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Series: Algebra Seminar

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).

Series: Algebra Seminar

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.

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.