ABSTRACTS FOR ANALYSIS TALKS - SPRING 2005 Friday, February 11, 2005, 4:00 p.m., VV B239 COLLOQUIUM Carlos Kenig (University of Chicago) Recent developments on the well-posedness of nonlinear dispersive equations Abstract: I will review some of the work on nonlinear dispersive equations done in the 90's and then discuss a class of interesting problems were these techniques do not apply and the recent partial progress in them. *********************************************************************** Friday, March 4, 2005, 1:20 p.m., VV B115 Stephen Yau (University of Illinois, Chicago) Holomorphic De Rham cohomology of strongly pseudoconvex CR manifolds. Abstract: In this talk we shall discuss the holomorphic DeRham cohomology of a compact strongly pseudoconvex CR manifold X. Holomorphic De Rham cohomology is derived from the Kohn-Rossi cohomology and is particularly interesting when X is real dimension 3 and the Kohn-Rossi cohomology is infinite dimesional. We shall use holomorphic De Rham cohomology to study the complex Plateau problem for a 3 real dimension compact strongly pseudoconvex CR manifold sitting in C^3. In general, for compact strongly pseudoconvex CR manifolds with trasversal holomorhpic S^1 action, we can relate the holomorphic De Rham cohomology to the puntured local holomorphic De Rham cohomology at the singularity in the variety V which X bounds. *********************************************************************** Friday, March 4, 2005, 4:00 p.m., VV B239 COLLOQUIUM Stephen Yau (University of Illinois, Chicago) CR manifolds and isolated singularities Abstract: CR manifolds are boundaries of complex manifolds or complex varities. In this talk we shall discuss the CR equivalent problem among CR manifolds, i.e. given two CR manifolds, how can we know whether they are CR biholomorphically equivalent. Recently we have introduced a new nonnegative Bergman function for any strongly pseudoconvex complex manifold. This Bergman function is invariant under biholomorphic maps and vanishes precisely on the exceptional set of the strongly pseudoconvex complex manifold. We shall show that it can be used to study the variations of CR structure of strongly pseudoconvex CR manifolds lying within an variety. In the joint work with H.S. Luk and X.J. Huang, we study the simultaneous embedding and filling problems for a CR family of CR manifolds. As a corollary, we showed that Buchweitz-Milson-Miyajima theorem is true even for singularities with dimension 3. Thus if (V,0) is a normal isolated singularity with dimension and depth at least 3, then the Kuranishi family of the link of (V,0) is realized as a real hypersurface of the versal family of deformation of (V,0). *********************************************************************** Svetlana Jitomirskaya (UC Irvine) will give two talks on Friday, March 11, 2005: Analysis Seminar: Friday, March 11. 2005, 1:20 p.m., VV B115 Title: The ten martini problem. Abstract: we will discuss the recent proof of Cantor spectrum for the almost Mathieu operator for all conjectured values of the parameters. COLLOQUIUM: Friday, March 11, 2005, 4:00 p.m., VV B239 Title: Spectral properties of quasiperiodic operators: the competition between order and chaos. Abstract: Up until the mid 70s the kind of spectra most people had in mind in the context of theory of Schrodinger operators were spectra occurring for periodic potentials and for atomic and molecular Hamiltonians. Then evidence started to build up that "exotic" spectral phenomena such as singular continuous, Cantor, and dense point spectrum do occur in mathematical models that are of substantial interest to theoretical physics. One area where such exotic phenomena are particularly abundant is quasiperiodic operators. They feature a competition between randomness (ergodicity) and order (periodicity), which is often resolved on a deep arithmetic level. Mathematically, the methods involved include a mixture of ergodic theory, dynamical systems, probability, functional and harmonic analysis. The interest in those models was enhanced by strong connections with some major discoveries in physics, such as integer quantum Hall effect, experimental quasicrystals, and quantum chaos theory. Quasiperiodic operators provide central or important models for all three. The talk will consist of an overview of certain topics related to this subject and of a glimpse into how one can fight small denominators in this setting. *********************************************************************** Friday, April 1, 2005, 4:00 p.m., VV B239 COLLOQUIUM Renormalization group approach to spectral problems and theory of radiation I.M. Sigal (Toronto and Notre Dame) Abstract: In this talk I will describe some analytical problems in Quantum Field Theory, viewed as Quantum Mechanics of infinitely many degrees of freedom or of extended objects (e.g. curves, surfaces, etc). I will present some recent results on the problem of radiation and describe a novel renormalization group technique used in proving these results. I will not assume any prior knowledge of the subject.