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 |