Organizers: Bruce Hughes and Mark Sapir

Wednesdays, 4:00pm-5:00pm in SC 1403 (unless otherwise noted)

• Date: 10 Sept 2003
  • Speakers: Mike Mihalik and Steve Tschantz, Vanderbilt University
  • Title:  Homotopy of ends/boundaries of CAT(0) groups
  • Abstract: For n greater than or equal to 0, we exhibit CAT(0) groups that are n-connected at infinity,
    and have boundary which is (n-1) connected, but this boundary has non-trivial homotopy in dimension n.
    In particular, we construct CAT(0) groups that are simply connected at infinity,
    but have a boundary with non-trivial fundamental group.

• Date: 18 Sept 2003 Mathematics Colloquium
  • Speaker: Vaughan Jones, University of California at Berkeley
  • Title: In and Around the Origin of Quantum Groups
  • Abstract:
    Quantum groups were invented largely to provide solutions of
    the Yang-Baxter equation and hence solvable models in 2-dimensional
    statistical mechanics and one-dimensional quantum mechanics. They
    have been hugely successful. But not all Yang-Baxter solutions
    fit into the framework of quantum groups. We shall explain how
    other mathematical structures, especially subfactors, provide a language
    and examples for solvable models. The prevalence of the Connes tensor
    product of Hilbert spaces over von Neumann algebras leads us to speculate
    concerning its potential role in describing entangled or interacting
    quantum systems.
     
• First East Coast Operator Algebras Symposium at Vanderbilt University, Saturday and Sunday, September 20-21, 2003
            See Vanderbilt ECOAS 2003 for details
   
• Date: 24 Sept 2003
  • Speaker: Ashot Minasyan, Vanderbilt University
  • Title: Some properties of subsets of word hyperbolic groups
  • Abstract:
    First, we will present some results about  quasiconvex subgroups of
    infinite index and their products. Next, we would like to extend the standard
    notion of a subgroup commensurator to an arbitrary subset of a group, and generalize
    some of the previously known results. In particular, we will discuss a connection
    between commensurator of a subset and the action of a hyperbolic group on its
    boundary.
    
     
• Date: 9 Oct 2003  Mathematics Colloquium
  • Speaker:  Steve Ferry, Rutgers University
  • Title: Characterizing euclidean spaces
  • Abstract:
    A compact connected metric space such that every pair of
    points disconnects the space is homeomorphic to a circle. A compact
    connected metric space containing more than one point such that no
    pair of points separates but such that every embedded circle separates
    is homeomorphic to a 2-sphere. We will discuss how efforts to extend these characterizations into
    higher dimensions have led to surprising positive and negative results.

• Date: 10 Oct 2003
  • Speaker:  Stanley Chang, Wellesley College
  • Title: On Conjectures of Mathai and Borel
  • Abstract

   

• Date: 15 Oct 2003
  • Speaker: Mozhgan Mirani, Vanderbilt University
  • Title: Trees and Ultrametric Spaces
  • Abstract:
     Ghys and de la Harpe show
    that a quasi-isometry of trees induces a Holder continuous and quasi-conformal
    homeomorphism on end spaces. By elaborating on their results I will show that
    there is a full and faithful functor from the category of locally finite trees
    and classes of quasi-isometries to the category of compact ultrametric spaces
    and quasi-conformal Holder homeomorphisms.

    
     

• Date: 22 Oct 2003
  • Speaker: Brent Everitt, University of York, England
  • Title: Coxeter groups and geometric manifolds
  • Abstract:  The rich theory of Coxeter groups can be used to provide algebraic constructions of finite volume geometric (particularly hyperbolic) n-manifolds. Combinatorial and arithmetic properties of these groups are used to establish basic properties of the resulting manifolds, such as their volume, size of automorphism group, orientability and so on. The main emphasis is on producing interesting examples, particularly in dimension four and higher.
     

   

• Date: 29 Oct 2003
  • Speaker: Denis Osin, Vanderbilt University
  • Title: Relatively hyperbolic groups I
  • Abstract:
    For any group  G and collection of subgroups H_1, H_2, ... of G we introduce
    the relative Dehn function of  G with respect to H_1, H_2, .... Our study of
    relative Dehn functions leads to the characterization of relative
    hyperbolicity of groups in the sense of Bowditch in terms of relative
    isoperimetric inequality.  We also obtain several new results about
    algebraic and algorithmic properties of relatively hyperbolic groups.

   

• Date: 5 Nov 2003
  • Speaker: Denis Osin, Vanderbilt University
  • Title: Relatively hyperbolic groups II
  • Abstract:

• Date: 6 Nov 2003 (Thursday) Mathematics Colloquium at 4:10pm in SC 1206
  • Speaker:  Nicolas Monod, University of Chicago
  • Title: Orbit equivalence rigidity
  • Abstract: Geometric group theory leads naturally to the notion of Measure Equivalence. This concept generalizes the classical Orbit Equivalence studied in ergodic theory. I will introduce these topics and present new superrigidity statements. We are in particular interested in "negatively curved groups", e.g. hyperbolic in Gromov's sense. This illustrates a new approach initiated in collaboration with Y. Shalom.

   

• Date: 20 Nov 2003 (Thursday) Mathematics Colloquium at 4:10 in SC 1206
  • Speaker: Nigel Higson, Penn State University
  • Title:  Index theorems and noncommutative geometry
  • Abstract: This talk will be aimed at graduate students and others seeking an introductory view of an important topic in noncommutative geometry. I will discuss the Fredholm index theory of elliptic differential operators and some of the basic tools of noncommutative geometry - cyclic cocycles and groupoid algebras - which can play a role in the formulation and proof of the Atiyah-Singer index theorem.

   

• Date: 3 Dec 2003
  • Speaker: Guoliang Yu, Vanderbilt University
  • Title: Novikov type conjectures and uniform convexity
  • Abstract:
    I will explain why Novikov type conjectures are interesting
    and how the geometric condition of uniform convexity can be
    used to study these conjectures. This talk should be accessible
    to nonexperts. This is joint work with Gennadi Kasparov.

   

• Date: 10 Dec 2003
  • Speaker: Greg Friedman, Yale University
  • Title: Cobordism of (high dimensional) knots
  • Abstract:
    The cobordism theory of smooth high-dimensional sphere knots was completely solved (or at least turned into an algebraic problem) in the 1960s through the work of Kervaire, who showed that all even-dimensional knots are cobordant, and J. Levine, who showed that the cobordism type of an odd-dimensional knot is determined by its Seifert matrix. In fact, two odd-dimensional knots are cobordant if and only if their Seifert matrices satisfy a certain algebraic cobordism equivalence.
    We study the cobordism theory of disk knots, i.e. PL locally-flat proper embeddings $D^{n-2}\into D^n$. Disk knots are very closely related to sphere knots with point singularities, and our study is motivated by an attempt to understand singular knots in general. We study both cobordism of disk knots and cobordism rel boundary, holding the boundary sphere knot fixed. It is fairly easy to classify cobordisms except in the odd-dimensional rel boundary case. In this case, we introduce a disk knot analogue of the Seifert matrix and show that it also provides a classification. Further, we study the realization of matrix invariants given a fixed boundary knot. The question is settled in all cases except those for which the middle-dimensional Alexander module of the boundary knot has 2-torsion. The solution is provided by defining a Blanchfield pairing for disk knots and determining close ties with Farber-Levine torsion pairing of the boundary knot (in fact the former determines the latter under certain connectivity assumptions).

   

Spring 2004

Date: TBA (tentative)

• Speaker: Gennadi Kasparov, Vanderbilt University
• Title: Novikov type conjectures and uniform convexity, continued
• Abstract: