Seminars and Colloquia by Series

Series

Recent progress on computing Groebner bases

Series: Algebra Seminar
Time: Monday, February 24, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Shuhong Gao – Clemson University

Buchberger (1965) gave the first algorithm for computing Groebner bases and introduced some simple criterions for detecting useless S-pairs. Faugere (2002) presented the F5 algorithm which is significantly much faster than Buchberger's algorithm and can detect all useless S-pairs for regular sequences of homogeneous polynomials. In recent years, there has been extensive effort trying to simply F5 and to give a rigorous mathematical foundation for F5. In this talk, we present a simple new criterion for strong Groebner bases that contain Groebner bases for both ideals and the related syzygy modules. This criterion can detect all useless J-pairs (without performing any reduction) for any sequence of polynomials, thus yielding an efficient algorithm for computing Groebner bases and a simple proof of finite termination of the algorithm. This is a joint work with Frank Volny IV (National Security Agency) and Mingsheng Wang (Chinese Academy of Sciences).

Tropical Laplacians and the Colin de Verdiere number of graphs

Series: Algebra Seminar
Time: Monday, February 17, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Eric Katz – University of Waterloo

Given a surface in space with a set of curves on it, one can ask whichpossible combinatorial arrangement of curves are possible. We give anenriched formulation of this question in terms of which two-dimensionalfans occur as the tropicalization of an algebraic surface in space. Ourmain result is that the arrangement is either degenerate or verycomplicated. Along the way, we introduce tropical Laplacians, ageneralization of graph Laplacians, explain their relation to the Colin deVerdiere invariant and to tensegrity frameworks in dynamics.This is joint work with June Huh.

Stable cohomology of toroidal compactifications of the moduli space of abelian varieties

Series: Algebra Seminar
Time: Friday, January 10, 2014 - 16:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Orsola Tomassi – Leibniz University Hannover

It is well known that the cohomology of the moduli space A_g of g-dimensional principally polarized abelian varieties stabilizes when the degree is smaller than g. This is a classical result of Borel on the stable cohomology of the symplectic group. By work of Charney and Lee, also the stable cohomology of the minimal compactification of A_g, the Satake compactification, is explicitly known.In this talk, we consider the stable cohomology of toroidal compactifications of A_g, concentrating on the perfect cone compactification and the matroidal partial compactification. We prove stability results for these compactifications and show that all stable cohomology is algebraic. This is joint work with S. Grushevsky and K. Hulek.

Alexander polynomials of curves and Mordell-Weil ranks of Abelian threefolds

Series: Algebra Seminar
Time: Friday, January 10, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Remke Kloosterman – Humboldt University Berlin

Let $C=\{f(z_0,z_1,z_2)=0\}$ be a complex plane curve with ADE singularities. Let $m$ be a divisor of the degree of $f$ and let $H$ be the hyperelliptic curve $y^2=x^m+f(s,t,1)$ defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of $H$ effectively. For this we use some results on the Alexander polynomial of $C$. This extends a result by Cogolludo-Augustin and Libgober for the case where $C$ is a curve with ordinary cusps. In the second part we discuss how one can do a similar approach over fields like $\mathbb{Q}(s,t)$ and $\mathbb{F}(s,t)$.

Tropical Scheme Theory

Series: Algebra Seminar
Time: Wednesday, January 8, 2014 - 11:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Noah Giansiricusa – UC Berkeley

I'll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring T, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over T. Upon passing to the set of T-valued points this yields Payne's extended tropicalization functor. We prove that the Hilbert polynomial of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.

The skeleton of the Jacobian and the Jacobian of the skeleton

Series: Algebra Seminar
Time: Monday, December 2, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Joseph Rabinoff – Georgia Tech – rabinoff@math.gatech.edu

Let X be an algebraic curve over a non-archimedean field K. If the genus of X is at least 2 then X has a minimal skeleton S(X), which is a metric graph of genus <= g. A metric graph has a Jacobian J(S(X)), which is a principally polarized real torus whose dimension is the genus of S(X). The Jacobian J(X) also has a skeleton S(J(X)), defined in terms of the non-Archimedean uniformization theory of J(X), and which is again a principally polarized real torus with the same dimension as J(S(X)). I'll explain why S(J(X)) and J(S(X)) are canonically isomorphic, and I'll indicate what this isomorphism has to do with several classical theorems of Raynaud in arithmetic geometry.

Two ways of degenerating the Jacobian are the same

Series: Algebra Seminar
Time: Monday, November 25, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Jesse Kass – University of South Carolina – kassj@math.sc.edu

The Jacobian variety of a smooth complex curve is a complex torus that admits two different algebraic descriptions. The Jacobian can be described as the Picard variety, which is the moduli space of line bundles, or it can be described as the Albanese variey, which is the universal abelian variety that contains the curve. I will talk about how to extend a family of Jacobians varieties by adding degenerate fibers. Corresponding to the two different descriptions of the Jacobian are two different extensions of the Jacobian: the Neron model and the relative moduli space of stable sheaves. I will explain what these two extensions are and then prove that they are equivalent. This equivalence has surprising consequences for both the Neron model and the moduli space of stable sheaves.

Families of lattice-polarized K3 surfaces

Series: Algebra Seminar
Time: Monday, November 18, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Wei Ho – Columbia University

There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.

The proetale topology

Series: Algebra Seminar
Time: Monday, November 18, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location: Skiles 006
Speaker: Bhargav Bhatt – Institute for Advanced Study

Abstract: (joint work with Peter Scholze) The proetale topology is a Grothendieck topology that is closely related to the etale topology, yet better suited for certain "infinite" constructions, typically encountered in l-adic cohomology. I will first explain the basic definitions, with ample motivation, and then discuss applications. In particular, we will see why locally constant sheaves in this topology yield a fundamental group that is rich enough to detect all l-adic local systems through its representation theory (which fails for the groups constructed in SGA on the simplest non-normal varieties, such as nodal curves).

An Algebraic Approach to Network Optimization

Series: Algebra Seminar
Time: Monday, November 11, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location: Skiles 005
Speaker: Dr. Sanjeevi Krishnan – University of Pennsylvania – sanjeevi@math.upenn.edu

Please Note: This talk assumes no familiarity with directed topology, flow-cut dualities, or sheaf (co)homology.

Flow-cut dualities in network optimization bear a resemblance to topological dualities. Flows are homological in nature, cuts are cohomological in nature, constraints are sheaf-theoretic in nature, and the duality between max flow-values and min cut-values (MFMC) resembles a Poincare Duality. In this talk, we formalize that resemblance by generalizing Abelian sheaf (co)homology for sheaves of semimodules on directed spaces (e.g. directed graphs). Such directed (co)homology theories generalize constrained flows, characterize cuts, and lift MFMC dualities to a directed Poincare Duality. In the process, we can relate the tractability and decomposability of generalized flows to local and global flatness conditions on the sheaf, extending previous work on monoid-valued flows in the literature [Freize].

Georgia Institute of Technology College of Sciences

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