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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)

Series: Algebra Seminar

We discuss the theory of symmetric Groebner bases, a concept allowing
one to prove Noetherianity results for symmetric ideals in polynomial
rings with an infinite number of variables. We also explain applications
of these objects to other fields such as algebraic statistics, and we
discuss some methods for computing with them on a computer. Some of this
is joint work with Matthias Aschenbrener and Seth Sullivant.

Series: Algebra Seminar

The construction of the Berkovich space associated to a rigid analytic
variety can be understood in a general topological framework as a type of
local compactification or uniform completion, and more generally in terms
of filters on a lattice. I will discuss this viewpoint, as well as
connections to Huber's theory of adic spaces, and draw parallels with the
usual metric completion of $\mathbb{Q}$.

Series: Algebra Seminar

The critical group of a graph G is an abelian group K(G) whose order is
the number of spanning forests of G. As shown by Bacher, de la Harpe
and Nagnibeda, the group K(G) has several equivalent presentations in
terms of the lattices of integer cuts and flows on G. The motivation for
this talk is to generalize this theory from graphs to CW-complexes,
building on our earlier work on cellular spanning forests. A feature of
the higher-dimensional case is the breaking of symmetry between cuts and
flows. Accordingly, we introduce and study two invariants of X: the
critical group K(X) and the cocritical group K^*(X), As in the graph
case, these are defined in terms of combinatorial Laplacian operators,
but they are no longer isomorphic; rather, the relationship between them
is expressed in terms of short exact sequences involving torsion
homology. In the special case that X is a graph, torsion vanishes and
all group invariants are isomorphic, recovering the theorem of Bacher,
de la Harpe and Nagnibeda. This is joint work with Art Duval
(University of Texas, El Paso) and Caroline Klivans (Brown University).

Series: Algebra Seminar

The Galois group of a problem in enumerative geometry is a subtle
invariant that encodes special structures in the set of solutions. This
invariant was first introduced by Jordan in 1870. In 1979, Harris showed
that the Galois group of such problems coincides with the monodromy
group of the total space. These geometric invariants are difficult to
determine in general. However, a consequence of Vakil's geometric
Littlewood-Richardson rule is a combinatorial criterion to determine if a
Schubert problem on a Grassmannian contains at least the alternating
group. Using Vakil's criterion, we showed that for Schubert problems of
lines, the Galois group is at least the alternating group.

Series: Algebra Seminar

In many applications in engineering and physics, one is interested in
computing real solutions to systems of equations. This talk will
explore numerical approaches for approximating solutions to systems of
polynomial and polynomial-exponential equations. We will then discuss
using certification methods based on Smale's alpha-theory to rigorously
determine if the corresponding solutions are real. Examples from
kinematics, electrical engineering, and string theory will be used to
demonstrate the ideas.

Series: Algebra Seminar

The central curve of a linear program is an algebraic curve specified by a hyperplane arrangement and a cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. I will discuss the algebraic properties of this curve and its beautiful global geometry, both of which are controlled by the corresponding matroid and hyperplane arrangement.

Series: Algebra Seminar

Let I be an ideal in a polynomial ring R := F[x_n,...,x_1] Let A := R/I be the corresponding quotient ring, and let Q(A) be its eld of fractions. The integral closure C(A, Q(A)) of A in Q(A) is a subring of the latter. But it is often given as a separate quotient ring, a presentation. Surprisingly, different computer algebra systems (Magma, Macaulay2, and Singular) choose to produce very different presentations. Some of these opt for presentations that have seductive forms, but miss the most important, namely a form that allows for determining when elements ofQ(A) are in C(A,Q(A)). This is called membership and is directly related to determining isomorphism.