Hosten, Khetan and Sturmfels. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement.

Algebraic algorithms for computing all complex solutions to the critical equations of the likelihood function were introduced by Hosten, Khetan and Sturmfels, with the aim of identifying the local maxima in the probability simplex.

In this talk, I will first introduce the notion of maximum likelihood degree of a statistical model. Then I will introduce some models used in Computational Biology to reconstruct phylogenetic trees from DNA sequences. Finally, I will present an application of these algebraic algorithms to compute the ML of some small phylogenetic trees. If time allows, I will compare our analytic methods to the numerical methods commonly used, e.g. fastDNAml, for estimating maximum likelihood phylogenetic trees from nucleotide sequences.
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Abstract:
We study the quot scheme which compactifies the space of rational curves on a Grassmannian. Our main result is a presentation of its equivariant cohomology using an extension of the formalism of Goresky-Kottwicz-Macpherson.
This is joint work in progress with Tom Braden and Linda Chen.
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