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]

**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.