Colloquia — Spring 2007
Friday, April 27, 2007
| Title |
Weighted approximation and interpolation on infinite intervals
(a survey) |
| Speaker |
J. Szabados
Renyi Mathematical Institute
HUNGARY |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Vilmos Totik |
Abstract
Weighted approximation and interpolation on infinite intervals means new challenges
compared to the unweighted case on a finite interval.
We give a survey of the problems encountered in this topic with emphasis on
Freud and generalized Laguerre weights. To establish Jackson type approximation
theorems in this setting and construct linear operators with good approximation
properties is difficult. We also list possible ways of defining suitable moduli
of continuity which can measure the rate of polynomial approximation.
Friday, April 27, 2007
| Title |
Szegő polynomials for the analytic weight: the Riemann-Hilbert
approach |
| Speaker |
Andrei Martinez-Finkelshtein
Universidad de Almeria
SPAIN |
| Time |
1:00-2:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Evguenii Rakhmanov |
Abstract
The steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert
characterization proposed by Fokas, Its and Kitaev, has proved to be a very
powerful technique for the study of the analytic properties of orthogonal polynomials.
We apply it to polynomials orthogonal with respect to a weight supported on
the unit circle in two cases.
First, we provide a complete asymptotic expansion
for the sequence of orthogonal polynomials when the weight is strictly positive
and analytic. These formulas are valid uniformly in the whole complex plane.
Second, we consider the situation when this weight is modified by some factors
containing zeros. As a consequence, in both cases we obtain results about the
distribution of zeros of the orthogonal polynomials, explaining and predicting
interesting behavior of these zeros.
This talk is based partially on a joint
work with K. T.-R. McLaughlin (U. Arizona, Tucson) and E. B. Saff (Vanderbilt
U.).
Wednesday, April 25, 2007
| Title |
Studying 3-Manifolds Using Knots |
| Speaker |
Michael McLendon
Washington College
Chestertown, Maryland |
| Time |
2:00-3:00 p.m. |
| Place |
LIF 272 |
| Sponsor |
Mohamed Elhamdadi |
Abstract
Given a knot in $\mathbb R^3$, one can compute a variety of polynomials using
the diagram of the knot. These polynomials are topological invariants because
equivalent knots are always associated with the same set of polynomials. Therefore,
if, say, the Jones polynomial of K1 is different from the Jones polynomial
of K2, then K1 and K2 must be different knots.
If we have a knot that is embedded into a 3-dimensional space other than $\mathbb
R^3$, we can still try to compute a knot polynomial. However, the process that
associated a single polynomial to a knot in $\mathbb R^3$ now gives us a (possibly
infinite) collection of polynomials. These polynomials give us information
not only about the knot that we started with, but also about the unusual 3-dimensional
space (i.e., the manifold) that we are working in.
The skein model of a 3-manifold is an algebraic object formed by the types
of knots and links that the manifold can contain. In the words of Józef
Przytycki, skein theory is “algebraic topology based on knots”. In
this talk, we will look at several 3-manifolds and study the structure of their
skein modules. In particular, we will look at the skein module of a 3-manifold
when the manifold is defined by gluing two solid 3-dimensional objects with
boundary together to form a 3-manifold without boundary (i.e., a Heegaard splitting).
Friday, April 20, 2007
| Title |
Local geometry of (quasi) Banach symetric function spaces |
| Speaker |
Anna Kaminska
University of Memphis |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Leslaw Skrzypek |
Abstract
We shall discuss the notions of type and cotype as well as order convexity
and concavity of Lorentz and Marcinkiewicz spaces. These spaces appear naturally
in the theory of interpolation of the linear operators and are solid subspaces
of the space of measurable functions. They are (quasi) Banach symmetric spaces
in the sense that the (quasi) norms of the element and its decreasing rearrangement
coincides.
Friday, April 20, 2007
| Title |
Lie bialgebras and the classical Yang-Baxter equation |
| Speaker |
Jörg Feldvoss
University of South Alabama |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Mohamed Elhamdadi |
Abstract
In this talk we will give an introduction to Lie bialgebras and the related
classical Yang-Baxter equation. The ultimate goal is to obtain a complete characterization
of those finite-dimensional Lie algebras which admit a non-trivial (quasi-)triangular
Lie bialgebra structure and some of its consequences. The proof uses several
important results from the structure and representation theory of Lie algebras
but also Poisson superbrackets, determinants, quadratic forms, and quaternion
algebras which all will be explained along the way. The only prerequisite needed
for this talk is some basic knowledge of linear algebra.
Friday, April 13, 2007
| Title |
Exit times for autoregressive processes |
| Speaker |
Göran Högnäs
Åbo Akademi University
Finland |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Arunava Mukherjea |
Abstract
Consider the autoregressive system X_{n+1}^\varepsilon = f(X_n^\varepsilon)
+ \varepsilon\xi_{n+1}, n = 0,1,2,… where f is a continuous contractive function
with fixed point at 0, the initial point X_0^\varepsilon = x_0 \in (-1,1) and
the \xi's form a sequence of i.i.d. standard normal random variables. The object
of study is the asymptotics of the exit time from the interval (-1,1), as \varepsilon
\to 0.
Define the exit time \tau^\varepsilon = \inf\lbrace n | X_n^\varepsilon \notin
(-1,1)\rbrace. We show, e.g., that in the case f(x) = ax,
|a|<1, \lim_{\varepsilon\to 0}\varepsilon^2\log
E\tau^\varepsilon = {{1-a^2}\over 2}.
This is joint work with doctoral student Brita Ruths.
Friday, April 6, 2007
| Title |
How cells make measurements |
| Speaker |
Jim Keener
University of Utah |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Yuncheng You |
Abstract
Bacteria need math, too. They need to count and make a variety of measurements
in order to survive in a constantly changing environment.
The purpose of this talk is to use mathematics (primarily ordinary differential
equations) to show how bacterial cells can extract quantitative information
from their environment. This will be illustrated with two specific examples:
Example 1: Bacterial populations of P. aeruginosa are known to make a decision
to secrete polymer gel on the basis of the size of the colony in which they
live. This process is called quorum sensing and only recently has the mechanism
for this been sorted out. It is now known that P. aeruginosa produces a chemical
whose rate of diffusion out of the cell provides quantitative information about
the size of the colony in which it exists, which when coupled with a positive
feedback biochemical network gives a hysteretic switch.
Example 2: Salmonella regrow their flagella if they are broken off, indicating
that the bacteria are able to measure the length of their flagella. They are
able to do this because of a biochemical network with negative feedback coupled
with a length dependent rate of efflux of a secreted molecule.
Friday, March 30, 2007
| Title |
Knotted Solitons in the Faddeev and Skyrme Models |
| Speaker |
Fang-Hua Lin
Courant Institute of the Mathematical Sciences
New York University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Nagle Lecture Series Committee |
Abstract
Here we give a brief survey on recent mathematical works concerning the Faddeev
and Skyrme models. One of the most fascinating phenomena described by these
models are the knotted topological soliton solutions which are fundamentally
different from many other well-known field theory models such as instantons
and monopoles in the Yang-Mills or the general gauge field theory, bubbles
in the nonlinear sigma models or ferromagnetisms or vortices in superconductors
and superfluids. In this lecture we shall illustrate some key features of these
models that lead to the existence of stable knotted solitons and to discuss
some possible implication in other problems.
Thursday, March 29, 2007
| Title |
Consistency and identifiability in stochastic regression
models |
| Speaker |
Christine Jacob
INRA
France |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 262 |
| Sponsor |
George Yanev |
Abstract
The abstract for this talk can be found here.
Wednesday, March 28, 2007
| Title |
Mixed Cauchy problems for analytic partial differential operators |
| Speaker |
Peter Ebenfelt
University of California, San Diego |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 267 |
| Sponsor |
Dmitry Khavinson |
Abstract
We consider analytic partial differential equations in real Euclidean space
such that the principal part of the operator is an iterated Laplacian. The
mixed Cauchy problem for the PDE consists of finding a solution that “interpolates” (with
the appropriate multiplicity) a data function on a given divisor P =
0. A particular example is the classical Cauchy problem in which the data is
posed along a non-singular hypersurface. The Cauchy-Kowalevsky theorem asserts
that the classical Cauchy problem always has a unique solution. In this talk,
we consider the possibility of posing data on more general, possibly singular
divisors P = 0.
Friday, March 23, 2007
| Title |
On the role of Rings and Modules in Algebraic Coding Theory |
| Speaker |
Sergio Lopez
Ohio University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Xiang-dong Hou |
Abstract
This talk is a rather idiosyncratic survey about the different ways in which
rings and modules appear in the study of Coding Theory. In particular, I mention
how they may appear both as ambients for certain families of codes (e.g., cyclic
codes are the ideals of F[x]/(xn-1)) or as alphabets for the codes themselves
(most famously, the Z/(4)-linear characterization of Kerdock and Preparata
Codes). I will mention the role of infinite rings (twisted polynomial rings)
in the study of convolutional codes and even the role of modules as alphabets
in some discussions of extensions of the MacWilliams Equivalence Theorems.
If time allows, we may talk about certain ring theoretic questions about finite
fields that have been brought to the forefront due to their relevance in Coding
Theory.
Wednesday, February 28, 2007
| Title |
Shift and Multiplication Operators on Holomorphic Spaces |
| Speaker |
Sherwin Kouchekian
University of South Alabama |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
Abstract
We will track back some classical results on the shift operator. Next we switch
to multiplication operators and show why their are interesting. Finally, we
move to the unbounded multiplication operators and state our obtained results. In
the last part of this talk, I will also briefly discuss the PDE type of problems
related to the potential distribution in scanning probe microscopy and the
obtained results there.
Monday, February 26, 2007
| Title |
Global well-posedness of a haptotaxis model with spatial
and age structure |
| Speaker |
Christoph Walker
Vanderbilt University |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
Abstract
A system of nonlinear partial differential equations modeling tumor invasion
into surrounding healthy tissue is analyzed. The model focuses on key components
involved in tumor cell migration and takes into account cell motility and haptotaxis,
that is, the directed migratory response of tumor cells to the extracellular
environment. Individual cell processes are modeled according to cell age. The
equation fo the tumor cell density thus incorporates second-order (parabolic)
terms representing diffusion and taxis as well as a first-order (hyperbolic)
part due to cell aging. Global existence and uniqueness of non-negative solutions
is shown.
Friday, February 23, 2007
| Title |
A Counterexample to the Bishop-Phelps Theorem in Complex
Spaces |
| Speaker |
Victor Lomonosov
Kent State University |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Boris Shekhtman |
Abstract
If S is a subset of a Banach space X, then a nonzero functional f is a support
functional for S and a point x in S is a support point of S if f attains maximum
of absolute value at the point x. We are going to present a construction of
a complex Banach space X with a closed bounded convex subset S such that the
set of the support points of S is empty.
Friday, February 23, 2007
| Title |
Cartan-type Estimates for Potentials With the Cauchy Kernel
and With Real Kernels |
| Speaker |
Vladimir Eiderman
University of Kentucky |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Dmitry Khavinson |
Abstract
Let ν be a (complex) Radon measure in $\mathbb{C}$ with compact support
and finite variation and let
$$
\mathcal{C}_\ast\nu(z)=\sup_{\varepsilon>0}\left|\int_{|\zeta-z|>\varepsilon}\frac{d\nu(\zeta)}{\zeta-z}\right|
$$
be the maximal Cauchy transform. We obtain sharp estimates of the Hausdorff
h-content of the set $\mathcal{Z}^\ast(\nu,P)=\{z\in\mathbb{C}:\
\mathcal{C}_\ast\nu(z)>P\}$,
where h is a measuring function and P > 0 is a given
number. In the case when $\nu$ consists of a finite number of unit changes
and h(t) = t this problem was posed by Macintyre
and Fuchs in 1940, and it was solved in 2005 by J. M. Anderson and the speaker
using a tool which appeared only in the last 10 years in connection with the
development of the theory of analytic capacity (Melnikov, Tolsa, Mattila, Nazarov,
Treil, Volberg and others).
We also consider the analogous problem for potentials with arbitrary real
non-increasing kernels and positive measures in $\mathbb{R}^m$, $m\ge1$. As
an application of the tool being used we obtain results on the connection between
the analytic capacity and Hausdorff measure (in particular, an analog of the
Frostman theorem on classical capacities).
Wednesday, February 21, 2007
| Title |
Application of Hirota's method and Pfaffians in soliton theory |
| Speaker |
Chunxia Li
Tsinghua University
Beijing, CHINA |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
Abstract
In 1971, Hirota developed an ingenious approach for obtaining multisoliton
solutions of the KdV equation, which has thereafter been used as a standard
method. The key point of Hirota's method is to transform nonlinear differential
equations into bilinear equations first and then solve the resulting bilinear
equations by the perturbation method. As by-products, bilinear Backlund transformations,
Lax pairs, infinitely many conservation laws and various special solutions
can be obtained.
Pfaffians are generated from determinants but have more varied properties
than determinants. Determinantal identities such as Pluecker relations and
Jacobi identities can be extended to and unified as Pfaffian identities. Many
interesting characteristics of Pfaffians were discovered through studies of
soliton equations.
Solutions to bilinear equations are normally presented by determinants and/or
Pfaffians. The KP hierarchy (the KdV is just a special reduction of the KP)
has determinant solutions; and the B-type KP hierarchy, the coupled KP hierarchy
and the D-type KP hierarchy have the Pfaffian representations of their solutions.
Besides an overview of Hirota's method and Pfaffians, I will discuss a (2+1)-dimensional
Lotka-Volterra equation by applying Hirota's basic idea, and analyze the integrability
of the Lotka-Volterra equation including the existence of Lax pair and the
construction of Pfaffian solutions, in particular, physically significant soliton
and dromion solutions.
Monday, February 19, 2007
| Title |
Pointwise Carleman estimates for Schrödinger equations
on Riemannian manifolds and control theoretic implications |
| Speaker |
Xiangjin Xu
University of Virginia |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
Abstract
In this talk, I'll present pointwise Carleman estimates without lower order
terms for general non-conservative Schrödinger equations defined on an n-dimensional
Riemannian manifold. As a consequence, we obtain the global uniqueness, continuous
observability and stabilization results for Schrödinger equations with
Dirichlet boundary condition or Neumann boundary condition. Results for the
Euler-Bernoulli equations with “hinged” boundary condition are
also discussed. Some future research on this ongoing program will be discussed.
The present work is part of the ongoing program with Professor Irena Lasiecka
and Professor Roberto Triggiani at the University of Virginia. The talk is
intended for a mathematically-literate audience.
Monday, February 12, 2007
| Title |
The Bellman function method in harmonic analysis: what it
is and what it does |
| Speaker |
Leonid Slavin
University of Connecticut |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
Abstract
The Bellman function method is a novel harmonic analysis technique with some
parallels to optimal stochastic control (hence the name). Originally used to
obtain explicit integral estimates (often sharp and/or dimensionless), it has
met with great success in the past decade and has now been employed to handle
a startlingly diverse array of questions.
Operator norms (e.g., Riesz and Beurling-Ahlfors transforms on various weighted
spaces), maximal functions, John-Nirenberg and reverse Holder inequalities,
duality, embedding theorems, exponential integrability, weak-form estimates...the
list goes on. Though all these applications have important unifying features,
defining the method in a teachable form has been a challenge. Recently, a new
version of the method has emerged. Using this approach, I will introduce the
technique from scratch, lay down the basics of pure-Bellman and Bellman-type
arguments, and show how some recent results fit into this framework.
No previous knowledge of the subject is assumed.
Friday, February 9, 2007
| Title |
The Morse-Thue Sequence and its Dynamical Friends |
| Speaker |
Ethan Coven
Wesleyan University |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Natasha Jonoska |
Abstract
The famous Morse-Thue sequence has the “no BBb” property: it contains
no block of the form
b1b2…bnb1b2…bnb1.
In the early 1900s, Axel Thue showed that the set of all doubly infinite sequences
having the no BBb property, called the Morse Minimal Set, is the closure of
the shift-orbit of the Morse-Thue sequence. So if you meet a sequence on the
street, you can tell whether or not it is a member of the Morse Minimal Set
by asking whether it has the no BBb property.
What if you meet a sequence on the street that is wearing a disguise, and
want to know whether or not it is a member of a minimal set that is topologically
conjugate to the Morse Minimal Set? (“Wearing a disguise” or “topologically conjugate” means that the names of the symbols have been changed in a perhaps very complicated,
but unknown, way.) I will discuss what questions to ask the sequence to find
out and also what questions to ask it if you want to know about membership
in the Toeplitz Minimal Set.
The Morse and Toeplitz Minimal Sets are the substitution minimal sets generated
by the two simplest constant length substitutions, 001, 110 (Morse) and 001,
100 (Toeplitz).
This is joint work with Mike Keane (Wesleyan) and Michelle LeMasurier (Hamilton
College).
Wednesday, February 7, 2007
| Title |
Analysis for Some PDEs From Fluid Dynamics and Related Areas |
| Speaker |
Tao Luo
Georgetown University |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
Abstract
Partial differential equations (PDEs) arising from fluid dynamics and related
areas are in general highly nonlinear. Solutions of these nonlinear equations
always exhibit very singular behavior. This makes mathematical analysis and
numerical computation of them challenging. In this talk, I will present some
results for those PDEs. The topics will include:
- Euler-Poisson equations of compressible fluids with self-gravitation.
- Shock waves of hyperbolic PDEs with stiff relaxation.
If time permits, I would also like to talk about
- Transport equations with non-smooth coefficients.
Friday, January 12, 2007
| Title |
Applications of Pfaffian Technique in Soliton Equations |
| Speaker |
Chunxia Li
Tsinghua University
CHINA |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Wen-Xiu Ma |
Abstract
Although the properties of determinants are well-known, most people know little
about pfaffians, which are more varied than those of a determinant. Determinantal
identities such as Plücker relations and Jacobi identities, are extended
and unified as pfaffian identities. There are many interesting features which
have been discovered (or rediscovered) through research into soliton equations.
In this lecture, we will talk about determinant solutions and pfaffian solutions
to the KP hierarchy, the coupled KP hierarchy, the B-type KP hierarchy, how
to derive coupled systems through pfaffianization, computativity of pfaffianization
and Bäcklund transformations, and applications of pfaffians in soliton
equations with sources, respectively.