SEMINAR IN ANALYSIS – Spring 2013

TO BE UPDATED SOON!

 

SEMINAR IN ANALYSIS – Fall 2012

 

Introduction: The seminar is targeted to expose graduate students to various topics in analysis. The faculty members (S. Datta,  F. Gao,  and L. Nguyen) will suggest topics and some guidance, and the students will read and present the material they choose (preferably from the list below).

Topics:

·      Frame/Wavelet Theory:

Reading list:

1.     J. Kovačević and  A. Chebira, Life beyond bases: the advent of frames (Part I), IEEE Signal Processing Magazine, 2007.

This is a tutorial style paper and an easy read. It is very well written and well developed. It develops a very nice motivation on why we care to bring in redundancy in bases and how frames are the way to go, along with nice instances of applications.

 

2.     J. Kovačević and  A. Chebira, Life beyond bases: the advent of frames (Part II) – same as above.

 

3.     R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Transactions of the AMS, 1952.

This is the paper where frames were first introduced and therefore this is a fundamental article. 

 

4.     P.G. Casazza, O. Christensen, S. Li, and A. Lindner, Density results for frames of exponentials, P.G. Casazza, O. Christensen, S. Li, and A. Lindner

This is a book chapter. It gives conditions under which a certain set of exponential functions might form a frame for the space L^2(-R, R). This is related to the study of reconstructing a function from (irregular) samples using frames. This paper is on a specific research problem rather than being a survey or tutorial.

 

·      PDEs: We study variational techniques in scattering theory.

Reading list:  

5.     C. S. Morawetz and D. Ludwig, An inequality for the reduced wave operator and the justification of geometrical optics. Comm. Pure Appl. Math 21, 187–203, 1968.

The paper contains some important identities and inequalities for scattering theory. The first part of the paper is quite easy to read and requires minimal background in PDEs.

 

6.     H.-M. Nguyen, and M-S. Vogelius, Full range scattering estimates and their application to cloaking, Archive for Rational Mechanics and Analysis 203, 769-807, 2012.

The paper provides an approach to estimate approximate cloaking. It requires some background in Analysis and PDEs. However, the reader can pick it up along the way. If the reader is interested in cloaking, more references will be provided.

 

·      Probability:

Reading list: 

1.    J. Bourgain, Sidon sets and Riesz Product.  Annles de l’institut Fourier, 1985.

 

 

DATE

TOPIC

SPEAKER

Thursday,  August 23

Organizational meeting

All

Tuesday,  August 28

An Introduction to Frame Theory

S. Datta

Tuesday,  Sept 4

Introduction to PDEs

L. Nguyen

Tuesday,  Sept 11

Wave in absorbing media

L. Nguyen

Tuesday,  Sept 18

Lacunary Fourier Series

F. Gao

Tuesday,  Sept 25

Brascamp-Lieb Inequality

F. Gao

Tuesday,  Oct 2

Brascamp-Lieb Inequality

J. Cokerham

Tuesday, Oct  9

Brascamp-Lieb Inequality

J. Cokerham

Tuesday, Oct  16

Nonharmonic Fourier Series

J. Oldroyd

Tuesday, Oct  23

Nonharmonic Fourier Series

J. Oldroyd

Tuesday, Oct  30

Unconditional Bases of Normed Spaces

X. Wu

Tuesday, Nov  6

Unconditional Bases of Normed Spaces

X. Wu

Tuesday, Nov 13

Frame Theory

P. Brown

Tuesday, Nov 20

Frame Theory

P. Brown