Colloquia -- Spring 2007

Friday, April 27, 2007

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

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

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

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

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

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

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

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

Abstract

The abstract for this talk can be found here.

Wednesday, March 28, 2007

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

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

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

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

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

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

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

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

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

Abstract

The famous Morse-Thue sequence has the “no BBb” property: it contains no block of the form

b₁b₂…b_nb₁b₂…b_nb₁.

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 Toeplizt Minimal Sets are the substitution minimal sets generated by the two simplest constant length substitutions, 0→01, 1→10 (Morse) and 0→01, 1→00 (Toeplitz).

This is joint work with Mike Keane (Wesleyan) and Michelle LeMasurier (Hamilton College).

Wednesday, February 7, 2007

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:

1. Euler-Poisson equations of compressible fluids with self-gravitation.
2. Shock waves of hyperbolic PDEs with stiff relaxation.

If time permits, I would also like to talk about

3. Transport equations with non-smooth coefficients.

Friday, January 12, 2007

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.

Please direct questions to mthmaster@nosferatu.cas.usf.edu.
Last updated: 25-Apr-2007.
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