Analysis   Seminar    

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

August  27th                                    Michio Seto,   Shimane University,  Japan 
                                                       A Rank Problem on Submodules in the Hardy Space over the Bidisk

September  11th                              Zhong-Jin Ruan,   University of Illinois at Urbana-Champaign 
(Thursday, Colloquium)                   Operator Spaces and Their Applications to Abstract Harmonic Analysis 
                                                       Abstract: The theory of operator spaces is a natural (noncommutative) quantization
                                                       of Banach space theory. It has already had many important applications to operator
                                                       algebras, abstract harmonic analysis and some other related areas. In this talk, I will
                                                       first review some basic notions in operator spaces. Then I will show a representation 
                                                       theorem related to the measure algebra M(G) and the completely bounded Herz-Schur 
                                                       multiplier algebra  McbA(G) on the operator space B(L2(G)), where G is a locally
                                                       compact group.         

September 17th                               Jingbo Xia,   SUNY at Buffalo
                                                       One Hundred Years of Diagonalization, Part I

                                                       Abstract: In 1909, Hermann Weyl published a theorem which, stated in modern terms, 
                                                       says that every self-adjoint operator A on a separable Hilbert space H can be diagonalized
                                                       modulo a compact operator. Since then this result has been improved and generalized 
                                                       in many different ways. Most noticeably, improvements and generalizations have come
                                                       in the direction of the simultaneous diagonalization of commuting tuples (A1, ..., An)
                                                       of self-adjoint operators modulo norm ideals. Modern-day diagonalization requires 
                                                       techniques and ideas from harmonic analysis, such as dyadic decomposition, stopping-time
                                                       argument, singular integral operators, etc. In these talks we will review the progress made
                                                       in the hundred years since Weyl's paper, culminating in the latest result.

September 24th                               Jingbo Xia,   SUNY at Buffalo
                                                       One Hundred Years of Diagonalization, Part II 
October 1st                                     Jingbo Xia,   SUNY at Buffalo 
                                                       One Hundred Years of Diagonalization, Part III

October 8th                                     Anthony Weston,    Canisius College
Determining lower bounds on the maximal p-negative type of finite metric spaces

                                                       Abstract: The metric notions of negative type (1910s) and generalized roundness (1960s)
                                                       were introduced to study isometric and uniform embeddings, respectively. More recently,
                                                       these notions (which are actually equivalent) have found serious applications in coarse
                                                       geometry and combinatorial optimization.
                                                                       The purpose of this talk will be to present a simple (but alarmingly effective)
                                                       method for determining lower bounds on the maximal p-negative type of finite metric spaces.
                                                       These bounds only depend upon the cardinality and the (scaled) diameter of the underlying
                                                       metric space. Moreover,  these bounds can easily be optimal, as we shall explain with
                                                       examples towards the end of the talk. I will also examine the special case of finite metric
                                                       trees in some detail.

October 15th                                   Joshua Isralowitz,    SUNY at Buffalo
                                                       Toeplitz operators with BMO symbols on the Segal-Bargmann space

                                                       Abstract: We show that  Zorboska's criterion for compactness of Toeplitz operators with
                                                       BMO1 symbols on the Bergman space of the unit disc holds, by a different proof, for the
                                                       Segal-Bargmann space of Gaussian square-integrable entire functions on Cn. We establish
                                                       some basic properties of BMOp for p1 and complete the characterization of bounded
                                                       and compact Toeplitz operators with BMO1 symbols.  Via the Bargmann isometry and
                                                       results of Lo and Engliš, we also give a compactness criterion for the Gabor-Daubechies
                                                       "windowed Fourier localization operators" on L2(Rn, dv) when the symbol is in a BMO1
                                                       Sobolev-type space. Finally, we discuss examples of the compactness criterion and
                                                       counter-examples to the unrestricted application of this criterion for the compactness of
                                                       Toeplitz operators. This is joint work with Lewis Coburn and Bo Li.

October 22nd                                  Ruhan Zhao,   SUNY at Brockport
                                                       An Excursion to Qp Spaces

                                                       Abstract: The Qp (0<p<) spaces were introduced by Aulaskari, Xiao and Zhao in 1995.
                                                       It generalizes the classical space BMOA and the Bloch space B in the sense that Q1=BMOA
                                                       and Qp=B for any p>1. For 0<p<1, Qp spaces are subspaces of BMOA. Since it was
                                                       introduced in 1995, many authors have studied these spaces and found connections with
                                                       many other classical function spaces. In this talk we will take a tour on these Qp spaces.
                                                       We will focus on some basic properties and some recent development on these spaces.
                                                       We will also look at the generalizations of Qp spaces to analytic functions on unit ball of
                                                       Cn and to real valued functions on Rn.

October 29th                                   Masoud Khalkhali,   University of Western Ontario
The Algebra of Formal Twisted Pseudodifferential Symbols and a Noncommutative Residue

                                                        Abstract: We extend the Adler-Manin trace on the algebra of pseudodifferential symbols to
                                                        a twisted setting.

November 5th                                  Tao Mei,     University of Illinois at Urbana-Champaign
Riesz Transforms in the Noncommutative Setting

In this talk I will review the boundedness of Riesz transforms in the classical setting
                                                        and P. A. Meyer's formulation of Riesz tranforms by semigroups of operators. Then I will
                                                        introduce a recent work (joint with M. Junge) on noncommutative Riesz transforms and
                                                        applications to quantum metic spaces.

November 12th                                Pinhas Grossman,   Vanderbilt University
Quadrilaterals of Factors
Abstract: A quadrilateral of factors is a pair of intermediate subfactors P, Q of an
                                                        irreducible finite-index inclusion of II1 factors NM such that P and Q generate M
                                                        and intersect in N. For certain noncommuting quadrilaterals, there is a rigidity to the inclusions
                                                        which imposes severe restrictions on invariants such indices and angles. This is joint work
                                                        with Masaki Izumi.
                                                                        Some of these quadrilaterals can also be used to estimate the strong-singularity
                                                        constants of Sinclair and Smith for subfactors. In particular, there is a subfactor which is singular
                                                        but not strongly singular. This is joint work with Alan Wiggins.

November 19th                                Quanlei Fang,   SUNY at Buffalo
Explicit transfer function realization for two-variable rational functions via zero/pole data

                                                        Abstract: A scalar-valued rational function of two complex variables, as the ratio of two
polynomials which are relatively prime, is determined by its pole curve and zero curve up to
multiplicity.   A transfer-function realization for the function (of Givone-Roesser or
                                                        Fornasini-Marchesini type), if minimal in the Popov-Belevitch-Hautus sense, determines
                                                        a linearization (i.e., a determinantal representation) for the pole curve and the zero curve. 
                                                        We discuss the converse question of constructing a realization (of either Givone-Roesser or
                                                        Fornasini-Marchesini type) for a function having prescribed zero and pole curvess. 
                                                        The basic idea follows the solution for the one-variable case due to Ball-Gohberg-Rodman,
but with additional ingredients from the theory of determinantal representations for algebraic
                                                        curves. This is a joint work with Joseph Ball and Victor Vinnikov.