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

August 27th

A Rank Problem on Submodules in the Hardy Space over the Bidisk

September 11th

(Thursday, Colloquium) Operator Spaces and Their Applications to Abstract Harmonic Analysis

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 M

compact group.

September 17th

One Hundred Years of Diagonalization, Part I

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 (A

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

One Hundred Years of Diagonalization, Part II

One Hundred Years of Diagonalization, Part III

October 8th

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

Toeplitz operators with BMO symbols on the Segal-Bargmann space

Abstract: We show that Zorboska's criterion for compactness of Toeplitz operators with

BMO

Segal-Bargmann space of Gaussian square-integrable entire functions on C

some basic properties of BMO

and compact Toeplitz operators with BMO

results of Lo and Engliš, we also give a compactness criterion for the Gabor-Daubechies

"windowed Fourier localization operators" on L

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

An Excursion to Q

Abstract: The Q

It generalizes the classical space BMOA and the Bloch space B in the sense that Q

and Q

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 Q

We will focus on some basic properties and some recent development on these spaces.

We will also look at the generalizations of Q

C

October 29th

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

Riesz Transforms in the Noncommutative Setting

Abstract: 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

Quadrilaterals of Factors

Abstract: A quadrilateral of factors is a pair of intermediate subfactors P, Q of an

irreducible finite-index inclusion of II

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

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.