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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.

Series: Algebra Seminar

Metric graphs arise naturally in tropical tropical geometry and Berkovich geometry. Recent efforts have extend conventional notion of divisors and linear systems on algebraic curves to finite graphs and metric graphs (tropical curves). Reduced divisors are introduced as an essential tool in proving graph-theoretic Riemann-Roch. In short, a q-reduced divisor is the unique divisor in a linear system with respect to a point q in the graph. In this talk, I will show how tropical convexity is related to linear systems on metric graphs, and define a canonical metric on the linear systems. In addition, I will introduce a generalized notion of reduced divisors, which are defined with respect to any effective divisor as in comparison a single point (effective divisor of degree one) in the conventional case.

Series: Algebra Seminar

Geometric modeling builds computer models for industrial design and manufacture from basic units, called patches, such as, Bézier curves and surfaces. The control polygon of a Bézier curve is well-defined and has geometric significance—there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. In this talk, I will present a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary. While our primary interest is to explain the meaning of control nets for the classical rational Bezier patches, we work in the generality of Krasauskas’ toric Bezier patches. Toric Bezier patches are multi-sided parametric patches based on the geometry of toric varieties and depend on a polytope and some weights. Our results rely upon a construction in computational algebraic geometry called a toric degeneration.

Series: Algebra Seminar

A matroid is a structure that captures the notion of "independence". For example, given a set of vectors in a vector space, one can define a matroid. Graphs also naturally give rise to matroids. I will talk about various simplicial complexes associated to matroids. These include the "matroid complex", the "broken circuit complex", and the "order complex" of the associated geometric lattice. They carry some of the most important invariants of matroids and graphs. I will also show how the Bergman fan and its refinement (which arise in tropical geometry) relate to the classical theory. If time permits, I will give an outline of a recent breakthrough result of Huh and Katz on log-concavity of characteristic (chromatic) polynomials of matroids. No prior knowledge of the subject will be assumed. Most of the talk should be accessible to advanced undergraduate students.

Series: Algebra Seminar

The talk will discuss the notion of Hilbert-Kunz multiplicity,
presenting its general theory and listing some of the outstanding open
problems together with recent progress on them.

Series: Algebra Seminar

In celestial mechanics a configuration of n point masses is called central if it collapses by scaling to the center of mass when released with initial velocities equal to zero. We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations in the Newtonian five-body problem with positive masses is finite, except for some explicitly given special choices of mass values. The proof will be computational using tropical geometry, Gröbner bases and sum-of-squares decompositions.This is joint work with Marshall Hampton.

Series: Algebra Seminar

[Note unusual day and time!]

In the last decades, path following methods have become a very popularstrategy to solve systems of polynomial equations. Many of the advances are due tothe correct understanding of the geometrical properties of an algebraic object, the so-called solution variety for polynomial system solving. I summarize here some of the mostrecent advances in the understanding of this object, focusing also on the certification andcomplexity of the numerical procedures involved in path following methods.

Series: Algebra Seminar

An ideal of a local polynomial ring can be described by calculating astandard basis with respect to a local monomial ordering. However if we areonly allowed approximate numerical computations, this process is notnumerically stable. On the other hand we can describe the ideal numericallyby finding the space of dual functionals that annihilate it. There areseveral known algorithms for finding the truncated dual up to any specifieddegree, which is useful for zero-dimensional ideals. I present a stoppingcriterion for positive-dimensional cases based on homogenization thatguarantees all generators of the initial monomial ideal are found. This hasapplications for calculating Hilbert functions.