Seminars and Colloquia by Series

Monday, October 29, 2012 - 15:05 , Location: Skiles 005 , Alexander Mueller , University of Michigan , Organizer:
Monday, October 8, 2012 - 15:05 , Location: Skiles 005 , Matthew Baker , Georgia Tech , Organizer: Matt Baker
A metrized complex of algebraic curves over a field K is, roughly speaking, a finite edge-weighted graph G together with a collection of marked complete nonsingular algebraic curves C_v over K, one for each vertex; the marked points on C_v correspond to edges of G incident to v. We will present a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical and tropical Riemann-Roch theorems, together with a semicontinuity theorem for the behavior of the rank function under specialization of divisors from smooth curves to metrized complexes. The statement and proof of the latter result make use of Berkovich's theory of non-archimedean analytic spaces. As an application of the above considerations, we formulate a partial generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type. This is joint work with Omid Amini.
Monday, September 24, 2012 - 15:05 , Location: Skiles 005 , Robert Krone , Georgia Tech , Organizer: Anton Leykin
A symmetric ideal in the polynomial ring of a countable number of variables is an ideal that is invariant under any permutations of the variables. While such ideals are usually not finitely generated, Aschenbrenner and Hillar proved that such ideals are finitely generated if you are allowed to apply permutations to the generators, and in fact there is a notion of a Gröbner bases of these ideals.  Brouwer and Draisma showed an algorithm for computing these Gröbner bases.  Anton Leykin, Chris Hillar and I have implemented this algorithm in Macaulay2.  Using these tools we are exploring the possible invariants of symmetric ideals that can be computed, and looking into possible applications of these algorithms, such as in graph theory.
Monday, September 17, 2012 - 15:05 , Location: Skiles 005 , David Zureick-Brown , Emory , Organizer: Matt Baker
Let a,b,c >= 2 be integers satisfying 1/a + 1/b + 1/c > 1. Darmon and Granville proved that the generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions; conjecturally something stronger is true: for a,b,c \geq 3 there are no non-trivial solutions and for (a,b,c) = (2,3,n) with n >= 10 the only solutions are the trivial solutions and (+- 3,-2,1) (or (+- 3,-2,+- 1) when n is even). I'll explain how the modular method used to prove Fermat's last theorem adapts to solve generalized Fermat equations and use it to solve the equation x^2 + y^3 = z^10.
Monday, August 27, 2012 - 15:00 , Location: Skiles 005 , Gregory G. Smith , Queens University , Organizer: Greg Blekherman
How does one study the asymptotic properties for the Hilbert series of a module?  In this talk, we will examine the function which sends the numerator of the rational function representing the Hilbert  series of a module to that of its r-th Veronese submodule.  As r tends to infinity, the behaviour of this function depends only on the multidegree of the module and the underlying multigraded polynomial ring.  More importantly, we will give a polyhedral description for the asymptotic polynomial and show that the coefficients are log-concave.
Monday, August 20, 2012 - 15:00 , Location: Skiles 005 , Daniel Plaumann , University of Konstanz , Organizer: Greg Blekherman
Hyperbolic polynomials are real polynomials that can be thought of as generalized determinants. Each such polynomial determines a convex cone, the hyperbolicity cone. It is an open problem whether every hyperbolicity cone can be realized as a linear slice of the cone of psd matrices. We discuss the state of the art on this problem and describe an inner approximation for a hyperbolicity cone via a sums of squares relaxation that becomes exact if the hyperbolic polynomial possesses a symmetric determinantal representation. (Based on work in progress with Cynthia Vinzant)
Monday, May 21, 2012 - 15:00 , Location: Skiles 006 , Chris Hillar , UC Berkeley , Organizer: Anton Leykin
We discuss the theory of symmetric Groebner bases, a concept allowing one to prove Noetherianity results for symmetric ideals in polynomial rings with an infinite number of variables. We also explain applications of these objects to other fields such as algebraic statistics, and we discuss some methods for computing with them on a computer. Some of this is joint work with Matthias Aschenbrener and Seth Sullivant.
Tuesday, April 24, 2012 - 14:00 , Location: Skiles 006 , Andrew Dudzik , UC Berkeley , Organizer: Anton Leykin
The construction of the Berkovich space associated to a rigid analytic variety can be understood in a general topological framework as a type of local compactification or uniform completion, and more generally in terms of filters on a lattice.  I will discuss this viewpoint, as well as connections to Huber's theory of adic spaces, and draw parallels with the usual metric completion of $\mathbb{Q}$.
Tuesday, April 17, 2012 - 14:05 , Location: Skiles 006 , Jeremy Martin , University of Kansas , jmartin@math.ku.edu , Organizer: Josephine Yu
The critical group of a graph G is an abelian group K(G) whose order is the number of spanning forests of G.  As shown by Bacher, de la Harpe and Nagnibeda, the group K(G) has several equivalent presentations in terms of the lattices of integer cuts and flows on G. The motivation for this talk is to generalize this theory from graphs to CW-complexes, building on our earlier work on cellular spanning forests. A feature of the higher-dimensional case is the breaking of symmetry between cuts and flows.  Accordingly, we introduce and study two invariants of X: the critical group K(X) and the cocritical group K^*(X), As in the graph case, these are defined in terms of combinatorial Laplacian operators, but they are no longer isomorphic; rather, the relationship between them is expressed in terms of short exact sequences involving torsion homology. In the special case that X is a graph, torsion vanishes and all group invariants are isomorphic, recovering the theorem of Bacher, de la Harpe and Nagnibeda.  This is joint work with Art Duval (University of Texas, El Paso) and Caroline Klivans (Brown University).
Tuesday, April 10, 2012 - 14:00 , Location: Skiles 006 , Abraham Martin del Campo , Texas A&M , Organizer: Anton Leykin
The Galois group of a problem in enumerative geometry is a subtle invariant that encodes special structures in the set of solutions. This invariant was first introduced by Jordan in 1870. In 1979, Harris showed that the Galois group of such problems coincides with the monodromy group of the total space. These geometric invariants are difficult to determine in general. However, a consequence of Vakil's geometric Littlewood-Richardson rule is a combinatorial criterion to determine if a Schubert problem on a Grassmannian contains at least the alternating group.  Using Vakil's criterion, we showed that for Schubert problems of lines, the Galois group is at least the alternating group.

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