Noncommutative Geometry & Operator Algebras Seminar
Spring 2004

Organizer: Dietmar Bisch

Tuesdays, 4:00pm-5:00pm in SC 1420

• Date: 1/15/04 (Thursday) Special Colloquium at 4:10pm in SC 1206
  • Speaker: Junhao Shen, University of Pennsylvania
  • Title: Free Probability, Free Entropy and the Generator Problem of von Neumann Algebras
  • Abstract: We will give a brief introduction to Voiculescu's Free Probablity theory, and explain how random matrices are connected to Free Probablity theory. Then the concept of Voiculescu's Free Entropy is introduced. We mainly focus on the application of Free Entropy theory to finite von Neumann algebras. The result we present is the computation of Free Entropy of the group von Neumann algebra associated to the special linear group SL(Z, 2n+1). After that we will show that this von Neumann algebra is generated by two selfadjoint operators, which settles a problem proposed by Voiculescu.

• Date: 1/20/04
  • Speaker: John Ratcliffe, Vanderbilt University
  • Title: Hyperbolic 4-manifolds and Quantum Cosmology
  • Abstract: In Stephen Hawking's theory of quantum cosmology, Einstein 4-manifolds with a totally geodesic boundary are used to model the creation of the universe. Hyperbolic manifolds are Einstein manifolds. In this talk, I will discuss some applications of the theory of hyperbolic 4-manifolds to our understanding of quantum cosmology.

• Date: 1/22/04 (Thursday) Special Colloquium at 4:10pm in SC 1206
  • Speaker: Remus Nicoara, UCLA
  • Title: Some Finiteness Results for Commuting Squares of Matrix Algebras
  • Abstract: We introduce a primeness condition for commuting squares of finite dimensional *-algebras. Such objects arise as invariants and construction data in Jones' theory of subfactors. We prove that the commuting squares satisfying the primeness condition are isolated among all commuting squares (modulo isomorphisms). We use similar methods to give an elementary proof to the finiteness of the number of Hopf C*-structures on a N-dimensional C*-algebra.

• Date: 2/10/04
  • Speaker: Tom Kephart, Vanderbilt University (Physics)
  • Title: Tight Knots/Links and their Applications
  • Abstract: We argue that tight knots and links may show up in many physical situations, with applications in fields ranging from biology to particle physics. As one example, we model the observed glueball mass spectrum in terms of energies for tightly knotted and linked QCD flux tubes. The data is fit well with one parameter. We predict additional glueball masses. (Work done in collaboration with Roman Buniy)

• Date: 2/24/04
  • Speaker: John Ratcliffe, Vanderbilt University
  • Title: Peaks in the Hartle-Hawking wave function from sums over topologies
  • Abstract: Recent developments in "Einstein Dehn filling" allow the construction of infinitely many Einstein manifolds that have different topologies but are geometrically close to each other. Using these results we show that the Hartle-Hawking wave function for a space-time with a negative cosmological constant develops sharp peaks at certain geometries of constant negative curvature.

• Date: 3/2/04
  • Speaker: Dietmar Bisch, Vanderbilt University
  • Title: A Notion of Free Product for Planar Algebras
  • Abstract: The standard invariant of a subfactor can be described as a planar algebra. We will explain a natural notion of free product for planar algebras and show how its dimension generating function can be computed using free probability theory. The key examples are the Fuss-Catalan algebras of Jones and myself which arise as free products of two Temperley-Lieb algebras (joint work with Vaughan Jones).

• Date: 3/9/04
  • no meeting, spring break

• Date: 3/16/04
  • Speaker: Bruce Hughes, Vanderbilt University
  • Title: A Geometric Interpretation of Birget's Representation of Thompson's Group into the Cuntz Algebra
  • Abstract: Birget recently gave a faithful representation of Thompson's group into the Cuntz algebra. I will interpret this geometrically in the context of groupoids. Every locally rigid subgroup of the group of local similarities of the end space of a tree gives rise to a locally compact groupoid. Moreover, there is a natural injection of the convolution algebra of the group into the convolution algebra of the groupoid. Thompson's group is the subgroup of a locally rigid group whose associated groupoid is the Cuntz groupoid. Renault has shown that the C*-algebra of the Cuntz groupoid is the Cuntz algebra; hence, there is a faithful representation as mentioned above.

• Date: 3/23/04, 4:10-5:00pm (joint with Topology & Group Theory Seminar)
  • Speaker: Vadim Kaimanovich, Universite de Rennes
  • Title: Probabilistic Aspects of the Boundary Theory on Groups
  • Abstract: Probabilistic methods provide a powerful tool for studying asymptotic properties of groups. The talk is devoted to a general survey of the theory of boundaries associated with random walks on groups (Poisson and Martin boundaries) and its applications.

• Date: 3/23/04, 5:10-6:00pm (joint with Topology & Group Theory Seminar)
  • Speaker: Vadim Kaimanovich, Universite de Rennes
  • Title: Double Ergodicity of the Poisson Boundary and Applications
  • Abstract: We prove that the Poisson boundary of any spread out non-degenerate symmetric random walk on an arbitrary locally compact second countable group $G$ is doubly $M^{sep}$-ergodic with respect to the class $M^{sep}$ of separable coefficient Banach $G$-modules. The proof is direct and based on an analogous property of the bilateral Bernoulli shift in the space of increments of the random walk. As a corollary we obtain that any locally compact $\s$-compact group $G$ admits a measure class preserving action which is both amenable and doubly $M^{sep}$-ergodic. This generalizes an earlier result of Burger and Monod obtained under the assumption that $G$ is compactly generated and allows one to dispose of this assumption in numerous applications to the theory of bounded cohomology.

• Date: 3/25/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1206
  • Speaker: Roger Smith, Texas A&M University
  • Title: Cohomology of von Neumann Algebras
  • Abstract: In the 1940's, Hochschild introduced cohomology groups for algebras, and these were adapted by Kadison and Ringrose in the 1970's to the functional analytic setting of von Neumann algebras, the weakly closed self-adjoint subalgebras of bounded operators on a Hilbert space. These groups Hⁿ(M,X) are defined in terms of a module X over the algebra M and can be used as isomorphism invariants. When X=M, they also act as obstruction groups whose vanishing gives structural information about the algebras. For example, the statement that H¹(M,M)=0 is, in different language, a celebrated theorem of Kadison and Sakai that derivations of a von Neumann algebra are always implemented by elements of the algebra. In their original work, Kadison and Ringrose conjectured that Hⁿ(M,M) should always vanish, and proved this in a number of cases. Further progress had to await the recently developed theory of completely bounded maps. In this talk we will survey the current state of affairs and describe some of our own work on this topic. The presentation will be aimed at a general audience with very little background assumed.

• Date: 3/26/03 (Friday) Special Talk, 3:00-4:00pm, SC 1432
  • Speaker: Roger Smith, Texas A&M University
  • Title: The Pukanszky Invariant for Group von Neumann Algebras
  • Abstract: Maximal abelian algebras (masas) play an important role in the theory of von Neumann algebras. We will discuss these algebras in the setting of finite von Neumann factors. Any two masas $A$ and $B$ are isomorphic since each is isomorphic to $L^{\infty}[0,1]$. However, if one asks whether there is an automorphism of the containing factor which moves $A$ to $B$ then the situation becomes much more complicated. To address this problem Pukanszky associated an invariant $Puk(A)$ to each masa $A$. This is a subset of $\mattbb{N}\cup \{\infty\}$, and masas with different invariants are then not conjugate to one another. What are the possible values of $Puk(A)$? While there is, as yet, no definitive answer to this question, the invariant can be computed in the common situation of a group von Neumann algebra $VN(G)$ and a masa $VN(H)$ arising from an abelian subgroup $H$. We will show how to determine the invariant from the group structure and then present diverse examples of $Puk(VN(H))$ for the hyperfinite factor. All terms appearing here will be explained.

• Date: 3/30/04
  • Speaker: Nick Wright, Vanderbilt University
  • Title: Variations of the coarse Baum-Connes conjecture
  • Abstract: The coarse Baum-Connes conjecture asserts that the "higher indices" of geometric operators can be computed from a topological invariant, namely the K-homology of a space. Most known examples of the conjecture are for classes of spaces which are closed under passing to subspaces. In this talk I will give a generalization of the conjecture for which this inheritance property can be proved, and I will prove a special case of the new conjecture. I will also give a construction which on the one hand indicates the problems with proving the original conjecture for subspaces, but on the other hand provides a new coarse invariant, and has implications for the Gromov-Lawson conjecture.

• Date: 4/13/04
  • Speaker: Alexis Alevras, US Naval Academy
  • Title: Cocycles of one-parameter flows on B(H)
  • Abstract: A natural invariant for a one-parameter semigroup $\alpha$ of endomorphisms of B(H) is the set of local $\alpha$-cocycles. In particular, the unitary local cocycles form a group that may be viewed as the automorphism group of the flow. This group has been computed by W. Arveson for the canonical, type I examples of flows arising through second quantization on the CAR/CCR algebra, while more general local cocycles for the same examples were computed by R. Bhat. In the talk we will present a computation of the local cocycles for more exotic, type II flows and obtain information about its multiplicative structure. This is joint work with Robert Powers and Geoffrey Price.

• Date: 4/20/04
  • Speaker: Florin Boca, University of Illinois at Urbana-Champaign
  • Title: The Statistics of the Linear Flow on a Punctured Torus
  • Abstract: Consider the linear flow on a punctured two-dimensional torus, with a disc of radius $r>0$ removed. Let $l_r$ denote the free path length (first exit time). I will show that the probability measures on $[0,\infty)$ associated with the random variable $r l_r$ converge as $r$ tends to zero in both cases when the flow starts at the center of the puncture, or when it starts at a randomly chosen point. The limit measures are explicitly computed in both cases. As a corollary one gets the asymptotic formula $h=-2\log r +2-3\log 2 +9\zeta(3)/4\zeta(2)+o(1)$ as $r$ tends to zero, which solves an open problem about the Sinai billiard.

Past NCGOA seminars