Harmonic Analysis and PDE Seminar 2007-2008

Fridays 3:00-4:00 p.m. Room Math-Phys 3314

All seminars are at 3:00 pm in Math-Phys 3314 unless otherwise noted.
Calendar	Speaker	Title	Affiliation
November 2	Shijun Zheng	Introduction to harmonic analysis and PDE (I)	Georgia Southern University
November 9	Shijun Zheng	Introduction to harmonic analysis and PDE (II)	Georgia Southern University
February 13 (4:30-5:30)	Steven Damelin	Dimension Reduction, Harmonic analysis and Non Euclidean Metrics	Georgia Southern University
February 14 (Colloquium)	Matthew Blair	Nonlinear wave equations on exterior domains	University of Rochester
February 29 (Colloquium)	Shijun Zheng	Spectral calculus, Besov spaces and dispersive equations	GSU
March 12 (4:30-5:30)	David Benko	Uniform approximation by weighed polynomials	Western Kentucky University
March 14 (4:00-5:00)	David Ragozin	The world of harmonicity defined without differentiation	University of Washington
March 24	TBA	Local smoothing estimates imply Bochner-Riesz conjecture	GSU
April 1 (GSU Distinguished Lecture 6:00-7:00)	George Andrews	Euler and the beginning of the theory of partitions	Penn State University
April 4 (Colloquium)	Chris Heil	Music, Time-Frequency Shifts, and Linear Independence	Georgia Institute Technology
April (Colloquium)	Ramona Anton	Nonlinear Schrödinger equations on domains with boundary	Université Paris XI and Johns Hopkins University
April (Colloquium)	Manoussos Grillakis	Impurity and quaternions in nonrelativistic scattering from quantum memory	University of Maryland
May 2 (Colloquium)	Shuanglin Shao	The restriction estimates for paraboloid in the cylindrically symmetric case	UCLA

UAIM Institute

Abstract of Talks

November 2 Shijun Zheng (GSU) Introduction to harmonic analysis and PDE (I)

The topic will be an introduction of harmonic analysis with applications in PDE theory. I will start from some of the fundamental theory and problems in harmonic analysis developed from classical Fouirer analysis, introducing basic methods and techniques that have been used till today (Calderon-Zygmund theory, Littlewood-Paley decomposition, Sobolev and Besov spaces, oscillatory integrals, singular integrals, Maximal functions). Problems involve Bochner-Riesz conjecture, Fourier Restriction phenomenon, Strichartz and Morawetz estimates, Kakeya set conjecture.

The main motivation of the development is from PDE arising in mathematical physics and therefore applications in PDEs will be discussed during the course. I may tend to keep the topic update with some of the advanced research as long as it fits the audience's interest. Since we anticipate some interesting discussions from a variety of background, the talk might be systematic while occasionally quite informal.

November 9 Shijun Zheng (GSU) Introduction to harmonic analysis and PDE (II)

We will continue to lecture on the introduction of harmonic analysis with applications in PDE. Last week we gave a general review of problems in Fourier analysis as well motivations from dispersive equations with linear or nonlinear perturbations that have physical background in quantum mechanics or nonlinear optics. In the second part of the talk I will be mainly following the Littlewood-Paley decomposition (dyadic) approach to address problems from harmonic analysis and PDE while many of the detailed results and proof will be based on Tao's lecture notes as well as my own.

February 13 S.B. Damelin Director UAIM (GSU); Professor (Elect), Applied and Computational Math (WITS). On Compression of Hyperspectral Data and its connections to Dimension Reduction, Harmonic analysis and Non Euclidean Metrics:

In this talk, I will discuss some recent joint work with M. Sears (Wits) and A. Hero (Michigan) on compression of Hyperspectral Data. We explore connections to Dimension Reduction, Harmonic Analysis and Metrics.

February 14 Matthew Blair (University of Rochester) Nonlinear wave equations on exterior domains

We consider certain semilinear wave equations posed on an exterior domain. While basic questions such as existence, uniqueness, and scattering of solutions have been answered in the Euclidean case, less is known in the case of an exterior domain. Here the presence of Dirichlet or Neumann boundary conditions can affect the flow of energy, complicating these issues considerably. We discuss recent progress in the area, including the development and applications of space-time integrability estimates for the wave equation ("Strichartz estimates"). This is a joint work with H. Smith and C. Sogge.

February 29 S. Zheng (GSU) Spectral calculus, Besov spaces and Dispersive equations

In this talk we consider Hörmander type spectral multiplier problem for Schrödinger operators with a critical potential. It is shown that the multiplier operator is bounded on $L^p$, Besov spaces and Triebel-Lizorkin spaces under the same sharp condition. We then derive Strichartz estimates that measure spacetime regularity for the corresponding wave equation. Our work is partially motivated by the standing wave problem for the quintic wave equation in 3+1 dimensions.

March 12 David Benko (Western Kentucky University) Uniform approximation by weighed polynomials

Let w(x) be a continuous non-negative weight on the real line which decays faster than 1/x. The topic of uniform approximation by weighted polynomials w(x)^nP_n(x) was introduced by Saff. He conjectured that w(x)^nP_n(x) (n=0,1,2,...) are dense on the support of the equilibrium measure, assuming log w(x) is concave. This was proved by Totik. In the talk we give other sufficient conditions on w(x) which imply denseness.

March 14 David Ragozin (University of Washington) The world of harmonicity defined without differentiation

We develop the world of harmonicity without differentiation and show how this natural and beautiful idea leads to many wonderful theorems on manifolds.

April 18 Chris Heil (Georgia Tech) Music, Time-Frequency Shifts, and Linear Independence

Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. But this representation is not local, and does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss Fourier series, and then present time-frequency representations, which are a type of local Fourier representation of signals. This gives us a mathematical model for representing music. While the model is crude for music, it is in fact a powerful mathematical representation that has appeared widely throughout mathematics (e.g., partial differential equations), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties.

April 25 2:00-3:00 Ramona Anton (Université Paris XI, Orsay) Nonlinear Schrödinger equations on domains with boundary

We are interested in proving global existence results in the energy space for the semi-linear Schrödinger equation on domains of dimension 2 or 3. The main ingredients are generalized Strichartz inequalities adapted to the domains, which have some loss of derivatives. We present the results and the strategy for three types of domains.

April 25 3:00-4:00 M. Grillakis (University of Maryland) Impurity and quaternions in nonrelativistic scattering from quantum memory

Models in quantum computing rely on transformations of states of quantum memory. We study mathematical aspects of a model proposed by Wu in which the memory state is changed via scattering of incoming particles. This operation causes the memory content to deviate from a pure state, i.e. induces impurity. For nonrelativistic particles scattered from a two-state memory and sufficiently general interaction potentials in 1 + 1 dimensions, we express impurity in terms of quaternionic commutators. I this context, pure memory states correspond to null hyperbolic quaternions. In the case of point interactions, the scattering process amounts to appropriate rotations of quaternions in the frequency domain. This point of view complements a previous analysis by Margetis and Myers (2006 J. Phys. A 39 11567-11581) and is in collaboration with D. Margetis.

May 2 3:00-4:00 S. Shao (UCLA) The restriction estimates for paraboloid in the cylindrically symmetric case

Restriction conjecture is one of the central problems in harmonic analysis. It was solved in 2-dimension but remains open in higher dimensions. In this talk, I will focus on the linear and bilinear versions of this conjecture for paraboloids in the cylindrically symmetric case. The main result is that we have further estimates available with this assumption which are sharp up to endpoints and turn out to be very useful in establishing the global wellposedness of certain mass-critical NLS in the radial case by Killip, Tao and Visan. Another consequence is that the restriction conjecture for the paraboloid is true in all dimensions in the cylindrically symmetric case.