FEBRUARY 2012
Tuesday, Feb. 21 - ROBERT JENSEN (Loyola University Chicago) - Second Order Elliptic and Parabolic Partial Differential Equations: A Short Guided Journey
Abstract: Over several lectures I plan to give a brief introduction to partial differential equations, concluding with the theory of viscosity solutions for 2nd order elliptic and parabolic partial differential equations. I will begin with the definitions of elliptic and parabolic partial differential equations, some examples, classical solutions, and the questions of existence and uniqueness. I will briefly discuss modern techniques and issues of regularity. I will conclude with a short discussion of the theory of viscosity solutions and connections with stochastic processes.

Tuesday, Feb. 28  - ROBERT JENSEN (Loyola University Chicago) - Second Order Elliptic and Parabolic Partial Differential Equations: A Short Guided Journey (part 2)

MARCH 2012
Tuesday, March 6 - No Seminar (Spring Break)

Tuesday, March 13  -  ROBERT JENSEN (Loyola University Chicago) - Second Order Elliptic and Parabolic Partial Differential Equations: A Short Guided Journey (part 3)

Thursday, March 22 - KAZIMIERZ GOEBEL (Maria Curie - Sklodowska University, Lublin, Poland) - Exotic constructions in Banach spaces
Abstract: The fact that the balls in infinitely dimensional Banach spaces are not compact causes many classical theorems of analysis to fail in this setting. This is usually shown via constructions of various examples.The aim of the talk is to present some samples of such situations. The main direction will be connected with the "optimal retraction problem". The unit ball in infinitely dimensional space can be mapped onto its boundary so that all the points on the sphere remain fixed. It can be done by uniformly continuous mappings. How regular can they be? The talk is addressed to a "general audience", whatever it means.

Tuesday, March 27  -  ROBERT JENSEN (Loyola University Chicago) - Second Order Elliptic and Parabolic Partial Differential Equations: A Short Guided Journey (part 4)

APRIL 2012
Tuesday, April 3 - ROBERT JENSEN (Loyola University Chicago) - Second Order Elliptic and Parabolic Partial Differential Equations: A Short Guided Journey (part 5)

Tuesday, April 10 - VLAD VICOL (University of Chicago) - Shape dependent maximum principles and applications
Abstract: We present a non-linear lower bound for the fractional Laplacian, when evaluated at extrema of a function. Applications to the global well-posedness of active scalar equations arising in fluid dynamics are discussed. This is joint work with Peter Constantin.

Friday, April 13 - PABLO PEDREGAL (Universidad de Castilla-La Mancha, Spain) - What are Young measures, and what are they good for?
Abstract: Through a non-technical motivation, I will try to explain why Young measures may be helpful in Analysis when one tries to understand the effect of non-linear functionals with respect to oscillatory behavior. This issue, in turn, is important for problems in models of material behavior, and in optimal design in continuous media, when minimizing sequences develop finer and finer spatial oscillations. I will dwell on a more detailed description (though still non-technical) of a typical situation in optimal design, where homogenization phenomena are involved. Some numerical simulations will be shown.

Tuesday, April 17 - RAFAL GOEBEL (Loyola University Chicago) - Conjugacy of convex and saddle functions and its application to dissipativity properties of uncertain linear dynamical systems
Abstract: To each convex function f there corresponds a convex conjugate function f* and the convex conjugate of f* is, under mild assumption, the original f. This relationship is one of the foundations of the duality theory of convex optimization problems and lets one study the properties of f through different properties of f*. The conjugacy idea extends --- with some twists --- to saddle functions, which are convex in some variables and concave in the other variables. Properties of linear dynamical and control systems can be characterized through the use of quadratic Lyapunov and storage functions. For uncertain linear systems, which can be characterized by linear differential inclusions or, equivalently, by switching between linear differential equations, quadratic Lyapunov and storage functions are of limited utility. Instead, one can rely on convex functions. It then turns out that pairs of conjugate convex Lyapunov functions can be used to characterize asymptotic stability properties of a linear system and its dual system. Surprisingly, pairs of conjugate saddle functions feature in characterizations of dissipativity properties of pairs of dual uncertain linear control systems.
The talks will be a fairly self-contained introduction to the topics outlined above.

Tuesday, April 24 - RAFAL GOEBEL (Loyola University Chicago) - Conjugacy of convex and saddle functions and its application to dissipativity properties of uncertain linear dynamical systems (part 2)

Friday, April 27 - DANIEL ONOFREI (University of Houston) - Passive and active designs for EM and acoustic cloaking
Abstract: Invisibility and cloaking have always fascinated the human imagination. In recent years there has been important progress in the direction of understanding the feasibility of an invisibility cloak. The initial approach to this problem was based on the transformation optics technique, where, by using suitable change-of-variables mappings and the invariance principle associated to the Maxwell system one could obtain different materials with the same boundary measurement map. This result will imply various exotic material designs for invisibility. In the first part of the talk I will describe the main ideas behind the transformation optics technique, highlight some important recent results on this problem, and discuss the advantages and challenges of this approach. In the second part of the talk I will offer a brief overview of other cloaking strategies and focus on the active exterior cloaking technique, where one uses suitable external sources (antennas) to actively control the field. Possible algorithms for approximation of solutions and their stability will be discussed and numerically supported, and future challenges of this approach will be discussed.