Monday, November 26, 2012 - 15:05 , Location: Skiles 005 , Saikat Biswas , Georgia Tech , Organizer: Matt Baker
We introduce a new invariant of an abelian variety defined over a number field, and study its arithmetic properties. We then show how an extended version of Mazur's visibility theorem yields non-trivial elements in this invariant and explain how such a construction provides theoretical evidence for the Birch and Swinnerton-Dyer Conjecture.
Monday, November 19, 2012 - 15:05 , Location: Skiles 005 , David Krumm , University of Georgia , Organizer: Matt Baker
We use a problem in arithmetic dynamics as motivation to introduce new computational methods in algebraic number theory, as well as new techniques for studying quadratic points on algebraic curves.
Monday, November 12, 2012 - 15:35 , Location: Note unusual start time for seminar. Skiles 005 , Florian Enescu , Georgia State University , Organizer: Matt Baker
The talk will discuss the concept of test ideal for rings of positive characteristic. In some cases test ideals enjoy remarkable algebraic properties related to the integral closure of ideals. We will present this connection in some detail.
Monday, November 5, 2012 - 15:05 , Location: Skiles 005 , Madhusudan Manjunath , Georgia Tech , firstname.lastname@example.org , Organizer: Josephine Yu
We describe minimal free resolutions of a lattice ideal associated with a graph and its initial ideal. These ideals are closely related to chip firing games and the Riemann-Roch theorem on graphs. Our motivations are twofold: describing information related to the Riemann-Roch theorem in terms of Betti numbers of the lattice ideal and the problem of explicit description of minimal free resolutions. This talk is based on joint work with Frank-Olaf Schreyer and John Wilmes. Analogous results were simultaneously and independently obtained by Fatemeh Mohammadi and Farbod Shokrieh.
Applying the Representation Theory of the Symmetric Group to Zeta Functions of Artin-Schreier CurvesMonday, 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)