Southern California Analysis
and PDE Conference

Saturday, November 7, 2009,  Donal Bren Hall 1500 10:00am-10.55am           P. Constantin (U Chicago) - Complex fluids: kinetics and and dynamics 11:00am-11.55am           A. Kiselev (U Wisconsin) - Regularity results for the surface quasi-geostrophic equation 12:00noon- 1.55pm         Lunch Break 2.00pm -2.55pm              N. Masmoudi (Courant) - Global existence for water waves in 3D 3:00pm-3.55pm               G. Ponce (UCSB) - Hardy uncertainty principle and unique continuation properties of nonlinear Schrodinger equations 4:00pm-4.30pm               Coffee  Break 4.30pm-5.25pm               R. Killip (UCLA) - Autocorrelation of the characteristic polynomial of a random matrix   Sunday, November 8, 2009, Rowland Hall 306 9.30am-10.25am          M. Christ (UCB) - Existence of extremals for the Tomas-Stein inequality 10.30am-11.25am        C. Remling (U Oklahoma) - Reflectionless Jacobi matrices 11.30am-11.55am        Coffee Break 12.00noon-12.55 pm    B. Simon (Caltech) - Natural boundaries and spectral theory     Abstract of the talk of M. Christ The Fourier transform maps L2(S2) to L4(R3). We show that there exist functions which extremize the associated inequality, and that any extremizing sequence of nonnegative functions has a convergent subsequence. This was previously known for paraboloids, where all extremizers are Gaussians and vice versa. Complex extremizers and extremizing sequences are related to nonnegative ones in a simple way. All critical points of the associated nonlinear functional are smooth. Constant functions are local extremizers, but we do not know whether they are global extremizers, nor whether extremizers are unique modulo symmetries of the problem. The proofs involve concentration compactness, convolution inequalities, Fourier integral operators, symmetrization, a characterization of approximate characters, a regularity theorem for a nonlocal equation, an idea from additive combinatorics, spherical harmonics and Gegenbauer polynomials, and several explicit computations. This is a joint work with Shuanglin Shao.   Abstract of the talk of Peter Constantin I will present a framework for the study of complex fluids that involve the study of gradient systems in metric spaces. I will present some of the ideas behind the analysis of the coupled Nonlinear Fokker Planck and Navier-Stokes PDEs and describe some of the many remaining challenges.   Abstract of the talk of R. Killip   We compute the expected value of a general product of the characteristic polynomial and its complex conjugate (evaluated at multiple points) in the microscopic thermodynamic limit. The result holds for arbitrary inverse temperature betta between zero and infinity. (Joint work with E. Ryckman).   Abstract of the talk of A. Kiselev We will review recent results on the global existence of regular  solutions to critical surface quasi-geostrophic equation and different methods developed to approach this problem.   Abstract of the talk of N. Masmoudi We prove the global existence of regular solutions to the water waves problem in 3D for small data. The proof is based on the combinaison of  energy estimates and dispersive estimates. (This is a joint work with Pierre Germain and Jalal Shatah).   Abstract of the talk of G. Ponce After reviewing some known unique continuation principles for nonlinear dispersive equations, we concentrate in the case of the Schrodinger equation. We explain the relation of this problem with uncertainty principles. We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schrodinger equations with non-constant coefficients. From this extension we return to the original deducing new continuation properties of solutions to semi-linear Schrodinger equations. (Joint work with L. Escauriza, C. E. Kenig, and L. Vega).   Abstract of the talk of C. Remling Reflectionless Jacobi matrices are of interest because they may be viewed as the basic building blocks of arbitrary Jacobi operators with non-empty absolutely continuous spectrum. I'll review these results and also discuss more recent attempts, largely unsuccessful so far, to understand specific reflectionless operators in more detail.   Abstract of the talk of B. Simon. This talk describes joint work with Jonathan Breuer. The last ten years has seen considerable understanding of the spectrum of general Jacobi matrices in terms of its right limits due to work of Last-Simon and especially Remling. We have discovered that analogs of these ideas can be used to understand when a power series (with bounded Taylor coefficients) has a natural boundary on the unit circle. One recovers and (within the class of bounded coefficients) improves many classical results. The main theorem depends on little more than the notions of right limit and reflectionless double power series (that we carry over from the theory of Jacobi matrices) and a clever lemma proven by M. Riesz in 1916 (using the maximum principle). This will be a colloquium-level talk that should be accessible to anyone with a good complex variables course.

Registration and local information

We will need an approximate head count, so unless you prefer to stand during the talks, please either arrive very early or register here. Contrary to what the registration form may say, there will be no banquet. However, there will be an informal dinner on Saturday night at a local restaurant.

The information about local hotels with UCI discounts can be obtained  here

For maps, and other travel and parking information click on the "Campus Map" link to the left or above. The parking lots closest to RH (building 400 on the map) are 12A and 12B. The parking  closest to  DBH (building 314 on the map) are 12B and the Engineering parking structure (building 653).

Funding

The SCAPDE is supported by a grant from the National Science Foundation

.  Funding is available for graduate students and postdocs who do not have NSF or other grant or school support. People interested in applying for travel funds should check the appropriate box in the electronic registration form and send an e-mail request to Hans Lindblad <lindblad@math.ucsd.edu>. The deadline for sending in requests for travel support is October 30.