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Mathematics & Statistics

**Title**

**Speaker**

**Time**

**Place**

**Sponsor**

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.

**Title**

**Speaker**

**Time**

**Place**

**Sponsor**

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.

**Title**

**Speaker**

**Time**

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**Sponsor**

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.

**Title**

**Speaker**

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**Sponsor**

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

CANADA

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.

**Title**

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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**

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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.

**Title**

**Speaker**

**Time**

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**Sponsor**

Potential Theory and Quadrature Domains

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.

**Title**

**Speaker**

**Time**

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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.

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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.

**Title**

**Speaker**

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**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.

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A new multi-component CKP hierarchy and its solutions

Yunbo Zeng

Tsinghua University

Beijing, CHINA

3:00pm-4:00pm

PHY 130

Wen-Xiu Ma

**Abstract**

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.