Department of Mathematics and Statistics

ANALYSIS SEMINAR

Location:

Organizers: Marian Bocea and Rafal
Goebel

Spring 2013 Analysis Seminar

Fall 2012 Analysis Seminar

Spring 2012 Analysis Seminar

Fall 2011 Analysis Seminar

Spring 2013 Analysis Seminar

Fall 2012 Analysis Seminar

Spring 2012 Analysis Seminar

Fall 2011 Analysis Seminar

Tuesday, September 17 -

Abstract: Asymptotic stability of an equilibrium of a dynamical system is an important property in the analysis and prediction of long-term behavior of the system. Biologists or economists may wish to know that the dynamical system they are analyzing, be it competing species or cooperating economies, behave in a stable and predictable way in the future. Engineers frequently fact the task of designing algorithms that create asymptotically stable equilibria in the system they need to control. A common tool for verifying asymptotic stability is a Lyapunov function. Roughly, it is a function that measures how far the state of the dynamical system is from an equilibrium and that is decreasing as the time goes by. This concept was pioneered by a Russian mathematician A. Lyapunov at the end of the 19th century, for the case of linear differential equations. The theory of Lyapunov functions has seen many developments since then, to nonlinear and multi-valued differential equations and far more general frameworks as well. My first talk will be a brief introduction to Lyapunov function theory, somewhat biased towards control engineering problems and set-valued analysis. I plan to keep things simple, so basic knowledge of differential equations and some linear algebra should be sufficient background. My second talk will present a twist on the Lyapunov function concept. It will be related to problems of consensus, where several agents need to arrive at a common position, and on the mathematical end, it will involve a ``decreasing'' set-valued mapping. A theme that will appear in both talks is how robustness of asymptotic stability to perturbations, for example to measurement error in a control situation, is related to regularity, for example continuity or differentiability, of a Lyapunov function.

Tuesday, September 24 -

Tuesday, October 1 -

Tuesday, October 8 -

Tuesday, October 15 -

Tuesday, October 22 -

Tuesday, October 29 - MARIAN BOCEA (Loyola University Chicago) -

Abstract: The original mass transport problem, formulated in 1781 by Gaspard Monge, asks to find the optimal volume-preserving map (allocation plan) between two given sets of equal volume, where optimality is measured against a given cost function. This turns out to be the prototype for a large class of questions arising in differential geometry, infinite-dimensional programming, and mathematical economics. In this series of lectures I will discuss approaches to solving Monge's problem, with an emphasis on the case of uniformly convex cost densities. Starting from Kantorovich's ideas based on relaxation and duality we will prove existence of optimal allocation plans, and we will discuss results dealing with their actual construction. The latter leads to some interesting connections with the study of existence and regularity of weak solutions (in the sense of Alexandrov) for the Monge-Ampere equation.

Tuesday, November 5 - MARIAN BOCEA (Loyola University Chicago) -

Tuesday, November 12 - MARIAN BOCEA (Loyola University Chicago) -

Tuesday, November 19 - ROBERT JENSEN (Loyola University Chicago) -

Abstract: In this talk I will discuss a couple of general techniques for probing properties of solutions of nonlinear elliptic PDEs. The main technique I'll discuss could be described as being analogous to the construction of useful test functions for calculus of variation problems or divergence structure elliptic PDEs.