This seminar meets Thursdays during QVNTS
off weeks.

Talks are held in Lafayette L307 from 10:00 AM-11:30 AM.

We meet will meet for lunch after the talk from 11:30 AM - 1:00 PM.

Talks are held in Lafayette L307 from 10:00 AM-11:30 AM.

We meet will meet for lunch after the talk from 11:30 AM - 1:00 PM.

The Spring 2023 unQVNTS schedule, speakers, titles and abstracts.

Jan 19: Ivan Perez Avellaneda, UVM, The Problem of Stationary Points of Chen-Fliess Series

Control systems that have state-space representation are ODEs driven by an input function that measure a certain aspect of the system. The set of outputs of a control system as a response to a set of inputs and initial conditions is called a reachable set. The computation of reachable sets of a control system has applications to stability and safety operations. The Chen-Fliess series formalism provides an input-output representation of a nonlinear affine control system. The smallest overestimating box containing a reachable set is obtained by optimizing the Chen-Fliess series. To perform the optimization, free monoids, a tool from formal language theory, is extended to differential monoids where a formal derivative acting on formal words is defined. Closed forms of the Gâteaux and Fréchet derivatives of a Chen-Fliess series are obtained in terms of the derivative of words. This provides homology-like sequences which could bring some light on the computation of stationary points of a Chen-Fliess series. On the other hand, for constant inputs and truncated Chen-Fliess series, the problem of finding the stationary points becomes the problem of solving polynomials in the field of the reals. In this talk, the problem of finding the stationary points of Chen-Fliess series and the mathematical questions it generates is presented.

Feb 2: Holly Paige Chaos, UVM, Torsion for CM Elliptic Curves Defined Over Number Fields of Degree 2p

Let E be an elliptic curve defined over a number field F. By the Mordell-Weil theorem we know that the points of E with coordinates in F can be given the structure of a finitely generated abelian group. We will focus on the subgroups of points with finite order. For a given prime p > 3 and an elliptic curve E defined over a number field of degree 2p, we would like to know exactly what torsion subgroups arise. Before discussing recent progress on this query, specifically in the case of elliptic curves with complex multiplication (CM), I will provide a brief overview on elliptic curves as well as outline some significant classical results.

Feb 23: Paige Helms, University of Washington, A Shared Extremal Property of Sphere Packing Lattices in R^2.

In this talk, we will construct the space of unimodular lattices and show that it naturally carries a Euclidean metric. Then, we demonstrate a shared extremal property of the Hexagonal lattice (Faulhuber, Steinerberger 2019) and the Square lattice Z^2 (H. 2022) with respect to this metric. Lastly, we will aim to discuss in any remaining time possible generalizations or interpretations of these results.

Mar 2: Daniel Vallieres, CSU Chico, Iwasawa theory and graph theory

Analogies between number theory and graph theory have been studied for quite some times now. During the past few years, it has been observed in particular that there is an analogy between classical Iwasawa theory and some phenomena in graph theory. In this talk, we will explain this analogy and present some of the results that have been obtained so far in this area.

Mar 30: Asimina Hamakiotes, UConn, Computing the proportion of sneaky primes for pairs of elliptic curves with and without CM

Apr 13: Mark Sing, Brown, A Dynamical Analogue of the Neron-Ogg-Shafarefich Criterion

Abstract 5

Apr 27: Everett Howe, Efficiently enumerating hyperelliptic curves over finite fields

There are many reasons why one might like to have a list of all of the hyperelliptic curves of a given genus over a given finite field --- perhaps to see whether there is any such curve with some interesting property, or to gather statistics to help guide research. There are about $2 q^{2g-1}$ hyperelliptic curves of genus $g$ over ${\mathbf F}_q$, and I will sketch an algorithm that, up to some logarithmic factors, takes time $q^{2g-1}$ to produce such a list. For hyperelliptic curves of genus $2$ there already exists such an algorithm, based on work of Mestre that produces a genus-$2$ curve from its Igusa invariants. The new algorithm that I will describe is hundreds of times faster than the existing one for reasonably sized $q$

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