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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.

Series: Algebra Seminar

This talk will describe some recent results using exact massformulas to determine all definite quadratic forms of small class number inn>=3 variables, particularly those of class number one.The mass of a quadratic form connects the class number (i.e. number ofclasses in the genus) of a quadratic form with the volume of its adelicstabilizer, and is explicitly computable in terms of special values of zetafunctions. Comparing this with known results about the sizes ofautomorphism groups, one can make precise statements about the growth ofthe class number, and in principle determine those quadratic forms of smallclass number.We will describe some known results about masses and class numbers (overnumber fields), then present some new computational work over the rationalnumbers, and perhaps over some totally real number fields.

Series: Algebra Seminar

For two polynomials G(X), H(Y) with rational coefficients, when does G(X) = H(Y) have infinitely many solutions over the rationals? Such G and H have been classified in various special cases by previous mathematicians. A theorem of Faltings (the Mordell conjecture) states that we need only analyze curves with genus at most 1.In my thesis (and more recent work), I classify G(X) = H(Y) defining irreducible genus zero curves. In this talk I'll present the infinite families which arise in this classification, and discuss the techniques used to complete the classification.I will also discuss in some detail the examples of polynomial which occur in the classification. The most interesting infinite family of polynomials are those H(Y) solving a Pell Equation H(Y)^2 - P(Y)Q(Y)^2 = 1. It turns out to be difficult to describe these polynomials more explicitly, and yet we can completely analyze their decompositions, how many such polynomials there are of a fixed degree, which of them are defined over the rationals (as opposed to a larger field), and other properties.

Series: Algebra Seminar

The ring of invariants for the action of the automorphism group of the projective line on the n-fold product of the projective line is a classical object of study. The generators of this ring were determined by Kempe in the 19th century. However, the ideal of relations has been only understood very recently in work of Howard, Millson, Snowden and Vakil. They prove that the ideal of relations is generated byquadratic equations using a degeneration to a toric variety. I will report on joint work with Benjamin Howard where we further study the toric varieties arising in this degeneration. As an application we show that the second Veronese subring of the ring of invariants admits a presentation whose ideal admits a quadratic Groebner basis.

Series: Algebra Seminar

Granville and Soundararajan have recently introduced thenotion of pretentiousness in the study of multiplicative functions ofmodulus bounded by 1, essentially the idea that two functions whichare similar in a precise sense should exhibit similar behavior. Itturns out, somewhat surprisingly, that this does not directly extendto detecting power cancellation - there are multiplicative functionswhich exhibit as much cancellation as possible in their partial sumsthat, modified slightly, give rise to functions which exhibit almostas little as possible. We develop two new notions of pretentiousnessunder which power cancellation can be detected, one of which appliesto a much broader class of multiplicative functions. This work isjoint with Junehyuk Jung.