Harmonic Analysis and PDE Seminar

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

Organizers: Shijun Zheng and Alex Stokolos



All seminars are at 3:00 pm in Math-Phys 3314 unless otherwise noted.
Calendar       
Speaker
   Title
    Affiliation

2007 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

2008 February 13 (4:30-5:30)

   Steven Damelin   Dimension Reduction, Harmonic analysis and Non Euclidean Metrics Georgia Southern University

2008 February 14 (Colloquium)

 Matthew Blair   Nonlinear wave equations on exterior domains  University of Rochester

2008 February 29 (Colloquium)

   Shijun Zheng  Spectral calculus, Besov spaces and dispersive equations  GSU

2008 March 12 (4:30-5:30)

   David Benko    Uniform approximation by weighed polynomials Western Kentucky University

2008 March 14 (4:00-5:00)

   David Ragozin   The world of harmonicity defined without differentiationUniversity of Washington

2008 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
2008 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
2008 April (Colloquium)

Manoussos Grillakis

  Impurity and quaternions in nonrelativistic scattering from quantum memory

   University of Maryland
2008 May 2 (Colloquium)

Shuanglin Shao

  The restriction estimates for paraboloid in the cylindrically symmetric case

   UCLA
2008 November 21 (Colloquium)

Jens G. Christensen

   A Wavelet Decomposition of Besov Spaces on the Forward Light Cone

   LSU
2009 Feb 6 (Colloquium)

Konstantin Oskolkov

  Schrödinger equation and Riemann's non-differentiable function

   USC
2009 Feb 25 6:00 The Fifth DISTINGUISHED LECTURE

Christopher Sogge

  Blowup rates for eigenfunctions and quasimodes

  Johns Hopkins University
2009 Feb 27 (Colloquium)

Christopher Sogge

   Abstract Strichartz estimates and existence theorems for nonlinear wave equations

   Johns Hopkins University
2009 October 9 (Colloquium)

Alexander Stokolos

   A gentle introduction in the Bellman Function technique I

   GSU
2009 October 23 (Colloquium)

Alexander Stokolos

   A gentle introduction in the Bellman Function technique II

   GSU
2009 October 30 (Colloquium)

Benjamin Dodson

  The I-method and global well-posedness for the defocusing nonlinear Schroedinger equation

  UC Riverside
2010 February 23 (3-4) MP1305 Alex Stokolos

  Bellman Functions and Monge-Amp`ere Equation VII

  GSU
2010 March 4 (2:00-3:00 MP3311) Armen Vagharshakyan

  Recovering Singular Integrals from Haar Shifts

   Geogia Tech
2010 March 5 (3:00-4:00) Colloquium Michael Lacey

   The Two Weight Inequality for the Hilbert Transform

   Geogia Tech
2010 March 12 (3:00-4:00) Colloquium Laura de Carli

   On the L^p norm of the Fourier transform of disconnected symmetric regions

   Florida International University
2010 March 9 (3:00-4:00) Shijun Zheng

  Nonlinear wave and Schroedinger equation I

   GSU
2010 April 6 (3:00-4:00) Shijun Zheng

  Nonlinear wave and Schroedinger equation II

   GSU
2010 April 22 (2:00-3:00) Xuwen Chen

  Classical proofs for Kato type smoothing estimates for the Schrödinger equation with quadratic potential

   University of Maryland
2010 April 23 (3:00-4:00) Colloquium Chengbo Wang

  Strauss conjecture for nontrapping obstacles

   Johns Hopkins University

 

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.

Feb 25, 6-7   C. Sogge (JHU)   Blowup rates for eigenfunctions and quasimodes

We study size estimates for eigenfunctions and quasimodes on compact Riemannian manifolds with and without boundary. We are interested in when these functions blowup at the maximal possible rate as measured by $L^p$ rates as their energy goes to infinity. We wish to characterize what sort of geometries must be present for maximal blowup and conversely what geometries lead to sub-maximal blowup rates. This is joint work with John Toth and Steve Zelditch and with Hart Smith.

 Feb 27, 3-4   C. Sogge (JHU)   Abstract Strichartz estimates and existence theorems for nonlinear wave equations

We shall show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with ``abstract Strichartz" estimates for the free wave equation to prove corresponding estimates for the perturbed wave equation when $n\ge3$. As we shall see, we are only partially successful in the latter endeavor when the dimension is equal to two, and therefore, at present, our applications to nonlinear wave equations in this case are limited.

 Feb 6, 3-4   K. Oskolkov (University of South Carolina) *  Schrodinger equation and Riemann's non-differentiable functions

 October 30 3:00-4:00   Benjamin Dodson (UC Riverside)   The I-method and global well-posedness for the defocusing nonlinear Schroedinger equation

The defocusing, nonlinear Schrodinger equation of power type nonlinearity
iu_t+\Delta u=|u|^\alpha u\\ u(0,x)=u_0\in H^1(R^n) has a global solution for \alpha < \frac{4}{n-2}. In this talk we discuss the I-method, which is used to prove global well-posedness for data in H^s(R^n), s<1. In particular, we will address the L^2-critical equation (\alpha=4/n) and the the cubic equation in three dimensions.

March 9, 3:00-4:00   S. Zheng (GSU) Nonlinear wave and Schroedinger equations

Nonlinear wave and Schroedinger equations model many physical phenomena ranging from optics, elasticity, quantum mechanics to general relativity. Their common feature is the propagating wave structure. In the linear case (free state), waves would spread out and decay. In the nonlinear case, the existence of the local or global waves is a consequence of the fight between linear and nonlinear interactions. In the lecture I will give a brief introduction to this broad subject, based on my own lecture notes. Some recent development may be reviewed during the course, including those frontier work by Bourgain, Tao and Grillakis. I will try to select the topics that can possibly be enjoyed by faculty members as well as graduate students.

April 22, 2:00-3:00   Xuwen Chen (UMD) Classical proofs for Kato type smoothing estimates for the Schrödinger equation with quadratic potential

In this talk, the speaker will use Hermite functions to give elementary proofs of Kato type smoothing estimates for the Schrodinger equation with quadratic potential. This is equivalent to proving an uniform L^2 to L^2 boundedness for the singularized Hermite projection operators.

April 23 3-4   Chengbo Wang (JHU)

In this talk, we discuss our recent work on the 2-dimensional Strauss conjecture for nontrapping obstacles. This is a joint work with H. Smith and C. Sogge. Recently, Hidano, Metcalfe, Smith, Sogge and Zhou proved the Strauss conjecture for nontrapping obstacles when the spatial dimension n equals 3 and 4. Their method is to prove abstract Strichartz estimates, including the |x|-weighted Strichartz estimates.
In the Minkowski spacetime, the |x|-weighted Strichartz estimates (also from the work of Fang and Wang) can be utilized to prove the Strauss conjecture with n = 2, 3, 4. The reason that they can only prove the general results for n = 3, 4 is that the abstract Strichartz estimates are proved only for the case with regularity s \in [(3-n)/2, (n-1)/2] (that is, s = 1/2 if n = 2). This restriction is essential for the general abstract Strichartz estimates. In this work, we overcome this difficulty for n = 2 by proving the generalized Strichartz estimates of the type L^q_t L^r_{|x|} L^2_\theta. The corresponding problem for n>4 are still open.