Seminars and Colloquia by Series

On the derived Witt groups of schemes

Series
Algebra Seminar
Time
Monday, December 3, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jeremy JacobsonUniversity of Georgia
The Witt group of a scheme is a globalization to schemes of the classical Witt group of a field. It is a part of a cohomology theory for schemes called the derived Witt groups. In this talk, we introduce two problems about the derived Witt groups, the Gersten conjecture and a finite generation question for arithmetic schemes, and explain recent progress on them.

Arithmetic of Abelian Varieties

Series
Algebra Seminar
Time
Monday, November 26, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Saikat BiswasGeorgia Tech
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.

Preperiodic points for quadratic polynomials

Series
Algebra Seminar
Time
Monday, November 19, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
David KrummUniversity of Georgia
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.

Strong test ideals

Series
Algebra Seminar
Time
Monday, November 12, 2012 - 15:35 for 1 hour (actually 50 minutes)
Location
Note unusual start time for seminar. Skiles 005
Speaker
Florian EnescuGeorgia State University
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.

Minimal Free Resolutions of the toppling ideal of a graph and its initial ideal

Series
Algebra Seminar
Time
Monday, November 5, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Madhusudan ManjunathGeorgia Tech
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.

Linear series on metrized complexes of algebraic curves

Series
Algebra Seminar
Time
Monday, October 8, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matthew BakerGeorgia Tech
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.

Algorithms for symmetric Gröbner bases

Series
Algebra Seminar
Time
Monday, September 24, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert KroneGeorgia Tech
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.

Explicit modular approaches to generalized Fermat equations

Series
Algebra Seminar
Time
Monday, September 17, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
David Zureick-BrownEmory
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.

Asymptotic Hilbert series

Series
Algebra Seminar
Time
Monday, August 27, 2012 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gregory G. SmithQueens University
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.

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