Colloquia, Spring 2005

Mathematics Colloquia, Spring 2005

Thursdays 4:10 pm in 1206 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center

January 13, 2005 
Sergei Treil, Brown University

Fun around Corona


The classical Carleson Corona Theorem states that
if bounded analytic in the unit disc functions
, then there exist bounded
functions such, that
. This is equivalent to the fact that the
unit disk is dense in the
maximal ideal space of the algebra of bounded analytic
functions, but the importance of
Corona Theorem goes much beyond the theory of maximal ideals of

It turned out that the Corona Theorem, and especially its
generalization, the so called
Matrix (Operator) Corona Theorem, play an important role in
theory (such as the angles between invariant subspaces, unconditionally
spectral decompositions, computation of spectrum, etc.).

In this talk I am going to discuss some new results related to the
Corona Problem,
describe connection with the operator theory and with the geometry of
holomorphic vector bundles, as well as state some new (and some old)
open problems.

It worth mentioning that “around corona” there are much more open
questions than answers. In particular, it is still an open problem to
find out if the Corona Theorem holds for a polydisc or a unit ball in
or for general domain in .

Contact person: Dechao Zheng

Friday, January 14, 2005
Brett Wick, Brown University

Holomorphic Vector Bundles, Analytic
Projections and the
Corona Problem

NOTE: 3:10-4 pm in room
Tea at 2:30 in room 1425

Abstract: The Corona problem is one of the more important problems
in complex
analysis and operator theory. Most of the proofs of this theorem
rely on the function theory behind the problem. However there is a
substantial geometric approach to the problem that provides an
elegant answer to the Corona problem. In this talk I will
discuss some aspects of the Corona problem and the relationship
with holomorphic vector bundles and analytic projections.

Contact person: Dechao Zheng

Tuesday, January 18, 2005 
Jeff Raven, Penn State University

Topological K-homology

NOTE: In room 1432

Abstract: In the 1980s Baum & Douglas defined a topological
K-homology group using cycles built from vector bundles on Spinc
manifolds. One might hope that these groups would agree with the
corresponding analytic K-homology, at least for finite CW-complexes;
however to get anywhere along these lines one must first show that the
topological K-homology groups are actually a homology theory (a result
whose proof is missing from the original paper). I will describe one means
of proving this, and then show how these arguments can be generalized to
the equivariant case.

January 20, 2005 
Greg Friedman, Yale University

Singular Knots and Stratified Spaces

Abstract: I will present an overview of some of my work on singular
knots and stratified spaces. Singular knots are codimension two embeddings
of spheres that may possess singularities – points at which the embedding
is not smoothable. I will discuss some of the properties of such knots and
how they differ from those of smooth knots, as well as provide some
motivation for the mathematical interest in such objects. From there, I
will generalize to stratified spaces – spaces that are not quite
manifolds, but which possess sets of singularities that can be filtered
into manifold strata. I will introduce intersection homology, a useful
algebraic tool for studying such spaces, and discuss some of my own work
on intersection homology theory.

Tuesday, January 25, 2005 
Robert Yuncken, Penn State University

Group C*-algebras and the

In room 1432

Abstract: In this talk I will discuss the problem of understanding
the representation theory of a locally compact group from the point of
view of “noncommutative topology”. I will begin by explaining what this
means. I will then turn my attention to connected Lie groups, in
particular the groups SL(2,C) and SL(3,C), and their discrete subgroups. I
will indicate how the differential complex of Bernstein, Gelfand and
Gelfand is pertinent to these examples.

February 10, 2005  (date reserved for biomath seminar)
February 17, 2005 
Liming Ge, University of New Hampshire

Factors and Groups

Abstract: The connections between (discrete) groups and their group von Neumann algebras will be the focus of the talk. Some of
the old and new problems in these two areas are discussed.

Contact person: Dietmar Bisch

February 24, 2005 
Alexander Powell, Princeton University

Finite Frames and Sigma-Delta Quantization

Abstract: Redundancy is a key to practical and reliable data representation in many settings. The standard example is sampling
theory for (bandlimited) audio signals, where oversampling is used to ensure numerical stability. Frame theory provides a mathematical
framework for stably representing signalsas linear combinations of “basic building blocks” that constitute an overcomplete collection.
Finite frames are the piece of this theory that is tailored for finite, but potentially high dimensional, data. We address the problem
of how to accurately and efficiently quantize redundant finite frame expansions. We analyze the performance of certain first and second
order Sigma-Delta quantization schemes in this setting, and derive rigorous approximation error estimates as well as new stability

Contact person: Phillip Crooke

March 3, 2005 
Graham Niblo, Southampton (UK)

Property T from a geometric group theorist’s point of view

Abstract: Kazhdan’s property T (which concerns the unitary representation theory
of a locally compact group) has a geometric interpretation for
countable discrete groups. We will describe this interpretation and
explore it in the context of groups acting on CAT(0) cubical complexes.

Contact person: Mark Sapir

March 17, 2005 
Xingxing Yu, Georgia Tech and Nankai University

Spanning double rays in infinite planar graphs

Abstract: Partially motivated by the Four Color Conjecture, Whitney proved
that every triangulation of the sphere has a Hamilton cycle. This
result was generalized to all 4-connected planar graphs by Tutte,
and later to locally planar graphs on other surfaces. In an attempt
to extend Tutte’s result to infinite planar graphs, Nash-Williams
in 1971 made two conjectures, concerning the existence of spanning
rays and double rays in infinite planar graphs. I shall give an
outline of proofs of these two conjectures. I shall then describe
the concept of a circle in an infinite graph, introduced by
Diestel, by associating with the infinite graph a topological space
(which is in a sense a compactification of the graph). In this
setting, Nash-Williams conjecture becomes a special case of a
larger conjecture.

Contact person: Mark Ellingham

March 24, 2005 
Craig Evans, UC Berkeley

Games and some degenerate nonlinear PDE

Abstract: I will discuss some simple nonlinear partial differential equations, depending on a parameter p lying between 1 and
infinity, and then explain that the most interesting cases occur in the singular limits as p tends to either 1 or infinity. These two
limit cases in fact have recently discovered game theoretic interpretations, which I will discuss and connect to other mathematical

Contact person: Gieri Simonett

March 31, 2005 
Yuri Bahturin, Memorial University (Canada) and the Moscow University (Russia)

Group Gradings on Simple Algebras and Superalgebras

Abstract: In this talk we will discuss new techniques that made it possible to
completely describe gradings by groups on finite dimensional simple
algebras from various classes: matrix algebras, classical simple Lie
algebras and superalgebras, etc. The talk will be accessible to a broad
mathematical audience (including graduate students).

Contact person: Alexander Olshanskiy

April 21, 2005 
Alex Travesset, Iowa State University

Ground States of Particles on a Frozen Topography

Abstract: Finding the configuration that minimizes the energy of a large number
of particles that are constrained to a particular topography is
a surprisingly difficult problem with many important realizations in the
traditional sciences. A classical example is the Thomson problem, consisting in
minimizing the energy of M particles constrained to move on the sphere
interacting with the 3 dimensional Coulomb potential. This problem has usually been
solved by numerical methods only for small (less than 200) number of particles. In
this talk, I will present a general approach to find the ground state in
the opposite limit of large number of particles, which applies to any geometry and
wide family of potentials. It will be shown that for many interaction potentials the energy of the
ground state is universal with respect to the geometry and consists of
defects (vertices that are five or seven coordinated) whose precise distribution can be predicted.
Explicit solutions will be given for the sphere and for particles interacting with
a short-ranged potential on a torus.

Contact person: Doug Hardin


Colloquium Chair (Spring 2005): Doug Hardin

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