Organizer: Dietmar Bisch

Tuesdays, 3:00pm-4:30pm in SC 1420

• Date:
  9/18/03 (Thursday) Mathematics Colloquium at 4:10pm in
  SC 1206
  • Speaker: Vaughan Jones, UC 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.

• Date: Saturday & Sunday, 9/20/03 & 9/21/03, SC 4309
  
  First East Coast Operator Algebras Symposium (at Vanderbilt University)

• Date: 9/23/03
  • Speaker: Narutaka Ozawa, University of Tokyo (visiting UC Berkeley)
  • Title: On solid von Neumann algebras
• Date: 9/30/03
  • Speaker: Doug Drinen, University of the South
  • Title: An introduction to C*-algebras of directed graphs
  • Abstract:
    In 1980, Cuntz and Krieger introduced a class of C*-algebras
    generated by a collection of partial isometries satisfying certain
    conditions determined by a square 0-1 matrix. In the mid-90s,
    Kumjian, Pask, Raeburn, and Renault defined, given a directed
    graph, a C*-algebra generated by collections of partial isometries
    and projections satisfying relations arising from the structure of
    the graph. They did so in a way that generalized the Cuntz-Krieger
    algebras. Further, they proved
    that certain properties of the C*-algebra could be deduced from the
    structure of the graph. This made it possible to rephrase theorems
    about Cuntz-Krieger algebras, the statements of which had been
    quite complicated in some cases, in much more friendly graphical
    terms.

    This talk will cover in very broad general terms the past, present,
    and future of graph algebras. We will start with Cuntz algebras,
    move on to graph algebras, and then touch upon the many ways in
    which graph algebras have been generalized.

• Date: 10/7/03
  • Speaker: Dietmar Bisch, Vanderbilt University
  • Title: Exotic subfactors I
  • Abstract:
    A subfactor is a group-like object which can encode what one might call
    the “generalized symmetries” of a quantum physical or mathematical
    situation, from which the subfactor was constructed, much like a group does.
    However a subfactor is a highly noncommutative, infinite dimensional
    mathematical object and the symmetries it captures are more general than
    group symmetries. Operator algebraic methods, which are both analytical
    and algebraic-combinatorial in nature, can then be used to analyze these
    symmetries.

    As in group theory there are many exotic subfactors some of which are
    not very well understood. Planar algebra techniques shed some light
    on the structure of theses subfactors.

• Date: 10/14/03
  • Speaker: Dietmar Bisch, Vanderbilt University
  • Title: Exotic subfactors II
• Date: 10/21/03
  • no meeting, fall break
• Date: 10/28/03
  • Speaker: Dechao Zheng, Vanderbilt University
  • Title: Toeplitz algebras on the disk
  • Abstract:
    Let $B$ be a Douglas algebra and let $\cal B$
    be the algebra on the disk generated by the harmonic extensions of the
    functions in $B$.
    In this talk I will show that $\cal B$ is generated by $H^{\infty}(D)$ and
    the complex conjugates of the harmonic extensions of the interpolating
    Blaschke products invertible in $B$. Every element
    $S$ in the Toeplitz algebra ${\cal T}_{B}$ generated by
    Toeplitz operators with symbols in $\cal B$ on the Bergman
    space has a canonical decomposition $S=T_{\tilde{S}}+\R$
    for some $\R$ in the commutator ideal ${\mathcal C}_{{\cal T}_{B}}$; and
    $S$ is in ${\mathcal C}_{{\cal T}_{B}}$
    iff the Berezin transform $\tilde{S}$ vanishes
    identically on the union of the maximal ideal space of the Douglas
    algebra $B$ and the set ${\cal M}_{1}$ of trivial Gleason
    parts. This is a joint work with S. Axler.

• Date:
  10/31/03 (Friday) Special Talk, 3:00pm-4:00pm,
  SC 1313
  • Speaker: Adebisi Agboola, UCSB
  • Title: Rational points on elliptic curves
  • Abstract:
    Mathematicians have been interested in finding integer solutions to
    polynomial equations in several variables for thousands of years e.g.
    Euclid gave a complete solution to the equation x^2 + y^2 = z^2. It is now
    known that, (thanks to the negative resolution of Hilbert’s
    10th Problem due to Matiyasevich (building on earlier work of Davis,
    Putnam and Robinson)), it is impossible to do this sort of thing in
    general.

    In special cases, however, one can nevertheless hope to say a great deal.
    For instance, when the equation in question defines an elliptic curve,
    e.g.

    y^2 = x^3 + 19x

    then an amazing conjecture due to Birch and Swinnerton-Dyer implies that
    the behaviour of the solutions is governed by the properties of an
    analytic object, namely the L-function associated to the elliptic curve.
    The mere existence of this L-function is itself an extremely deep problem
    which was only recently resolved by work of Wiles, Taylor, Diamond,
    Conrad, and Breuil.

    In my talk I shall explain some of the ideas that go into formulating the
    conjecture of Birch and Swinnerton-Dyer, and I shall describe some aspects
    of our current state of knowledge. I shall not assume any previous
    knowledge of the subject.

• Date:
  11/6/03 (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:
  11/20/03 (Thursday) Mathematics Colloquium at 4:10pm 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: 11/25/03
  • no meeting, Thanksgiving break

• Date:
  12/3/03 (Wednesday) Topology & Group Theory Seminar,
  4:00pm-5:00pm, SC 1403
  • 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.