Organizer: Dietmar Bisch

Tuesdays, 4:00pm-5:00pm in SC 1432

• Date: 8/31/04
  • Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
  • Title: The classification of noncommutative tori
  • Abstract:
    The irrational rotation algebras, which can be thought of as
    noncommutative versions of the ordinary two dimensional torus, have been
    one of the central examples in the theory of C*-algebras, particularly
    the parts of the theory motivated by thinking of it as “noncommutative
    topology”. Smooth versions of them have played a similar role in Connes’
    noncommutative differential geometry.

    The simple higher dimensional noncommutative tori are one of the
    most natural generalizations. Their structure and properties have been
    rather more difficult to determine than in the two dimensional case. For
    example, it has been known since about 1980 that two irrational rotation
    algebras are isomorphic if and only if they have the same ordered
    K-theory. For the higher dimensional case, this result could be
    completely proved only very recently. Furthermore, unlike for the two
    dimensional case, isomorphism of the C*-algebras does not imply
    isomorphism of the smooth algebras.

    In these two talks, I will begin by describing the irrational
    rotation algebras and their higher dimensional analogs, as well as the
    smooth versions. Then I will explain what is known about the isomorphism
    classification in both the smooth and C* cases, and I will explain the
    connection of the C* classification with the Elliott classification
    program. Finally, I will describe some of the steps in the proof of the
    isomorphism classification in the C*-algebra case. Key roles are played
    by Lin’s classification result for simple C*-algebras with tracial rank
    zero, and by versions of the Rokhlin property for group actions on
    simple C*-algebras.

• Date: 9/7/04
  • Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
  • Title: The classification of noncommutative tori, continued
• Date: 9/14/04
  • no meeting
• Date: 9/21/04
  • Speaker: Guoliang Yu, Vanderbilt University
  • Title: Group actions on Banach spaces and operator K-theory
  • Abstract:
    In this talk, I will discuss groups actions on Banach spaces
    and its applications to K-theory of group C*-algebras (joint work with
    Gennadi Kasparov).
• Date: 9/28/04
  • Speaker: Remus Nicoara, Vanderbilt University
  • Title: A rigidity result for irrational rotation HT factors
  • Abstract:
    We present a result on the irrational rotation HT factors
    M_\alpha (\Gamma)=L_\alpha Z^2) \rtimes \Gamma, where \Gamma
    are arbitrary non-amenable subgroups of SL(2,Z) and
    \alpha=e^{2\pi it}, t irrational, showing that for each fixed
    \Gamma there exists no separable II_1 factor that contains M_\alpha
    (\Gamma) for uncountably many \alpha’s. In particular,
    \{ M_\alpha (\Gamma)\}_\alpha are non-isomorphic modulo countable sets.
    
    (Joint work with Sorin Popa, UCLA and Roman Sasyk, Purdue
    University).
• Date: 10/5/04
  • Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
  • Title: Classifiability of crossed products by minimal homeomorphisms
  • Abstract:
    This is joint work with Huaxin Lin.

    Let $X$ be an infinite compact metric space with finite covering
    dimension and let $h \colon X \to X$ be a minimal homeomorphism.
    We show that the associated crossed product C*-algebra
    $A = C^* ({\mathbb{Z}}, X, h)$ has tracial rank zero whenever
    the image of $K_0 (A)$ in ${\mathrm{Aff}} (T (A))$ is dense.
    No assumption about either smoothness or the number of extreme
    tracial states is necessary.
    As a consequence, these crossed product C*-algebras are
    classifiable, and are isomorphic to
    simple AH~algebras with real rank zero.

• Date: 10/12/04
  • Speaker: Dietmar Bisch, Vanderbilt University
  • Title: Intermediate subfactors
  • Abstract: It is well-known that the Temperley-Lieb algebras
    are the basic algebras of symmetries associated to a subfactor.
    If an intermediate subfactor is present Jones and I showed a few
    years ago that another tower of algebras – what we called the
    Fuss-Catalan algebras – capture the additional symmetry given
    by the intermediate subfactor. What happens if two or more intermediate
    subfactors occur? The story becomes a lot more complicated and
    fascinating. We will discuss some recent work on this question.

• 
  10/16-10/17/04 Southeastern Section AMS Meeting at Vanderbilt University
  
  
  
  Schedule of talks, Special Session “Index theory and the topology of
  manifolds”
  
  
  Schedule of talks, Special Session “Operator theory on function spaces”
  
  
  Schedule of talks, Special Session “Von Neumann algebras and noncommutative
  ergodic theory”
  
  
  
  Click here for the meeting program

• Date:
  10/19/04 (Tuesday) Special Talk during the Fall Break,
  4:00pm-5:00pm in SC 1432
  • Speaker: Jesse Peterson, UCLA
  • Title: A 1-cohomology characterization of property (T) in von Neumann
    algebras
  • Abstract:
    We obtain a characterization of property (T) for von Neumann
    algebras in terms of 1-cohomology, similar to the Delorme-Guichardet
    Theorem for groups.

• Date:
  10/20/04 (Wednesday) Special Talk in the Topology & Group Theory Seminar, 4:10-5:00pm in SC 1424
  • Speaker: Sorin Popa, UCLA
  • Title: Some calculations of 1-cohomology for actions of groups
  • Abstract:
    For each group $G$ having an infinite normal subgroup with relative
    property (T) and each countable abelian group $\Gamma$ we
    construct measure-preserving actions $\sigma_\Gamma$ of $G$
    on the probability space such that the 1’st cohomology group
    of $\sigma_\Gamma$, $H^1(\sigma_\Gamma,G)$, is
    equal to Char$(G) \times \Gamma$. We deduce that $G$ has uncountably
    many non stably orbit equivalent actions. We also calculate 1-cohomology
    groups and show existence of “many” non stably orbit equivalent
    actions for free products of groups as above.
• Date: 10/26/04
  • Speaker: Remus Nicoara, Vanderbilt University
  • Title: Some finiteness results for commuting squares
  • Abstract:
    Commuting squares are squares of inclusions of matrix algebras,
    which arise naturally as invariants in Jones’ theory of subfactors.
    Commuting squares can also be used as construction data for subfactors, and
    many explicit examples of subfactors have been obtained this way. By
    using a combination of algebraic-combinatorial and analytic methods, we
    prove some finiteness results for commuting squares and find a good notion
    of primeness (in the sense of isolation) for these objects. As a
    consequence, elementary proofs are obtained for theorems such as D. Stefan’s
    theorem on the finiteness of the number of Hopf structures on a finite
    dimensional matrix algebra. The isolation results obtained also suggest
    methods of constructing new families of non-isomorphic subfactors.

• Date:
  10/28/04 (Thursday) Mathematics Colloquium,
  4:10pm-5:00pm in SC 1206
  • Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
  • Title: Rational cohomology for Banach algebras
  • Abstract:
    Let A be a commutative unital Banach algebra. Its maximal ideal space
    Max (A) is a compact Hausdorff space which plays a role somewhat similar
    to that of the space Spec (R) in algebraic geometry. In particular, A can
    be represented, not necessarily faithfully, as an algebra of continuous
    functions on Max (A). The Taylor problem asks for a construction of the
    (Cech) cohomology of Max (A) “directly from A”. For example, H^1 (Max (A); Z)
    is the quotient of the invertible group of A by the image of the exponential
    map. This, and related descriptions of H^0 (Max (A); Z) and H^2 (Max (A); Z),
    have been known since the 1970s, but the program seemed to stop there.
    In this talk, after giving some background, we describe a solution for
    the _rational_ (Cech) cohomology H^s (Max (A); Q) for arbitrary s, in terms
    of the rational homotopy groups of the spaces of last columns of invertible
    n by n matrices over A for suitable n. The approach has the promise of
    giving something interesting for noncommutative Banach algebras as well.
    This is joint work with Greg Lupton, Claude Schochet, and Samuel Smith.
• Date: 11/2/04
  • Speaker: Alex Furman, University of Illinois at Chicago
  • Title: Ergodic equivalence relations, measure equivalence and higher
    rank lattices
  • Abstract:
    We relate the notion of Orbit Equivalence between group actions in
    ergodic theory,
    the notion of Measure Equivalence – a measure theoretical analogue of
    quasi-isometries between groups, and discuss aspects of rigidity for
    actions of higher rank lattices.
• Date: 11/9/04
  • Speaker: Jeff Raven, Penn State University
  • Title: An equivariant bivariant Chern character
  • Abstract:
    Using notions from homological algebra and sheaf theory Baum and Schneider
    defined a bivariant equivariant cohomology theory for spaces which shares many
    of the properties of equivariant KK-theory. When the group under consideration
    is profinite (that is, compact totally disconnected) they then showed that
    this theory is naturally isomorphic to complexified equivariant KK-theory,
    and conjectured that the same should be true for all totally disconnected
    groups. In my talk I’ll survey some of the background material and
    discuss how to construct an isomorphism when the group is discrete.
• Date: 11/16/04
  • Speaker: Vrej Zarikian, University of Cincinnati
  • Title: Operator spaces, multipliers, and ideals
  • Abstract:
    After reviewing the main ideas of operator space
    theory, I will discuss Blecher’s theory of
    one-sided operator space multipliers. Then I will
    present an extremely useful “geometric” characterization
    of these multipliers, obtained jointly with
    Blecher and Effros. Applications to non-self-adjoint
    operator algebras, dual operator spaces, and one-sided
    M-ideals will be explained.

• Date:
  11/19/04 (Friday) Special Seminar Talk,
  3:00pm-4:00pm in SC 1308
  • Speaker: Jon Bannon, University of New Hampshire
  • Title: Transitive families of projections in factors of
    type II_1
  • Abstract:
    In this talk we will explore a bit of operator theory relative to
    a type II_1 factor. We introduce a notion of transitive family of subspaces
    relative to a type $II_1$ factor, and hence a notion of transitive family of
    projections in such a factor. We show that whenever M is a factor of type
    $II_1$ and $M$ is generated by two self-adjoint elements, then
    $M\otimesM_2(\mathbb{C})$ contains a transitive family of 5 projections.
    Finally, we exhibit a free transitive family of 12 projections that generate
    a type $II_1$ factor.

• Date: 11/23/04
  • no meeting, Thanksgiving break
• Date: 12/7/04
  • Speaker: Piotr Nowak, Vanderbilt University
  • Title: Coarse embeddings and uniform homeomorphisms in the geometry of
    Banach spaces
  • Abstract:
    Coarse embeddings were defined by Gromov in the study of the
    large scale geometry of groups and turned out to have important
    applications to some major conjectures in non-commutative geometry
    and topology. I will present recent results on coarse embeddings
    into Hilbert and, more generally, L_p-spaces and explain the close
    connection with uniformly homoemorphic embeddings studied in the 1980’s by
    Enflo, Aharoni, Lindenstrauss and others in the context of Banach spaces.

• Date:
  12/9/04 (Thursday) Mathematics Colloquium,
  4:10pm-5:00pm in SC 1206
  • Speaker: Victor Nistor, Penn State University
  • Title: Analysis on polyhedral domains
  • Abstract:
    The analysis of elliptic partial differential operators on smooth, bounded domains is well understood and has numerous applications, many outside mathematics. By contrast, the behavior of these operators on non-smooth domains can be quite different from the one on smooth domains and is much less understood. In my talk, I will first review some known results on boundary value problems on non-smooth domains. Then I will present an approach to analysis on polyhedral domains that is based on a modification of the usual Sobolev spaces, yielding the so called “Sobolev spaces with weights.” One can, for instance, obtain a regularity theorem within these Sobolev spaces with weights that is completely analogous to the usual elliptic regularity on smooth domains. This result, joint work with C. Bacuta and L. Zikatanov, has potential applications to numerical methods. I will also briefly discuss at the end some connections with Operator Algebras, more precisely, with groupoid C^*-algebras. The talks is meant to be accessible to a mathematical literate audience, including graduate students.