Colloquia — Summer 2019

Wednesday, June 12, 2019

Title: Two applications of the theory of finite fields
Speaker: Giacomo Micheli
Time: 2:00pm‐3:00pm
Place: CMC 130


In the first application I describe a new method to produce pseudorandom number generators using the algebraic theory of projective automorphisms (this is a joint work with F. Amadio Guidi and S. Lindqvist). Our construction beats asymptotically the classical inversive congruential generator in terms of computational cost, for the same discrepancy bounds.

In the second application, I give a brief introduction to powerline communication and permutation codes, and explain how to use classical coding theory to obtain new improved lower bounds for the size of an \((n,d)\) permutation code (this is a joint work with A. Neri).

Friday, May 10, 2019

Title: Tutte's integer flow conjectures
Speaker: Cun-Quan (CQ) Zhang, West Virginia University
Time: 3:00pm‐4:00pm
Place: CMC 130

Sponsor: Lesɫaw Skrzypek


Let \(G\) be a graph with an orientation \(D\). A mapping \(f:E(G)\to\{\pm1,\pm2,\dotsc,\pm(k-1)\}\) is called a nowhere-zero \(k\)-flow if, for every vertex \(v\in V(G)\), \[ \sum_{e\in E^+(v)}f(e)=\sum_{e\in E^-(v)}f(e). \]

The integer flow problem is a dual of the vertex coloring problem: it is pointed out by Tutte that a planar graph \(G\) admits a nowhere-zero \(k\)-flow if and only if \(G\) is \(k\)-face-colorable. Tutte proposed several important conjectures about integer flows, such as, 3-flow, 4-flow and 5-flow conjectures. Those conjectures, though there are some breakthrough in last 40 years, remain widely open. This talk will introduce not only some history but also some basics of Tutte's flow theory.

Wednesday, May 8, 2019

Title: Occupation Kernel Methods for System Identification and Motion Tomography
Speaker: Joel Rosenfeld
Time: 2:00pm‐3:00pm
Place: CMC 130


In this talk I will discuss two approximation problems that appear in dynamical systems theory. First, we will examine the gray box system identification problem, where the goal is to obtain a collection of parameters for a parameterization of a dynamical system using observed trajectories from the system. The second problem concerns flow field estimation using trajectories obtained from very simple “gliders.” Using the difference between the endpoints of the anticipated trajectory and the actual trajectory, the flow field estimation problem can be treated with the tools of motion tomography.

Each of these examples utilize occupation kernels in different ways. The system identification uses occupation kernels indirectly to obtain constraints on the parameters, whereas the motion tomography problem uses occupation kernels directly as basis functions for the estimation of the flow field. Because of the intimate connection between occupation kernels and the problems discussed, we will also spend some time talking about reproducing kernel Hilbert spaces and the estimation of the occupation kernels themselves.