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Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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 Curves

Series: Algebra Seminar

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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)