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Mathematics & Statistics

# Colloquia — Spring 2008

## Friday, April 18, 2008

Title
Speaker

Time
Place

Fixed Points, Semigroups and Rigidity of Holomorphic Mappings
David Soikhet
The Galilee Research Center for Applied Mathematics
ORT College Braude
and
The Technion-Israel Institute of Technology
3:00pm-4:00pm
PHY 130
Dmitry Khavinson

Abstract

There is a long history associated with the problems of iterating nonexpansive and holomorphic mappings and finding their fixed points, with the modern results of K. Goebel, W. A. Kirk, T. Kuczumow, S. Reich, W. Rudin and J. P. Vigué being among the most important.

Historically, complex dynamics and geometrical function theory have been intensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathematicians with many applications to nonlinear ananlysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dynamical system: $$\frac{dx}{dt}+f(x)=0,$$ where $$x$$ is a variable describing the state of the system under study, and $$f$$ is a vector-function of $$x$$. The study of such systems when $$f$$ is a monotone or an accretive (generally nonlinear) operator on the underlying space has rec3ently been the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems.

In this talk we give a brief description of the classical statements which combine the celebrated Julia Theorem of 1920, Carathéodory's contribution in 1929 and Wolff's boundary version of the Schwarz Lemma of 1926 with their modern interpretations for discrete and continuous semigroups of hyperbolically nonexpansive mappings in Hilbert spaces. We also present flow-invariance conditions for holomorphic and hyperbolically monotone mappings.

Finally, we study the asymptotic behavior of one-parameter continuous semigroups (flows) of holomorphic mappings. We present angular characteristics of the flows trajectories at the Denjoy-Wolff points, as well as at their regular repelling points (whenever they exist). This enables us by using linearization models in the spirit of functional Schroeder's and Abel's equations and eigenvalue problems for composition operators to establish new rigidity properties of holomorphic generators which cover the famous Burns-Krantz Theorem.

## Friday, April 11, 2008

Title
Speaker

Time
Place

Boundaries of analytic varieties, and the projective hull
John Wermer
Brown University
Providence, RI
3:00pm-4:00pm
PHY 130
Dmitry Khavinson

Abstract

Let $$\gamma$$ be a closed real curve in the complex manifold $$X$$. When does there exist a Riemann surface in $$X$$ having $$\gamma$$ as its boundary? When $$X=C^n$$, the answer involves the polynomial hull of $$\gamma$$. When $$X$$ is complex projective space, this does not work. A recent treatment of that problem, by Harvey and Lawson, is based on a generalized notion of a “hull” which they call the “projective hull”. We shall talk about questions concerning this.

## Friday, April 4, 2008

Title
Speaker

Time
Place

Dickson Polynomials Over Finite Fields: Two Different Perspectives
Gary Mullen
Penn State University
3:00pm-4:00pm
PHY 130
Xiang-dong Hou

Abstract

If $$a$$ is in $$F_q$$, the finite field of order $$q$$, the Dickson polynomial of degree $$n$$ and parameter $$a$$ is defined by $$D_n(x,a)=\sum_{i=0}^{\lfloor n/2\rfloor} \frac{n}{n-i}\begin{pmatrix}n-i \\ i\end{pmatrix} (-a)^i x^{n-2i}$$ Dickson polynomials over finite fields have many very interesting properties. I will first briefly discuss some of these properties, hoping to generate further interest in these very fascinating polynomials. Some of the properties of Dickson polynomials are related to questions of permutations of finite fields. In previous work, the parameter $$a$$ has been fixed and the variable $$x$$ then runs through the field $$F_q$$. In some current work with James Sellers (Penn State) and Joe Yucas (Southern Illinois), we reverse these roles, and fix $$x$$ in $$F_q$$, and then allow $$a$$ to run through the elements of the field $$F_q$$. It appears that once again, we have an interesting, though far from understood, class of polynomials.

## Friday, March 21, 2008

Title

Speaker

Time
Place

Modeling Disease Spread Involving Structured Populations Using Non-Local Delayed Reaction Diffusion Equations
Jianhong Wu
Center for Disease Modeling
Department of Mathematics & Statistics
York University
3:00pm-4:00pm
PHY 130
Yuncheng You

Abstract

We discuss the interaction of time delay and spatial dispersal and its implication for modeling the evolution of biological systems using non-local delayed reaction diffusion or lattice differential equations. We present two case studies about spread of rabies and West Nile virus to illustrate how understanding the model's nonlinear dynamics assists disease management.

## Friday, February 15, 2008

Title
Speaker

Time
Place

From Approximation Theory to Compressive Sampling via Banach Spaces Geometry: a Computational Tour
Simon Foucart
Vanderbilt University
4:00pm-5:00pm
PHY 108
Boris Shekhtman

Abstract

Starting with an issue on computation stability, I will introduce the notion of condition number for a system spanning a normed space $$V$$. I will show how optimization techniques can be used to calculate the minimum of these condition numbers. The latter is an intrinsic constant of the space $$V$$, and I will examine its connections with the projection constant of $$V$$. In particular, I will raise a question — formulated only in terms of projections — related to the $$P_\lambda$$-problem. The arguments will lead me to the new and exciting field of Compressive Sampling. The paradigm that only few information on a signal is necessary for its reconstruction will be illustrated by some striking yet simple results, including a proof of Kashin's theorem on widths as a byproduct. All along, an eye will be kept on the computational aspect of the theory.

Title
Speaker

Time
Place

Finite Pseudorandom Binary Sequences and Their Applications in Cryptography
A. Sarkozy
University of Memphis/
Eotvos University
3:00pm-4:00pm
PHY 130
Vilmos Totik

Abstract

Finite pseudorandom binary sequences play a crucial role in cryptography. The pseudorandomness is usually characterized in terms of complexity theory. First the limitations of this approach will be analyzed. Then another, more constructive approach developed in the last decade by Mauduit, Sarkozy and others will be described. Finally, constructions of large families of binary sequences with strong pseudorandom properties will be presented.

## Monday, February 11, 2008

Title
Speaker

Time
Place

Tomas Sjödin
Royal Institute of Technology
Stockholm, SWEDEN
3:30pm-4:30pm
LIF 263
Dmitry Khavinson

Abstract

In this talk we will discuss the theory of quadrature domains from a potential-theoretic point of view. We will briefly outline some of its history and mention connections to other areas of research.

After this some of the later developments will be discussed. In particular, we will deal with the theory of partial balayage and its applications to Hele-Shaw flows from fluid mechanics, as well as the classical exterior inverse problem related to uniqueness questions for quadrature domains.

## Friday, February 8, 2008

Title
Speaker

Time
Place

Asymptotic Representation and Zero Distribution of Carleman Orthogonal Polynomials
Erwin Miña-Diaz
Indiana University-Purdue University
Fort Wayne, IN
3:00pm-4:00pm
PHY 130
E. Rakhmanov

Abstract

We will discuss several results about the asymptotic behavior of Carleman polynomials, i.e., polynomials $$P_n(z)$$, $$n=0,1,2,\dotsc$$ ($$p_n$$ of exact degree $$n$$), that are orthonormal with respect to area measure over the interior of an analytic Jordan curve $$L$$. We shall show that each $$P_n$$ of sufficiently large degree can be expanded in a series (that depends on $$n$$) of certain recursively generated integral transforms. The asymptotic behavior of $$P_n$$ is then easily obtained by analyzing that of the series as $$n$$ tends to infinity. In particular, one obtains at once Carleman's formula describing the strong asymptotic behavior of $$P_n$$ on the exterior of $$L$$, as well as an integral representation for $$P_n$$ inside $$L$$. We then show how this integral representation can be also obtained via the reproducing kernel, which is the key for extending (by means of a nice inductive type of argument) the validity of Carleman's formula toward a maximal domain $$D$$. The domain $$D$$ is maximal in the sense that each of its boundary points is an accumulation point of the zeros of the orthogonal polynomials. Finally, we discuss fine results on the location, limiting distribution and accumulation points of the zeros, valid for quite general sets $$D$$ having piecewise analytic boundary.

## Friday, February 1, 2008

Title
Speaker

Time
Place

Integrable Decompositions of Soliton Equations and Their Applications
Ruguang Zhou
Xuzhou Normal University
CHINA
3:00pm-4:00pm
PHY 130
Wen-Xiu Ma

Abstract

In this talk we will give a brief introduction to integrable decompositions of soliton equations. The method of nonlinearization of spectral problem and its various generalizations will be discussed, along with their applications. Some recent results on integrable decompositions of the nonlinear Schröder equation will be shown.

## Monday, January 14, 2008

Title
Speaker

Time
Place
Note

The Laplacian Growth: Physics, Mathematics, and Algorithms for Shape Recovery
Mark Mineev-Weinstein
Los Alamos National Laboratory
11:00am-12:00pm
ENB 313
Dmitry Khavinson
This Colloquium is joint with Computer Engineering.

Abstract

This talk is a short and informal introduction into the class of natural, industrial, and laboratory processes, called the 'Laplacian Growth'. A remarkable mathematical structure lying behind these processes will be introduced and discussed. Unexpectedly strong connections of this structure with various branches of mathematics, physics, and medicine will also be addressed. In addition, the numerically stable algorithms of solving the inverse problem of shape recoveries using implicit data will be introduced and discussed in detail.

## Friday, January 11, 2008

Title
Speaker

Time
Place
We construct a new multi-component CKP hierarchy based on the eigenfunction symmetry constraint. It provides an effective way to find new first type and second type of CKP equation with self-consistent sources (CKPSCS) and their Lax representations. Also it admits reductions to $$k$$-constrained CKP hierarchy and to a $$(1+1)$$-dimensional soliton hierarchy with self-consistent source, which gives rise to new first type and second type of Kaup-Kuperschmidt equation with self-consistent sources (KPSCS) and of bi-directional Kaup-Kuperschmidt equation with self-consistent sources (bdKPSCS). By using the solutions of the CKP and KK equations and their corresponding eigenfunctions, $$N$$-soliton solutions for CKPSCS and KKSCS are constructed by means of method of variation of constant, respectively.