Monday, February 11, 2013 - 15:00 , Location: Skiles 006 , Michael Burr , Clemson University , Organizer: Anton Leykin
Many real-world problems require an approximation to an algebraic variety (e.g., determination of the roots of a polynomial). To solve such problems, the standard techniques are either symbolic or numeric. Symbolic techniques are globally correct, but they are often time consuming to compute. Numerical techniques are typically fast, but include more limited correctness statements. Recently, attention has shifted to hybrid techniques that combine symbolic and numerical techniques. In this talk, I will discuss hybrid subdivision algorithms for approximating a variety. These methods recursively subdivide an initial region into smaller and simpler domains which are easier to characterize. These algorithms are typically recursive, making them both easy to implement (in practice) and adaptive (performing more work near difficult features). There are two challenges: to develop algorithms with global correctness guarantees and to determine the efficiency of such algorithms. I will discuss solutions to these challenges by presenting two hybrid subdivision algorithms. The first algorithm computes a piecewise-linear approximation to a real planar curve. This is one of the first numerical algorithms whose output is guaranteed to be topologically correct, even in the presence of singularities. The primitives in this algorithm are numerical (i.e., they evaluate a polynomial and its derivatives), but its correctness is justified with algebraic geometry and symbolic algebra. The second algorithm isolates the real roots of a univariate polynomial. I will analyze the number of subdivisions performed by this algorithm using a new technique called continuous amortization. I will show that the number of subdivisions performed by this algorithm is nearly optimal and is comparable with standard symbolic techniques for solving this problem (e.g., Descartes' rule of signs or Sturm sequences).
Monday, December 3, 2012 - 15:05 , Location: Skiles 005 , Jeremy Jacobson , University of Georgia , Organizer: Josephine Yu
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.
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.