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Mathematics & Statistics
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  Colloquium Archive

Colloquia — Summer 2019

Friday, May 10, 2019

Title
Speaker

Time
Place
Sponsor

Tutte's integer flow conjectures
Cun-Quan (CQ) Zhang
West Virginia University
3:00pm-4:00pm
CMC 130
Lesɫaw Skrzypek










Abstract

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
Speaker
Time
Place

Occupation Kernel Methods for System Identification and Motion Tomography
Joel Rosenfeld
2:00pm-3:00pm
CMC 130

Abstract

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.