NDSU Applied Mathematics Seminar - Spring 2010
Loyola University Chicago
Department of Mathematics and Statistics

ANALYSIS SEMINAR


Time: Thursdays, 4:00PM-4:50PM  
Location:
BVM Hall - Room 1102 (6349 N. Kenmore Avenue, Chicago, IL 60626) [map]


Organizers: Marian Bocea and Rafal Goebel

Fall 2015 Analysis Seminar
Spring 2015 Analysis Seminar
Fall 2014 Analysis Seminar
Fall 2013 Analysis Seminar
Spring 2013 Analysis Seminar
Fall 2012 Analysis Seminar
Spring 2012 Analysis Seminar
Fall 2011 Analysis Seminar

SPRING 2016 SCHEDULE


FEBRUARY 2016
Thursday, February 11 - RAFAL GOEBEL (Loyola University Chicago) -  A (mildly biased) survey of mathematical models of hybrid systems (Part II)
Abstract: Classical approach to dynamical systems categorizes each dynamical system as a continuous-time dynamical system or a discrete-time dynamical system. For example, systems of planets that evolve under Newton's laws or biological systems where populations of species grow or decay exponentially are continuous-time systems and have solutions parameterized by time. Your savings account balance that increases every month, recursively-defined sequences, binary values in an algorithm, etc. are examples of discrete-time systems and have solutions parameterized by natural numbers that count the number of transitions. Broadly understood, hybrid dynamical systems are dynamical systems that exhibit features characteristic of continuous-time dynamical systems and features characteristic of discrete-time systems. Modeling of cyber-physical systems that combine analog and digital components; of mechanical systems with impacts, where velocities may change instantaneously; of biological systems with impulses; etc. motivate the interest in hybrid systems in control theory and control engineering literature. There, one can find piecewise affine systems, complementarity systems, differential or hybrid automata, dynamical systems on time scales, hybrid inclusions, impulsive differential equations, measure-driven differential equations, switched systems, etc., all of which have hybrid features.
The talk will present several of these modeling frameworks, discuss some concepts of solutions to the models, and relate selected frameworks to hybrid inclusions. Part II of this talk is fairly independent of Part I, and little background beyond differential equations will be necessary.

Friday, February 26 - AARON YIP (Purdue University) - Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media
Abstract:
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.


MARCH 2016
Thursday, March 16 - BRIAN SEGUIN (Loyola University Chicago) -  The benefits of symmetry in the homogenization of linearly elastic materials
Abstract: I will present a result on the connection between material symmetry and the symmetries of the periodic microstructure of an elastic material using homogenization theory.  The talk will begin with a short description of how I came across this topic and why knowing the material symmetry group of a material with periodic microstructure is useful from a numerical point of view.  Then, after describing the basic ideas in homogenization theory and material symmetry, I will describe the main result.  Of crucial importance is the idea of a periodic elastic structure and the symmetries associated with this structure.  The main result will be illustrated though several examples, one of which shows that a material can be anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Finally, I’ll say a few words about materials with a locally-periodic microstructure and when they are materially uniform.

Thursday, March 31 - RAFAL GOEBEL (Loyola University Chicago) - Going beyond Zeno times in a hybrid dynamical system through a pointwise asymptotically stable set of equilibria
Abstract:
Even for simple differential equations, the limits of solutions as time goes to infinity - assuming they exist - may depend wildly on initial conditions. In hybrid dynamical systems, passing to a limit may be required before the usual time goes to infinity, due to an accumulation of discontinuous behaviors: jumps, switches, impacts, etc. This so-called Zeno phenomenon has puzzled both the ancient Greek philosophers and modern control engineering researchers. For the latter, one important question is how to extend a solution to a hybrid dynamical system past its Zeno time, i.e., the time at which the discontinuous behaviors accumulate. This talk will outline a scenario where certain stability properties of the set of equilibria of a hybrid dynamical system let one extend the solutions past their Zeno times in a way that preserves reasonable dependence of solutions on initial conditions. Sufficient conditions for these stability properties will be given and illustrated by examples.

APRIL 2016  
Thursday, April 14 - MARIAN BOCEA (Loyola University Chicago) - Wasserstein distance, metric derivatives, and the Benamou-Brenier theorem (part I)
Abstract:
In this series of talks we survey some results surrounding the Benamou-Brenier theorem from the theory of optimal transport. We begin by defining the p-Wasserstein distance on the space of probability measures with finite p-moments and then discuss the induced topology. Next, we define the notions of metric derivatives and constant speed geodesics on general metric spaces, and we show that the space of probability measures endowed with the p-Wasserstein distance is a geodesic space. Finally, we describe some connections of these ideas with a continuum mechanics formulation of the optimal mass transfer problem due to Benamou and Brenier.

Thursday, April 21 - ANDREI TARFULEA (University of Chicago) - Front propagation and symmetrization for the fractional Fisher-KPP equation
Abstract:
We prove strong gradient decay estimates for solutions to the multi-dimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front flattens and quantifiably circularizes, losing its initial structure. Joint work with Jean-Michel Roquejoffre.

Thursday, April 28 - MARIAN BOCEA (Loyola University Chicago) - Wasserstein distance, metric derivatives, and the Benamou-Brenier theorem (part II)
Abstract:
In this series of talks we survey some results surrounding the Benamou-Brenier theorem from the theory of optimal transport. We begin by defining the p-Wasserstein distance on the space of probability measures with finite p-moments and then discuss the induced topology. Next, we define the notions of metric derivatives and constant speed geodesics on general metric spaces, and we show that the space of probability measures endowed with the p-Wasserstein distance is a geodesic space. Finally, we describe some connections of these ideas with a continuum mechanics formulation of the optimal mass transfer problem due to Benamou and Brenier.