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) Ekaterina
Shemyakova (ekaterina dot
shemyakova at utoledo dot
edu) |
|
Time: |
Tuesday,
4:00-5:00PM |
|
Location: |
UH
2210 |
Spring Semester, 2021
|
Talk: |
Relaxation
oscillations in predator-prey systems |
|
Speaker: |
Professor Shangbing Ai, University of
Alabama |
|
Date: |
4:00-5:00PM, Tuesday, April 27, 2021 |
|
Abstract: |
We study certain
classes of predator-prey systems with a small parameter. The systems are fast-slow
planar ODE systems about the populations of predators and preys. We are
interested in the limit cycles of these systems which exhibit relaxation
oscillations and approach singular closed orbit that consist of piecewise
smooth curves as the small parameter approaches zero. In this talk I will talk about some recent results on the
existence of single and multiple relaxation oscillations. The existence of
such a solution is obtained by finding a single
closed orbit and then constructing a positively or negatively invariant thin
annular domain near this single closed orbit and applying the Poincare-Bendixson theorem. |
Spring Semester, 2020
|
Talk: |
Generalized Givental's theory for flat F-manifolds |
|
Speaker: |
Professor Alessandro Arsie, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, March. 3, 2020 |
|
Abstract: |
I will present some
of the results contained in the first part of a joint work with Sasha Buryak, Paolo Lorenzoni and
Paolo Rossi (``Semisimple flat F-manifolds in
higher genus" ). In particular, after having
recalled few points about the geometry of flat F-manifolds around a semisimple point, I will start sketching a generalized Givental's theory for them. This controls essentially the
genus zero part of any F-Cohomological Field Theory
(but this will be addressed in subsequent talks). |
|
Talk: |
The boundary strata
of the moduli spaces of pointed stable curves |
|
Speaker: |
Professor Alessandro Arsie, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Feb. 18, 2020 |
|
Abstract: |
I will present an
elementary description of the combinatorial and inductive structures of the
boundary strata of the Deligne-Mumford moduli
spaces of pointed stable curves. This description is based on graphs with
"legs" and it is essential in constructing a full-fledged F-Cohomological Field Theory in all genera, starting from
topological data in genus zero and some additional data from the compactified moduli space of elliptic curves as we have
done in https://arxiv.org/abs/2001.05599 |
Fall Semester, 2019
|
Talk: |
Dirichlet Problem for Some Singular Perturbed Elliptic
Equations |
|
Speaker: |
Dr. Zhiwei Chen,
The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Nov. 19, 2019 |
|
Abstract: |
It is well known that there is a close
relation between the theory of second-order differential equations and Markov
processes with continuous trajectories. In this talk, I will use the
second-order elliptic equation that is singular perturbed as an example, to
show how it can be associated to asymptotic problems arising in the random
perturbation of dynamical systems. Particular attention will be given to
those cases that involve study of the effect of random perturbations on large
time intervals, where the small perturbations essentially influence the behavior
of the dynamical system. |
|
Talk: |
Endpoint
Strichartz estimate for the kinetic transport
equation in one dimension |
|
Speaker: |
Dr. Jingchun Chen,
Visiting Assistant Professor, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Oct. 29, 2019 |
|
Abstract: |
We will discuss the problems of endpoint Strichartz estimates for the kinetic equation in one
dimension. We mainly talk about that the endpoint Strichartz
estimate does not hold for the kinetic transport equation in the mixed norm
Lebesgue spaces by an explicit counterexample and duality argument. We also
mention some open problem about the Strichartz
estimates. |
|
Talk: |
Bifurcation
analysis of a predator-prey system (II) |
|
Speaker: |
Chanaka Kottegoda,
, graduate student, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Oct. 22, 2019 |
|
Abstract: |
In this talk I will analyze the predator-prey
system with Holling type IV functional response and
Allee effect on the prey population. I will show
the existence of three limit cycles by analyzing the Hopf
bifurcation of codimension 3. Also, I will prove
that the predator-prey system shows Bogdanov-Takens
bifurcation of codimension 3, which gives some rich
dynamics for the system. This is joint work with Alessandro Arsie and Chunhua Shan. |
|
Talk: |
Bifurcation
analysis of a predator-prey system (I) |
|
Speaker: |
Chanaka Kottegoda,
, graduate student, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Oct. 1, 2019 |
|
Abstract: |
In this talk I will analyze the predator-prey
system with Holling type IV functional response and
Allee effect on the prey population. I will show
the existence of three limit cycles by analyzing the Hopf
bifurcation of codimension 3. Also, I will prove
that the predator-prey system shows Bogdanov-Takens
bifurcation of codimension 3, which gives some rich
dynamics for the system. This is joint work with Alessandro Arsie and Chunhua Shan. |
|
Talk: |
Tight Contact
Structures on Some Small Seifert Fibered Spaces (II) |
|
Speaker: |
Kursat Yilmaz, graduate student, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Sept. 24, 2019 |
|
Abstract: |
I will continue with
surgery description of small Seifert fibered spaces. If time permits, I would
like to constract the tight contact structures on
one of them, which will indeed give a lower bound for the number of tight
contact structures. Finally, by using convex surface theory, I will show the
upper bound is exactly the same. |
|
Talk: |
Tight Contact
Structures on Some Small Seifert Fibered Spaces (I) |
|
Speaker: |
Kursat Yilmaz, graduate student, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Sept. 10, 2019 |
|
Abstract: |
Small Seifert
fibered space is a Seifert fibered space with three exceptional fibers. There
is an invariant of Seifert fibered spaces which is called Euler number (e_0).
In this talk, the classification of tight contact structures on some small
Seifert fibered 3-manifolds will be given. The classifications are based on
understanding the interaction between different techniques and theories known
as Dehn surgery, contact surgery, bypass technique,
and convex surface theory. The complete classification of the tight contact
structures on small Seifert fibered spaces having e_0 is less than or equal
to -3, and greater than or equal to 1 is given by Wu. The case where e_0 is equal
to 0 is also completed by Ghiggini, Lisca and Stipsicz. The case
when e_0 is -1 or -2 is still open. I will give some partial results when e_0
is equal to -1 by using the work of Mark and Tosun. |
|
Talk: |
Symplectic Topology |
|
Speaker: |
Professor Alessandro Arsie, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Sept. 3, 2019 |
|
Abstract: |
The origins of symplectic topology lie in classical dynamics, and the
search for periodic orbits of Hamiltonian systems. It is now understood to
arise naturally in algebraic geometry, in low-dimensional topology, in
representation theory and in string theory. Following seminal ideas of Gromov and Floer from the
1980s, several of the most powerful tools in symplectic
topology revolve around invariants counting pseudoholomorphic
curves. An important theme in recent years has been that holomorphic curve
invariants are not independent, but are bound together and governed by very
rich algebraic structures, making connections to integrable
systems, and to the theory of A-infinity and L-infinity algebras. |
Spring Semester, 2019
|
Talk: |
Dynamics
of the $\lambda \tan z^2 $ family (II) |
|
Speaker: |
Dr. Santanu Nandi,
Visiting Assistant Professor, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, April 16, 2019 |
|
Abstract: |
In my second talk,
I'll discuss some topological properties of the dynamical plane ($z$-plane)
and a combinatorial structure of the parameter plane (\lambda-plane) of $
\lambda \tan z^2.$ In the dynamical plane, I'll prove that there is no
Herman ring and the Julia set is a Cantor set for the map when the
parameter is in the central capture component. Julia set is
connected for the maps when the parameters are in other hyperbolic
components. In the parameter plane, I'll show that the capture components are
simply connected and there are always four hyperbolic shell components
attached to a virtual center. The capture components and the periodic shell
components of the period greater than one are
bounded. |
|
Talk: |
Dynamics
of the $\lambda \tan z^2 $ family (I) |
|
Speaker: |
Dr. Santanu Nandi,
Visiting Assistant Professor, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, April 9, 2019 |
|
Abstract: |
In my first talk, I'll give an
introduction to complex dynamics. I'll give a short description of the
dynamics of the quadratic polynomials in the complex plane and Mandelbrot
set. My work is a motivation to study the dynamics of the family of meromorphic maps, in particular, the dynamics of the
$\lambda \tan z $ family where $ \lambda $ is a parameter in the complex
plane. I'll review some important results to describe the combinatorial
structure of both the dynamical plane($z$-plane) and the parameter plane(z-plane)
of this family. |
|
Talk: |
A
generalized Floquet theory result for analytic
linear delay differential equations |
|
Speaker: |
Paduma Eranga, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, April 2, 2019 |
|
Abstract: |
In
the classical Floquet Theory, there exists a smooth
change of variables that reduces a given linear periodic system to a linear
system with constant coefficients. In this talk, I discuss a generalized Floquet theory result for analytic linear delay differential
equations. I provide a simpler proof of the asymptotic reducibility that
bypasses the method of accelerated convergence and it is instead based on
more transparent KAM-type estimates. |
|
Talk: |
Using
symmetries to solve differential equations |
|
Speaker: |
Professor Ryad Ghanam, Virginia Commonwealth University
at Qatar |
|
Date: |
4:00-5:00PM, Tuesday, March 12, 2019 |
|
Abstract: |
In this talk we will show how Lie symmetries
play a major role in solving differential equations. We will talk about the
origin of symmetries, how to calculate them and use them to solve
differential equations. |
|
Talk: |
Ideal
Fluid Flow Past A Body In Space |
|
Speaker: |
Xueliang Lu, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Feb. 26, 2019 |
|
Abstract: |
This talk is about the problem of an irrotational and incompressible flow around a body in
space. The basic existence is proved by formulating the problem into a
variational problem. It is also shown that the
solution is unique, and that the maximum speed is attained on the
body's boundary. |
|
Talk: |
Topological
degree theory and boundary value problems (I) |
|
Speaker: |
Meetra Nouri, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Feb.12, 2019 |
|
Abstract: |
I will review Rafael Ortega's lectures on
topological degree theory and boundary value problems. I will discuss few
elementary examples and then I will talk about topologica
degree in R and R^2, in order to generalize Bolzano's theorem to higher
dimensions (Poincare'-Miranda theorem) and to pave the way for dealing with
less elementary examples. |
|
Talk: |
Bogdanov-Takens bifurcation of codimension
2 |
|
Speaker: |
Chanaka Kottegoda,
The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Feb.5, 2019 |
|
Abstract: |
In this talk I will introduce the concepts of
structural stability and bifurcation of a vector field. Especially I will
talk about Bogdanov-Takens bifurcation of
co-dimension 2, all possible topological structures we can see from this
bifurcation and the proof of it. Finally I will talk about the prey-predator
model and the existence of Bogdanov-Takens
bifurcation in this model. |
Fall Semester, 2018
|
Talk: |
Turkowski's 1990 paper on
6-dimensional Lie algebras and beyond |
|
Speaker: |
Professor Gerard Thompson, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Nov. 27, 2018 |
|
Abstract: |
I will review Turkowski's 1990 paper on 6-dimensional solvable Lie
algebras, which leaves quite a few details. I will provide a few structure
theorems and discuss the prospects for extending the results to 7 dimensions. |
|
Talk: |
Sheaves
and Algebraic Geometry |
|
Speaker: |
Professor Alessandro Arsie, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Nov. 20, 2018 |
|
Abstract: |
I will start introducing sheaves (and
possibly their cohomology) and some basic ideas in
algebraic geometry (using tools from complex analysis rather than commutative
algebra) using my graduate school notes. |
|
Talk: |
Connection
from mediating maps of pullbacks |
|
Speaker: |
Professor Geoffrey Martin, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Nov. 13, 2018 |
|
Abstract: |
This will be a somewhat elementary talk on an
approach to connection theory I have found useful. The approach makes
constructions more functorial. I give some simple
applications and if I will I can show why the second structure equation is
rightly call the second structure equation. |
|
Talk: |
Regularized
Solutions for Backward Heat Equation with Time-Dependent Veritable
Coefficient |
|
Speaker: |
Dr. Sujeewa Hapuarachchi, Visiting Assistant Professor, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, Nov. 6, 2018 |
|
Abstract: |
Backward heat equation with time-dependent
variable coefficient is severely ill‐posed in the sense of Hadamard,
so we need regularization. We consider Backward heat equation with
time-dependent variable coefficient, and by small perturbing, we obtain an
approximation problem. We show this approximation problem is well‐posed with small parameter. Also, we show
this approximation system converges to the original problem when parameter
goes to zero. Here, we use modified‐quasi boundary value method to regularize
this problem. |
|
Talk: |
Classification and the dynamics of surface mapping classes on Teichmuller space |
|
Speaker: |
Professor Funda Gultepe, The University
of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, October 30, 2018 |
|
Abstract: |
In this introductory
talk, I will discuss the Nielsen-Thurston classificiation
of surface mapping classes in analogy with classification of isometries of
the upper half plane. If time permits, I will also talk about their dynamical
behavior on the Teichmuller space and give sketch
of proof of the classification theorem. This will be a self-contained
talk, previous knowledge is helpful but not required to follow. |
|
Talk: |
Introduction to Mapping class group of a surface |
|
Speaker: |
Professor Funda Gultepe, The University
of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, October 23, 2018 |
|
Abstract: |
Study on the mapping
class group goes back to works of Dehn and Nielsen
in 1920's. During 1970's, Thurston gave this study a geometric flavor
and used it to understand hyperbolic 3- manifolds. Today mapping
class group is used in many areas to understand topological and geometric
properties of 3 manifolds as well as to solve moduli problems related to
curves. As a consequence, it is one of the main topics of the geometric group
theory. I will give a basic introduction to this group which will include
first examples. The talk will be expository and will not
assume background in the area. |
|
Talk: |
New
approaches to integrable hierarchies of topological
type (II) |
|
Speaker: |
Professor Alessandro Arsie,
The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, October 9, 2018 |
|
Abstract: |
I will survey a large class of systems of
partial differential equations which on one hand appear in classical problems
in mathematical physics and on the other hand provide an efficient tool for
description of enumerative invariants in algebraic geometry. Particular
attention will be paid to new approaches to these systems. |
|
Talk: |
New
approaches to integrable hierarchies of topological
type (I) |
|
Speaker: |
Professor Alessandro Arsie,
The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, September 25, 2018 |
|
Abstract: |
I will survey a large class of systems of
partial differential equations which on one hand appear in classical problems
in mathematical physics and on the other hand provide an efficient tool for
description of enumerative invariants in algebraic geometry. Particular
attention will be paid to new approaches to these systems. |
|
Talk: |
Stable
curves, Moduli Spaces and Cohomological
Field Theories |
|
Speaker: |
Professor Alessandro Arsie,
The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, September 11, 2018 |
|
Abstract: |
I will review the notion of stable curves and
their moduli spaces. I will also introduce axiomatically Cohomological Field Theories following Manin and Kontsevich and I will
provide some examples. |
Spring Semester, 2018
|
Talk: |
Relativistic
Treatment of Confined Hydrogen Atoms via Numerical Approximations |
|
Speaker: |
Jacob Noon, The
University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, April 10, 2018 |
|
Abstract: |
The study of particles and atoms confined to spherically
symmetric regions have been used to illustrate the differences between classical
and quantum systems since Erwin Schrodinger famous equation was published in
1926. Paul Dirac later added his own equation, the Dirac Equation (1928),
which integrated the quantum principles that had been developing in the
decades prior to Schrodinger equation with the principles of Einstein Special
Theory of Relativity. The literature on Hydrogen atoms confined to spherically
symmetric regions (using the Schrodinger equation) is abundant, with
consensus on the results. To the best of our knowledge there exists no
relativistic treatment to this problem (i.e., using the Dirac equation). Some
reasons for this are the complexity of the Dirac equation, its solutions, and
problems that arise when trying to satisfy the boundary conditions. |
|
Talk: |
A revisit to the
Jordan canonical form for a complex square matrix |
|
Speaker: |
Professor Biao Ou, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, March 13, 2018 |
|
Abstract: |
I will first look at the canonical form of two by two and three
by three matrices via a rank one matrix. Next, I look at the process in which
we see the Jordan canonical form by first applying a much easier Schur's theorem and then considering a system of linear
differential equations. I will also apply the Jordan canonical form to a
sequence of numbers satisfying a linear iterative equation. |
|
Talk: |
Factorization of
differential operators on the algebra of densities on the line (II) |
|
Speaker: |
Professor Ekaterina Shemyakova, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, February 20, 2018 |
|
Abstract: |
I shall show that for differential
operators on the algebra of densities on the line (unlike the familiar
setting) there are obstructions for factorization. I shall analyze
these obstructions. In particular, for the "generalized Sturm-Liouville" operators acting on the algebra of
densities on the line, I shall show our criterion of factorizabily
in terms of solution of the classical Sturm-Liouville
equation. I shall also show the possibility of an incomplete
factorization. |
|
Talk: |
Factorization of differential
operators on the algebra of densities on the line (I) |
|
Speaker: |
Professor Ekaterina Shemyakova, The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, February 13, 2018 |
|
Abstract: |
I shall speak about my new work where I explore factorization
problem for differential operators on the algebra of densities. This work is
a starting point for one of the directions of my research program, namely
study of differential operators acting on geometric quantities and their Darboux transformations in classical and super setting. |
|
Talk: |
Flat $F$-manifolds, Miura
invariants and integrable systems of conservation
laws |
|
Speaker: |
Professor Alessandro Arsie,
The University of Toledo |
|
Date: |
4:00-5:00PM, Tuesday, February 6, 2018 |
|
Abstract: |
In this talk, I will present the extension to the case of
systems of integrable conservation laws of some of
the results proved for scalar equations in Arsie,
Moro, Lorenzoni (Integrable
viscous conservation laws, Nonlinearity 2015) and in Arsie,
Moro, Lorenzoni (On Integrable
Conservation Laws, Proceedings of the Royal Society A, 2014). |
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, Tuesday, 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, Tuesday, 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
model will be studied. |
|
Talk: |
Representation
homology of spaces |
|
Speaker: |
Professor 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: |
Professor Alexandre
Odesski, Brock University, Canada |
|
Date: |
4:00-5:00PM, Tuesday, October 12, 2017 |
|
Abstract: |
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: |
Professor 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: |
Professor 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, 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: |
Professor 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 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 equation and discuss
the relevant analysis towards this phenomenon. |
|
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: |
𝝅 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: |
Professor Chunhua Shan, The University of
Toledo |
|
Date: |
4:00-5:00PM, Tuesday,
February 14, 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: |
Professor 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: |
Professor 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. |