Analysis   Seminar    


Unless specified, all seminars are Wednesday 4-5pm at  250 Math Building.                      

January 28th                            Hanfeng Li,    SUNY at Buffalo
                                               Turbulence, representations, and trace-preserving actions

                                               Abstract: I will give criteria for tubulence in spaces of C*-algebra representations, and indicate
                                               how this helps to establish results of nonclassifiability by countable structures, for group actions
                                               on a standard atomless probability space and on the hyperfinite II1 factor. This is a joint work
                                               with David Kerr and Mikaël Pichot. 
                                   
February 11th                          David Sherman,    University of Virginia  
                                                Relative operator theory

                                                Abstract: Many statements from operator theory still make sense when B(H) is replaced with
                                                a different von Neumann algebra. Some classical theorems generalize perfectly, others fail miserably;
                                                most valuable are those whose truth depends on the structure of the von Neumann algebra. I will
                                                survey several recent results and open problems in this "relative" operator theory.                              
 
March  4th                               Joshua Isralowitz,    SUNY at Buffalo
                                                Schatten class Toeplitz and Hankel operators on the Segal-Bargmann and Bergman spaces

                                                Abstract: We discuss and compare Schatten p class membership for 0 < p < ∞ of Toeplitz and
                                                Hankel operators on the Segal-Bargmann space and the Bergman space of the unit ball in Cn.  
                                                In particular, we discuss the cut-off phenomenon that occurs when characterizing Schatten p class membership
                                                of Toeplitz and Hankel operators on the Bergman space of the unit ball, but which does not occur
                                                when characterizing Schatten class Toeplitz and Hankel operators on the Segal-Bargmann space.
                                                This is partly joint work with K. Zhu.

March 11th                              Mikaël Pichot,    IPMU,   University of Tokyo
(Canceled)                               Isoperimetry and dynamical systems

                                                Abstract: I will describe the behavior of isoperimetric type constants for certain measurable
                        dynamical systems. This is joint work with R. Lyons and S. Vassout.

March 18th                              Shuzhou Wang,   University of Georgia
                                                Simple Compact Quantum Groups

                                                Abstract: In the theory of compact quantum groups, two classes of examples have been
                        intensively studied. The first are obtained as various deformations of compact Lie groups
                                                and were main objects of study in the theory of quantum groups in the later 1980s and 1990s.
                                                The second are constructed as universal objects in appropriate categories of quantum
                                                transformation groups and are in general not obtainable as deformations of compact Lie groups.
                        The latter class of quantum groups have been focus of study in recent years. Because of
                                                the existence of the latter class of quantum groups, it is a fundamental problem to classify
                                                simple compact quantum groups.

                                                               In this talk, I will explain the notion of simple compact quantum groups and
                                                indicate how to prove simplicity of (1) the universal orthogonal groups, (2) the quantum
                                                automorphism group of finite spaces and (3) deformations of simple Lie groups.

March 25th                              Xiaoqing Li,     SUNY at Buffalo
                        A spectral mean value theorem for GL(3) automorphic forms

                                                Abstract: Assuming the Langlands functoriality conjectures, it is known the first Fourier coefficient
                                                of a Maass form on GL(n) (normalized such that its L2 norm is 1) is essentially bounded;
                                                without assuming the Langlands functoriality conjectures, we will prove the essentially boundedness
                                                of the first Fourier coefficient of a GL(3) Maass form is true on the average.

April 1st                                   Quanlei Fang,    SUNY at Buffalo
                                                Schatten class membership of Hankel operators  on the unit sphere, I

                        Abstract: Let Hf be a Hankel operator on the Hardy space of the unit sphere in Cn, n≥2. A key
                                                feature of this investigation is that we consider all possible symbol functions f in the L2 of the sphere.
                        We completely determine the membership of Hf in the Schatten class Cp. In the case p>2n,
                        Hf ∈Cp if and only if Hf maps the constant function 1 into the Besov space Bp. In the case p≤2n,
                                                the membership Hf ∈Cp implies Hf = 0.

April 8th                                   Jingbo Xia,     SUNY at Buffalo
                                                Schatten class membership of Hankel operators  on the unit sphere, II

April 22nd                                Erik van Erp,    University of  Pennsylvania
                                                Contact structures of higher codimension: an analytic perspectives
                                                               
                                                                Abstract: A contact manifold is a manifold equipped with a codimension one distribution that satisfies
                                                a certain "maximal non-integrability" condition. Contact manifolds appear naturally as boundaries of
                                                complex domains. Operators of interest on contact manifolds are the tangential Cauchy-Riemann operator
                                                and the Szego projector. The Heisenberg calculus is the appropriate pseudodifferential calculus for
                                                the analysis of these operators.

                                                We propose a simple condition on a distribution of higher codimension that generalizes this geometric structure.
                                                Geometrically, our condition is satisfied, for example, for the Carnot Caratheodory structure on the boundary
                                                of hyperbolic space, or for a natural codimension 3 distribution on the boundary of domains in hyperkahler manifolds.
                                                We characterize these higher codimension "contact structures" in terms of a property of the appropriate
                                                Heisenberg calculus. We construct a natural generalization of the tangential CR operator, and prove that
                                                it has similar spectral properties as its classical counterpart. There is a Szego projector, and the Boutet de Monvel
                                                index theorem holds for the associated Toeplitz operators.