**Seminar
on Differential Equations and Nonlinear Analysis**

**Organizers**:
Alessandro Arsie (alessandro dot arsie at utoledo dot edu)

Chunhua
Shan (chunhua dot shan at utoledo dot edu)

**Time**: Tuesday, 4:00-5:00PM

**Location**: UH 4170

• All are welcome to attend.

• If you would like to present your work or an interesting paper you
have read, please contact me or Dr. Arsie.

• If you have any suggestions on this seminar, please also let us
know. Thanks.

**Fall Semester, 2017**

**Talk:**** ** **Lie symmetries of the canonical Lie group connection**

Speaker: Professor Gerard Thompson, The University of Toledo

Date: 4:00-5:00PM, Thursday, December 5, 2017

Abstract: It is well known that any Lie group carries a canonical symmetric although usually not metric linear connection - Cartan's so-called "0"-connection. In this talk we investigate Lie symmetries of the canonical connection. We shall focus particularly on the codimension one abelian nilradical case for which many symmetries and first integrals may be written down explicitly.

**Talk:**** ** **Hopf**** bifurcation of planar systems**

Speaker: Chanaka Kottegoda,
The University of Toledo

Date: 4:00-5:00PM, Thursday, November 28, 2017

Abstract: In this talk, we will review the Hopf bifurcations of planar systems. The definition of Hopf bifurcation and the Hopf
bifurcation Theorem will be introduced. As an application, periodic solutions
of Selkov’s model will be studied.

** **

**Talk:**** ** **Representation homology of spaces**

Speaker: Yuri Berest, Cornell University

Date: 4:00-5:00PM, Thursday, November 16, 2017

Abstract: Let $ G $ be an affine algebraic group defined
over a field $ k $. For any (discrete) group $ \pi $, the set of all
representations of $ \pi $ in $ G $ has a natural structure of an algebraic
variety (more precisely, affine k-scheme) called the representation variety $ Rep_G(\pi)
$. If $ X $ is a (based) topological space, the representation variety of its
fundamental group $ Rep_G[π_1(X)] $ is an
important geometric invariant of $X$ that plays a role in many areas of
mathematics. In this talk, I will present a natural homological extension of
this construction, called representation homology, that takes into account a
higher homotopy information on $ X $ and has good functorial properties. The representation homology turns
out to be computable (in terms of known invariants) in a number of interesting
cases (simply-connected spaces, Riemann surfaces, link complements, lens spaces, ...), some of which I will examine in detail. Time
permitting, I will also explain the relation of representation homology to other
homology theories associated with spaces, such as higher Hochschild
homology, $ S^1$-equivariant homology of free loop spaces and the (stable) homology of
automorphism groups of the free groups $ F_n $.

**Talk:**** ** **Lie Symmetries of Differential Equations (II)**

Speaker: Dr. Jeongoo Cheh, The University of
Toledo

Date: 4:00-5:00PM, Tuesday, November 7, 2017

Abstract: It is well known to most students that
differential equations are usually studied with tools provided by some kind of
analysis -- real analysis, complex analysis, functional analysis, harmonic
analysis, etc.. A very different approach is to treat
differential equations as submanifolds of jet bundles
and employ geometric tools to study their symmetries. In fact, it was this
geometric approach to differential equations that led historically to the
genesis of the vast central industry of Lie groups and Lie algebras. In this
introductory talk into the area, we will start by recalling a few necessary
basics on manifolds and group actions, proceed to define
Lie (point) symmetries of differential equations, construct symmetry algebras
and symmetry groups, and then conclude with specific examples including an
application to the Hopf-Cole transformation.

**Talk:**** ** **Lie Symmetries of Differential Equations (I)**

Speaker: Dr. Jeongoo Cheh, The University of
Toledo

Date: 4:00-5:00PM, Tuesday, October 31, 2017

Abstract: It is well known to most students that
differential equations are usually studied with tools provided by some kind of
analysis -- real analysis, complex analysis, functional analysis, harmonic
analysis, etc.. A very different approach is to treat
differential equations as submanifolds of jet bundles
and employ geometric tools to study their symmetries. In fact, it was this
geometric approach to differential equations that led historically to the
genesis of the vast central industry of Lie groups and Lie algebras. In this
introductory talk into the area, we will start by recalling a few necessary
basics on manifolds and group actions, proceed to define
Lie (point) symmetries of differential equations, construct symmetry algebras
and symmetry groups, and then conclude with specific examples including an
application to the Hopf-Cole transformation.

**Talk:**** ** **Coﬁnite**** graphs and their proﬁnite
completions**

Speaker: Dr. Amrita Acharyya, The
University of Toledo

Date: 4:00-5:00PM, Tuesday, October 24, 2017

Abstract: We generalize the idea of coﬁnite
groups due to B. Hartley. First we deﬁne coﬁnite spaces. Then, as a special situation,
we study coﬁnite graphs and their uniform
completions. The idea of constructing a coﬁnite
graph starts with deﬁning a uniform topological graph $\Gamma$, in an appropriate fashion. We endow abstract graphs
with uniformities corresponding to separating ﬁlter bases of equivalence
relations with ﬁnitely many equivalence classes over $\Gamma$. It is
established that for any coﬁnite graph there
exists a unique Profinite
completion.

**Talk:**** ** **Integrable**** structures of dispersionless systems
and differential geometry**

Speaker: Dr. Alexandre Odesski, Brock
University, Canada

Date: 4:00-5:00PM, Tuesday, October 12, 2017

Abstact: We develop the theory
of Whitham type hierarchies integrable
by hydrodynamic reductions as a theory of certain differential-geometric
objects. As an application we construct Gibbons-Tsarev
systems associated to moduli space of algebraic curves of arbitrary genus and
prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.

**Talk:**** ** **Recovery of initial conditions for some classes of PDEs using discrete
time samplings (II)**

Speaker: Dr. Alessandro
Arsie, The University of
Toledo

Date: 4:00-5:00PM, Tuesday, October 3, 2017

Abstract: I will present some results about using discrete time samplings to
recover in an optimal way and in suitable functional spaces the initial
conditions for some classes of linear evolutive PDEs,
using discrete time samplings at a fixed location. We will also provide some
insights about a question posed by DeVore (Texas A&M) and Zuazua (Basque
Foundation for Science) about the dependence of the optimal sampling strategy
on the details of the spectrum of a linear operator. It turns out that for the
class of PDEs we analyzed, the dependence of the optimal strategy on the
spectrum is really weak. If time allows, I will talk about some open problems
involving nonlinear PDEs (both in the integrable and
non-integrable cases) and linear non-autonomous
evolutionary PDEs. This is a joint paper with Roza Aceska (Ball State
University) and Ramesh Karki (Indiana University
East).

**Talk:**** ** **Recovery of initial conditions for some classes of PDEs
using discrete time samplings (I)**

Speaker: Dr. Alessandro Arsie,
The University of Toledo

Date: 4:00-5:00PM, Tuesday, September 26,
2017

Abstract: I will present some results about
using discrete time samplings to recover in an optimal way and in suitable
functional spaces the initial conditions for some classes of linear evolutive PDEs, using discrete time samplings at a fixed
location. We will also provide some insights about a question posed by DeVore (Texas A&M) and Zuazua (Basque Foundation for Science) about the dependence
of the optimal sampling strategy on the details of the spectrum of a linear
operator. It turns out that for the class of PDEs we analyzed, the dependence
of the optimal strategy on the spectrum is really weak. If time allows, I will
talk about some open problems involving nonlinear PDEs (both in the integrable and non-integrable
cases) and linear non-autonomous evolutionary PDEs. This is a joint paper with Roza Aceska (Ball State
University) and Ramesh Karki (Indiana University
East).

**Talk:**** ** **A Reducibility Theorem for
Smooth Quasi periodic Linear Systems**

Speaker: Paduma Eranga,
The University of Toledo

Date: 4:00-5:00PM, Tuesday, September 5, 2017

Abstract: In this talk, I'll explain an iterative procedure for finding a
change of variables to reduce a quasi-periodic linear system into an autonomous
system worked done by G.C. O'Brien. This process called the accelerated
convergence method. A quasi periodic linear system is a linear system of
ordinary differential equations

\begin{align*}
x' & = Ax + P(\varphi)x
\\ \varphi' & = \omega , \end{align*}

where $x \in \mathbb{R}^n, \varphi \in \mathbb{R}^m, \, A
$ is a constant $n\times n$ matrix, $\omega $ is a constant vector
in $\mathbb{R}^m. $ P(\varphi)$
is periodic in $\varphi_i$ with period $2\pi$
for $i =1, \dots, m$.

In this
discussion, we are going to obtain a quasi-periodic transformation which
transform above system into the system with constant coefficients.

------------------------------------------------------------------------------------------------------------------------

**Spring Semester, 2017**

**Talk:**** ** **Analysis of a Pseudo-Harmonic Tubular Bell**

Speaker: Dr. Douglas Oliver, The University of Toledo

Date: 4:00-5:00PM, Tuesday April 18, 2017

Abstract: Tubular bells,
or chimes are used for ambient sounds as well as serious music. Unlike most
wind or stringed instruments, a tubular bell does not have a harmonic set of
overtones. The lack of harmonious overtones
creates a problem with using tubular bells for serious music: there is not
unanimity regarding the pitch, or musical note associated with a particular
tubular bell.

The Euler-Bernoulli model for vibrating thin beams was used to
derive a mathematical model for vibrations of a tubular bell. Using this model,
an analysis of the natural frequencies of a modified tubular bell was
presented. One or more ends of the tubular bell were weighted with a mass. This
mass changes the boundary conditions, and hence the ratio of the natural
frequencies of the tubular bell.

Values for the ratio of the mass of weight(s) to the mass of the
tube were identified such that the ratio of the frequency of the first overtone
to the second overtone was 2. Under these conditions, the
these overtones are one octave apart. The frequency ratios predicted by
the model have been compared with experimental results of a frequency analysis
of the sound produced by two physical tubes. The experimental results were in
good agreement with the theoretical predictions.

**Talk:**** ** **The Lavrentiev Phenomenon**

Speaker: Dr. Dean A. Carlson, Mathematical Reviews, American
Mathematical Society, Ann Arbor, MI

Date: 4:00-5:00PM, Tuesday, April 4, 2017

Abstract: In 1926 M. Lavrentiev gave an example
of a free problem in the calculus of variations for which the infimum over the
class of functions in $W^{1,1}[t_1,t_2]$ satisfying prescribed end point
conditions was strictly less than the infimum over the dense subset $W^{1,\infty}[t_1,t_2]$ of admissible functions in
$W^{1,1}[t_1,t_2]$. This property is now referred to as the Lavrentiev
phenomenon. After Lavrentiev's discovery L.~Tonelli and B. Mania gave
sufficient conditions under which this phenomenon does not arise. After
these results, the study of the Lavrentiev phenomenon
lay dormant until the 1980s when a series of papers by Ball and Mizel and by Clarke and Vinter
gave a number of new examples for which the Lavrentiev
phenomenon occurred. Also in 1979, T. S. Angell showed that the Lavrentiev phenomenon did not occur if the integrands
satisfy a certain analytic property known as property (D). Moreover, he showed
that the conditions of Tonelli and Mania insured that the analytic property (D)
was satisfied. In this talk we will begin by presenting B.~Mania's elementary example to illustrate that the
phenomenon exists and discuss Angell's property (D) to give a general theorem
that avoids Lavrentiev's phenomenon and show briefly
that some more recent results can be viewed as corollaries to Angell's result
in that the conditions assumed imply property (D).

**Talk:**** ** **Some classes of nonlinear integral operators and existence results via Schauder's fixed point theorem**

Speaker: Dr. Alessandro Arsie,
The University of Toledo

Date: 4:00-5:00PM, Tuesday, March 28, 2017

Abstract: I will discuss three examples of
nonlinear integral operators that are completely continuous on some spaces of
continuous functions (they are Volterra integral
operators, Fredholm integral
operators and integral operators with delay). By means of Schauder's
fixed point theorem, I will discuss existence of continuous solutions for the
integral equations associated to these operators.

**Talk:**** ** **Mathematical Modelling for Parametric Resonance**

Speaker: Dr. Zhiwei Chen, The University of Toledo

Date: 4:00-5:00PM, Tuesday, March 21, 2017

Abstract: When a physical parameter in an
oscillatory system is modulated to vary in time, it may cause a dynamic
instability associated with the system. This phenomenon is referred to as parametric resonance. The mathematical models
amenable to such phenomena are differential equations with periodic
coefficients, specifically, the Mathieu’s equation. In this talk, I will
discuss some parametrically excited systems and their characteristics in
resonance. I will derive some simple schemes of electrical circuits into the
Mathieu’s equation and discuss the relevant analysis towards this
phenomena.

**Talk:**** ** **Circumference over diameter; the different universes of pi** (𝝅 Day Colloquium)

Speaker: Dr. Nate Iverson, The
University of Toledo

Date: 4:00-5:00PM, Tuesday, March 14, 2017

Abstract: Pi is the ratio of circumference to
diameter in a circle. We define a circle to be a set of points equidistant from
a common point. When the method of measuring distance is changed different
ratios are possible. This talk will discuss the ratio of circumference to
diameter in all p-norms including p=1, the taxicab norm, and p=∞,
infinity the supremum norm. Results dating to 1932 using the Minkowski functional norms will also be discussed along
with further generalizations.

**Talk:**** ** **Predator-prey models with Holling types of
functional responses (II)**

Speaker: Dr. Chunhua Shan, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 15, 2017

Abstract: Predator-prey system has been
extensively studied by biologists and mathematicians. In this talk I will
introduce the classical predator-prey models of Holling
types of functional responses. Dynamics of
predator-prey system with Holling type II functional
response will be reviewed by qualitative analysis and bifurcation theory.

**Talk:**** ** **Predator-prey models with Holling types of
functional responses (I)**

Speaker: Dr. Chunhua Shan, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 7, 2017

Abstract: Predator-prey system has been
extensively studied by biologists and mathematicians. In this talk I will
introduce the classical predator-prey models of Holling
types of functional responses. Dynamics of
predator-prey system with Holling type II functional
response will be reviewed by qualitative analysis and bifurcation theory.

**Talk:**** ** **Floquet**** Theory and periodic linear differential equations**

Speaker: Paduma Eranga,
The University of Toledo

Date: 4:00-5:00PM, Tuesday, January 31, 2017

Abstract: In this talk I'll discuss a main theorem
in Floquet Theory, which appear in the study of
periodic linear differential equations, of the form $x' = A(t)x , A(t+T)= A(t), T>0 $ where
$A(t)$ is a matrix of complex continuous functions. That main theorem; Floquet theorem due to Gaston Floquet(1883)
gives a representation of a fundamental matrix solution $\Phi(t)$, as the
product of periodic nonsingular matrix $P(t)$ with the same period $T$ and a
constant matrix $R$ such that $\Phi(t) = P(t)e^{tR}$.
As a result we can transform the periodic system into a usual linear system
with constant coefficients.

**Talk:**** ** **A proof of uniformly boundedness principle**

Speaker: Dr. Alessandro Arsie,
The University of Toledo

Date: 4:00-5:00PM, Tuesday, January 24, 2017

Abstract: In this talk I'll discuss a
main theorem in Floquet Theory, which appear in the
study of periodic linear differential equations, of the form $x' = A(t)x , A(t+T)= A(t), T>0 $ where $A(t)$ is a matrix of complex continuous functions. That main
theorem; Floquet theorem due to Gaston Floquet(1883)** **gives a representation of a fundamental matrix solution
$\Phi(t)$, as the product of periodic nonsingular matrix $P(t)$ with the same
period $T$ and a constant matrix $R$ such that $\Phi(t) = P(t)e^{tR}$. As a result we can transform the periodic system into
a usual linear system with constant coefficients.