Analysis   Seminar    

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

August 28th                 Raffael Hagger,   Reading University
                                     Fredholm theory and localization on metric measure spaces


September 25th            Mariusz Tobolski,     Institute of Mathematics Polish Academy of Sciences  
                                      Towards the classification of locally trivial noncommutative principal bundles

                                      Abstract: A classical result from topology states that if X is a principal G-bundle over a paracompact space M, then there exists a map from M to the
                                      classifying space BG, and all isomorphic principal G-bundles are classified by the homotopy class of such a map. The aim of this talk is to find an
                                      analog of this result in the realm of noncommutative topology, where instead of topological spaces and groups, we consider C*-algebras and
                                      quantum groups respectively. First, we introduce the notion of a locally trivial noncommutative principal bundle in the setting of compact quantum
                                      group actions on C*-algebras. Then, for a compact quantum group G, we define the C*-algebra of functions on the noncommutative classifying
                                      space C(BG) and prove that it classifies all locally trivial noncommutative principal G-bundles.

October 2nd                  Yi Wang,     SUNY at Buffalo 
                                       A sharp inequality of Hardy-Littlewood type via derivatives

                                      Abstract: In this talk, I will introduce a sharp inequality relating a parameterized set of weighted Bergman norms and the Hardy norm on the unit disk.
                             The original form of this inequality can be traced back to 1921, when Carleman provided a complex analytic proof of the famous isoperimetric theorem.
                             In recently years, the inequality has regained attention because of its application in number theory. By taking a close examination of the derivatives of
                             the norms with respect to the parameter, we obtain some sufficient conditions for the inequality to hold. This is joint work with Hui Dan and Kunyu Guo.

October 9th                   Xiaocheng Li,   University of Wisconsin-Madison
                                      An Estimate for Spherical Functions on SL(3,R)

                                      Abstract: We prove an estimate for spherical functions \phi_\lambda(a) on SL(3,R), establishing uniform decay in the spectral parameter \lambda
                                      when the group parameter a is restricted to a compact subset of the abelian subgroup A. In the case of SL(3,R), it improves a result by
                                      J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that a should remain regular. As in their work, we estimate the
                                      oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference
                                      is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the
                                      parameters \lambda and a vary.

October 16th                Quanlei Fang,    City University of New York
                                      Revisiting Arveson's Dirac operator of a commuting tuple

: About twenty years ago, Arveson introduced an abstract Dirac operator based on Taylor spectrum and functional calculus. He showed that
                                      every Dirac operator is associated with a commuting tuple. The Dirac operator of a commuting tuple has inspired several interesting problems in
                                      multivariable operator theory. In this talk, we will revisit the Dirac operator and discuss some related problems.

October 23rd                Han Li,   Wesleyan University
                                     Masser's conjecture on equivalence of integral quadratic forms

                                    Abstract: A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent. Formulated in
                                    terms of matrices the problem asks, for given symmetric n-by-n integral matrices A and B, whether there is a unimodular integral matrix X satisfying
                                    A=X'BX, where X' is the transpose of X. For definite forms one can construct a simple decision procedure. Somewhat surprisingly, no such procedure
                                    was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s D. W. Masser conjectured for n at least 3, there exists
                                    a polynomial search bound for X in terms of the heights of A and B. In this talk we shall discuss our recent resolution of this problem based on a joint
                                    work with Professor Gregory A. Margulis, and explain how ergodic theory is used to understand integral quadratic forms.

October 30th                 Chunlan Jiang,   Hebei Normal University
                                       Similarity invariants of essentially normal Cowen-Douglas operators and Chern polynomials

                                      Abstract: In this talk, I will discuss our resent work on a class of essentially normal operators by using the geometry method from the Cowen-Douglas
                                      theory and a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental
                                      forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.

November 6th                Alexandru Chirvasitu,   SUNY at Buffalo
                                       Loosely embeddable metric spaces

                                      Abstract: Embedding finite metric spaces isometrically into Hilbert spaces has elicited some interest outside of pure mathematics due to applications
                                      to fields like computer vision, machine learning, the structure of networks and other such areas.

                                      In the talk I will introduce a weaker notion of embeddability motivated by the study of "quantum symmetries" for metric spaces and Riemannian
                                      manifolds. I will mention some results on the generic behavior of "most" compact metric spaces and list a number of open questions.

November 20th              Hui Dan,      Fudan University
                                       The cyclicity of composite functions in the Hardy space over the infinite polydisc

                                       Abstract: The cyclic vector problem on the Hardy space over the infinite polydisc is the analytic function space version of Wintner and Beulring's
                                       periodic dilation completeness problem. In this talk, I will mainly concentrate on characterizing cyclic vectors in terms of composition operators.
                                       Also, in order to study composition operators, dilation theory for doubly commuting sequence of C.0 contractions is involved.                

Past Analysis Seminar