WESTERN MICHIGAN UNIVERSITY ANALYSIS SEMINAR
  SPRING 2007

Wednesday at  10 a.m. - 10:50 a.m.
Alavi Commons Room, Everett Tower


Comments, questions?  Contact   Yuri Ledyaev ,  phone (269)  387-4557
 

SEMINAR'S ARCHIVE: Spring 2007Fall 2006Spring 2006 , Spring 2005


Date
Speaker, Title and Abstract

April 11
 Dr. Qiji J. Zhu
Department of Mathematics
Western Michigan University

HOW TO MANAGE YOUR PORTFOLIO USING CONVEX ANALYSIS

In 1990, Harry Markowitz received the Nobel Price in Economics for developing the theory of portfolio choice in early 1950s. More than one half century later, Markowitz'es theory is still one of the most popular in portfolio management. In this talk we give a convex analyst's introduction to Markowitz'es pioneering work. Time permits, we will also discuss some recent developments.

April 4

Dr.John Petrovic
Department of Mathematics 

Western Michigan University

 ON SPECTRAL RADIUS ALGEBRAS AND NORMAL OPERATORS

It is well known that, if N is a normal operator, then its commutant {N}' (the set of all operators that commute with N) has an invariant subspace. We will introduce the so-called spectral radius algebra B_N associated to N and establish a strenghtening of the mentioned result. Namely, we'll show that B_N also has an invariant subspace. In addition, B_N properly contains the commutant {N}', so the result is, indeed, stronger than the classical one.

March 28

  Yuri S.Ledyaev
Department of Mathematics 
Western Michigan University

  INTRODUCTION TO  NON-LINEAR FOURIER TRANSFORM
 (after Terence Tao)
This is a continuation of the previous talk. We discuss some standard material on the  nonlinear Fourier  transform which is the scattering theory analogue of the more familiar linear Fourier transform. Our exposition closely follows Terence Lao's (2006 Fields Medal winner) survey on this subject.

March 21

  Yuri S.Ledyaev
Department of Mathematics 
Western Michigan University

  INTRODUCTION TO  NON-LINEAR FOURIER TRANSFORM
 (AFTER TERENCE TAO)
We discuss some standard material on the  nonlinear Fourier  transform which is the scattering theory analogue of the more familiar linear Fourier transform. Our exposition closely follows Terence Lao's (2006 Fields Medal winner) survey on this subject.

March 14

Anirban Dutta
Department of Mathematics

Western Michigan University

FOURIER ANALYSIS ON LOCAL COMPACT ABELIAN GROUPS

This is the gourth talk of a srudent-oriented mini-course on generalization of Fourier Transforms  in R^n to any Locally Compact Abelian  (LCA) group. We'll  review  briefly some basic concepts of measure theory and functional analysis that are relevant to the topic. Our  final goal is generalization of classical important results (namely, Inversion Theorem, Plancherel Theorem, Pontryagin Duality Theorem and Bohr Compactification Theorem) of classical Fourier Transform to the case of locally compact group. These results are used in modern physical theories. This course is primarily intended for graduate students who have some exposure to graduate level analysis.

February 28

Anirban Dutta
Department of Mathematics

Western Michigan University

FOURIER ANALYSIS ON LOCAL COMPACT ABELIAN GROUPS

This is the third talk of a srudent-oriented mini-course on generalization of Fourier Transforms  in R^n to any Locally Compact Abelian  (LCA) group. We'll  review  briefly some basic concepts of measure theory and functional analysis that are relevant to the topic. Our  final goal is generalization of classical important results (namely, Inversion Theorem, Plancherel Theorem, Pontryagin Duality Theorem and Bohr Compactification Theorem) of classical Fourier Transform to the case of locally compact group. These results are used in modern physical theories. This course is primarily intended for graduate students who have some exposure to graduate level analysis.

February  21

Vinod Kumar.P.B
Department of Mathematics
Rajagiri School of Engineering and Technology
India.

INTRODUCTION TO CHAOS AND FRACTALS

There are events occurring in nature which are predictable and unpredictable. If such unpredictability arises because of sensitivity then chaos occurs. The theory of chaos was originated in the work of a meteorologist Edward Lorenz  in 1960's. But an interest to mathematical aspects of chaos was stimulated by  the work of Tien Yin Li and James Yorke in 1975. In this talk we'll discuss and illustrate some forms of chaos.
Most of the objects in the nature are irregular. They do not have a definite shape or size. Fractals are irregular objects. Their theory was originated by the work of Benoit Mandelbrot in 1980's and it became popular  with the introduction of self similarity concept by John Hutchinson in 1985. We'll provide a brief introduction to self-similarity in this talk.

February 14

Anirban Dutta
Department of Mathematics

Western Michigan University

FOURIER ANALYSIS ON LOCAL COMPACT ABELIAN GROUPS

This is the second talk of a srudent-oriented mini-course on generalization of Fourier Transforms  in R^n to any Locally Compact Abelian  (LCA) group. We'll  review  briefly some basic concepts of measure theory and functional analysis that are relevant to the topic. Our  final goal is generalization of classical important results (namely, Inversion Theorem, Plancherel Theorem, Pontryagin Duality Theorem and Bohr Compactification Theorem) of classical Fourier Transform to the case of locally compact group. These results are used in modern physical theories. This course is primarily intended for graduate students who have some exposure to graduate level analysis.

February 7

Anirban Dutta
Department of Mathematics

Western Michigan University

FOURIER ANALYSIS ON LOCAL COMPACT ABELIAN GROUPS

This is a first talk of a srudent-oriented mini-course on generalization of Fourier Transforms  in R^n to any Locally Compact Abelian  (LCA) group. We'll  review  briefly some basic concepts of measure theory and functional analysis that are relevant to the topic. Our  final goal is generalization of classical important results (namely, Inversion Theorem, Plancherel Theorem, Pontryagin Duality Theorem and Bohr Compactification Theorem) of classical Fourier Transform to the case of locally compact group. These results are used in modern physical theories. This course is primarily intended for graduate students who have some exposure to graduate level analysis.


January 31
 Dr. Qiji J. Zhu
Department of Mathematics
Western Michigan University

HOW MANY TRICKS DOES A VARIATIONAL ANALYST HAVE

It is said that even a great mathematician has only a few tricks. We will explain that many important tools in convex and nonsmooth analysis are different facets of two basic tricks of variational methods: a variational principle and a decoupling method. We'll also discuss interesting new directions of research.


January 24

Yuri S.Ledyaev
Department of Mathematics 
Western Michigan University

 
FIXED POINTS AND ZEROES OF MAPPINGS AND INVARIANCE
(second part of the January 17th talk)


January  17

  Yuri S.Ledyaev
Department of Mathematics 
Western Michigan University

 
FIXED POINTS AND ZEROES OF MAPPINGS AND INVARIANCE

This is a student-oriented talk intended to provide them with a survey of classical (and not so classical) results on existence of  fixed points and zeroes of mappings f: K->R^n on closed subsets K of R^n. Such results are  important in various fields of applied mathematics including  existence of solutions of partial differential equations or existence of equilibria  in mathematical economics models.
We start with  classical fixed points theorems by Brouwer and Kakutani and relate them to results on existence of zeroes of mappings on convex compact sets under some boundary conditions for these mappings. We use invariance properties of some flows generated by these mappings to obtain generalization of existence theorems for zeroes of mappings for nonconvex sets.


Last modified :April 24, 2007