Monday, March 11, 2013 - 16:05 , Location: Skiles 006 , Dustin Cartwright , Yale University , Organizer: Josephine Yu
A tropical complex is a Delta-complex together with some additional numerical data, which come from a semistable degeneration of a variety. Tropical complexes generalize to higher dimensions some of the analogies between curves and graphs. I will introduce tropical complexes and explain how they relate to classical algebraic geometry.
Monday, March 4, 2013 - 15:00 , Location: Skiles 005 , Greg Blekherman , Georgia Tech , Organizer: Greg Blekherman
I will explain and draw connections between the following two theorems: (1) Classification of varieties of minimal degree by Del Pezzo and Bertini and (2) Hilbert's theorem on nonnegative polynomials and sums of squares. This will result in the classification of all varieties on which nonnegative polynomials are equal to sums of squares. (Joint work with Greg Smith and Mauricio Velasco)
Monday, February 25, 2013 - 15:05 , Location: Skiles 005 , Julio Andrade , ICERM , Organizer:
Monday, February 18, 2013 - 15:05 , Location: Skiles 005 , Eric Katz , Waterloo , Organizer: Matt Baker
In a recent work with June Huh, we proved the log-concavity of the characteristic polynomial of a realizable matroid by relating its coefficients to intersection numbers on an algebraic variety and applying an algebraic geometric inequality. This extended earlier work of Huh which resolved a long-standing conjecture in graph theory. In this talk, we rephrase the problem in terms of more familiar algebraic geometry, outline the proof, and discuss an approach to extending this proof to all matroids. Our approach suggests a general theory of positivity in tropical geometry.
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.