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 Abstract: The classical Carleson Corona Theorem states that if bounded analytic in the unit disc functions satisfy , then there exist bounded analytic 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 the 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 operator theory (such as the angles between invariant subspaces, unconditionally convergent 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 1206 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 BGG-resolution 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 theorems. 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 issues. 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