ECOAS06 Abstracts

Christoph Bergbauer: Hopf and Lie algebras in perturbative renormalization and Dyson-Schwinger equations: We briefly review the Connes-Kreimer approach to perturbative renormalization in terms of Hopf and Lie algebras of Feynman graphs which capture the combinatorial aspects of the renormalization procedure. Important properties can be traced back to 1-cocycles in the Hochschild cohomology of these Hopf algebras. At the same time these 1-cocycles provide the building blocks of Dyson-Schwinger equations and thus a link to non- perturbative results. We finally discuss new ideas on the structure and towards actual solutions of these Dyson-Schwinger equations.

Marius Dadarlat: The K-theory of Continuous Fields of C*-algebras: We give a characterization of the invertible elements of the parameterized Kasparov group KK(X;A,B), for a compact, metrisable, finite dimensional space X and separable, nuclear and continuous C(X)-algebras A and B. This leads to a series of automatic trivialization results for continuous fields of Kirchberg algebras.

Pinhas Grossman: Forked Temperley-Lieb Algebras and Intermediate Subfactors: We consider noncommuting pairs P,Q of intermediate subfactors of an irreducible, finite-index inclusion N in M of II_1 factors such that P and Q are supertransitive with Jones index less than 4 over N. We show that up to isomorphism of the standard invariant, there is a unique such pair corresponding to each even value [P:N] = 4 cos2(pi/2n) but none for the odd values [P:N] = 4 cos2(pi/(2n+1)). The proof uses both the planar algebra apparatus for intermediate subfactors and an algebraic construction of a "forked" version of the Temperley-Lieb algebras, which captures the structure of noncommuting supertransitive subfactors with small index.

Adrian Ioana: Amalgamated Free Products of w-rigid Factors and Calculation of their Symmetry Groups: We present several unique decomposition results (á la Bass-Serre) for amalgamated free products of w-rigid factors. We apply this to prove rigidity results for isomorphisms between such factors and to explicitly calculate their symmetry groups, i.e. their fundamental groups and automorphism groups. This is joint work with Jesse Peterson and Sorin Popa.

Piotr Nowak: Quasi-isometric invariants associated to exact groups: Property A is a metric "non-equivariant" amenability condition introduced by G.Yu in his work on the Coarse Baum- Connes Conjecture. Results of Guentner, Kaminker and Ozawa say that for a discrete group G with the word length metric, Property A is equivalent to exactness of the reduced C*-algebra of G. In the talk we will introduce a quasi-isometry invariant related to Property A, it can be viewed as a measure of "how exact" the group is. We will show a relation of our invariant to two other invariants arising from amenability (introduced by Vershik in 70's) and asymptotic dimension (Gromov, 1993). As it turns out, amenable groups are far from being the "most exact" ones in this sense.

Jesse Peterson: L²-rigidity in von Neumann algebras: I will present a new approach for showing primeness in von Neumann algebras. Specifically I will apply Popa's deformation/rigidity techniques in the context of Sauvageot's deformations arising from closable derivations to conclude that all free product II₁ factors, as well as all group factors arising from groups with positive first L²-Betti number are prime. These techniques also give a new approach to Ozawa's result that all nonamenable subfactors of a free group factor are prime.

Mihai Pimsner: Graded Groups in K-theory: We introduce the notion of graded groups and their restricted crossed products. This includes as a particular case graded crossed products with 2-cocycles. The reason for doing this is to be able to use equivariant KK-theory for the computation of the K-groups of these algebras, in the same way as for ordinary crossed products. This is useful in the ungraded case too.

Fred Shultz: K-theoretic invariants for interval map dynamical systems: Each piecewise monotonic interval map generates an (irreversible) dynamical system. A C*-algebra can be associated with this system, and its K-theory computed, providing invariants for the dynamical system. If the map is transitive, one of the K-groups (with certain distinguished elements and a canonical automorphism) determines the system up to conjugacy. The invariants that arise can be described explicitly for unimodal and bimodal maps.

Lin Shan: An equivariant higher index theory of elliptic operators: In this talk, I will introduce an equivariant higher index theory of elliptic operators on noncompact manifolds and discuss its applications. In particular, I will outline a proof for a non-vanishing theorem of the equivariant higher indices for nonpositively curved manifolds.

Stephen Summers: Tomita-Takesaki Modular Theory in Quantum Field Theory: In this talk an overview of some of the recent uses of Tomita-Takesaki modular theory in mathematical quantum field theory is given. Of particular interest is the modular structure of inclusions of von Neumann algebras with a common cyclic and separating vector. Algebraic and other relations between the modular objects of such inclusions encode information about the inclusion. Modular objects associated to small numbers of such inclusions can encode surprisingly rich structures, which, in applications to quantum field theory, contain crucial physical information about the quantum system.

Roman Tessera: Compression of uniform embeddings of groups into Hilbert spaces: In this talk, we will show that for amenable groups, there exists a relation between the compression functions associated with uniform embedding of the group into a Hilbert space and the first eigenvalue of the Laplacian in balls. For a certain class of groups, including all amenable Lie groups, this relation is "optimal" in an asymptotic sense. We also characterize quantitatively a 3-regular tree can be uniformly embedded into a Hilbert space.

Andreas Thom: L²-invariants and rank metric: We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L²-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invariance of L²-Betti numbers under orbit equivalence.