## The following contains a list of speakers and abstracts of their talks:

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Eitan Angel

Cyclic cocycles on twisted convolution algebras:

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu have provided a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.

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Pierre Bieliavsky

"Oscillating" quantum groups, pentagon equations and multiplicative unitaries:

We present a general notion of oscillating integral for tempered Lie groups allowing to define Fréechet multiplier Hopf-algebras affiliated to Lie groups that provides a nice non-formal topological framework for Drinfel'd twists. In the case of non-standard Drinfeld twists based on a Kählerian Lie group, this notion of "oscillating twists" underlies multiplicative unitaries in the sense of Baaj and Skandalis that non- trivially deform the classical Kac-Takesaki operator on the group. The construction is entirely explicit and applies to Hermitean type semi-simple Lie groups as well. We end the talk with an application to Minkowski spaces. Part of this work is joint with Ph. Bonneau, F. D’Andrea, V. Gayral and Y. Voglaire.

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Alain Connes

Modular Curvature for Noncommutative Two-Tori.

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Joachim Cuntz

C*-algebras generated by regular representations of semigroups:

Recently there is a renewed interest in the study of endomorphisms of C*-algebras and in related constructions such as crossed products by semigroups or the C*-algebras associated with the left or right regular representation of a semigroup. This new interest is mainly due to intriguing explicit examples provided by structures from other fields such as number theory or ergodic theory. In this talk we will discuss a number of such examples.

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Michel Dubois-Violette

Operations of Hopf algebras in graded differential algebras:

A noncommutative version of H. Cartan's operations.

After a short summary of the notion of Cartan operations, algebraic connections and of the Weil algebra, we describe a noncommutative generalization of the above notions. This noncommutative framework is the result of a joint work with Gianni Landi.

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Uwe Franz

ad-invariant Lévy processes on free quantum groups and their potential theory:

We classify Lévy processes on the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$ introduced by Wang, whose distributions and Markov semigroups are invariant under the adjoint action. Then we study potential theory of these processes, i.e. their Markov semigroups and Dirichlet forms and the derivations associated to them by Cipriani and Sauvageot's construction. This also leads to candidates for spectral triples on these quantum groups. Joint work with Anna Kula and Fabio Cipriani.

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Olivier Gabriel

Compact quantum groups and fixed points of the Cuntz algebra

In this talk, we will show how some representations of compact quantum groups define unital Kirchberg algebras which are classified by their K-theories. A general theorem ensures that a unitary representation of dimension $N$ of compact quantum group $(H, \Delta)$ induces an action on the Cuntz algebra $\Oo_N$. This talk will be devoted to the study of the fixed point algebras of this action. We will see that under mild assumptions on the unitary representation $\alpha$ we start with, this fixed point algebra $\Oo^\alpha$ is Kirchberg and in the bootstrap class $\UCTclass$ -- hence classified by its $K$-theory. Moreover, one can prove that this $K$-theory only depends on $H$ through its representation category and its fusion rules. This theorem applies in particular to the case of $SU_q(N)$ and its natural N dimensional representation. If time permits, I will give an overview of the proof and consider the case of $SU_q(2)$, in which the fixed points can be identified with the infinite Cuntz algebra $\Oo_\infty $.

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Robin Hillier

"K-theory, KK-theory, and universal C*-algebras for conformal nets":

We discuss the representation theory of a completely rational conformal net A over S from a K-theoretical point of view. To this end we introduce several suitable C*-algebras associated to A. It turns out that the fusion algebra of representations of A acts faithfully on the K-theory of these algebras, so that the latter contains information about the representation theory of A. We close by illustrating how KK-theory enters into the scene.

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Martin Lindsay

Quantum Brownian motion on a noncommutative manifold:

The aim of this talk is to show how quantum Brownian motion may be defined on a spectral triple of finite compact type which is admissible in the sense of Connes. The key ingredients are: a general theory of L\'evy processes on compact quantum groups which goes beyond the bounded generator case in the universal setting, developed with my former student Adam Skalski; the quantum isometry group of a noncommutative manifold of the above type, due to Goswami; and the theory of quantum stochastic cocycles and their generation via quantum stochastic differential equations in the sense of Hudson and Parthasarathy.

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Fedele Lizzi

Spectral data, spectral action, renormalization, the Higgs and the Dilaton:

We describe, in a way hopefully comprehensible to mathematicians, our point of view of the emergence of the standard model of particle interactions from the spectral data of an almost commutative geometry. The key role for this is played by the Chamseddine Connes spectral action, and the regularization and the renormalization and the anomalies of a quentum field theory.

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Roberto Longo

How to add a boundary condition in Conformal Quantum Field Theory:

Abstract: An operator algebraic procedure to consistently add a boundary condition in any 2D CFT net (without affecting the algebraic structure away from the boundary). There are finitely many solutions, and they appear all together in our construction.

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Yoshiaki Maeda

Deformation quantizations and spectral analysis:

We study the star exponential functions via deformation quantizations and observe how it play to investigate the spectral analysis.

**** Richard Melrose

Automorphisms of algebras of pseudodifferential operators:

Duistermaat and Singer identified the automorphism group of the algebra of pseudodifferential operators on a manifold as the group of invertible Fourier integral operators. This has been refined (and applied) in work with Mathai Varghese. Here further extensions to algebras of operators on compact manifolds with corners -- in particular the b-pseudodifferential operators -- obtained with Ubertino Battisti will be described.

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Gerardo Morsella

QFT on quantum (curved) spacetime and cosmology:

Cosmology is the natural arena for testing physical applications of models of QFT on quantum spacetime, for which one would need a general notion of curved quantum spacetime. We attempt some first steps in this direction. We first show that the DFR analysis of operational limitations to localizability of events can be carried out in a general spherically symmetric spacetime. Then we discuss the backreaction in a quantum FRW spacetime, and show that the horizon problem of standard cosmology disappears. (Joint work with S. Doplicher and N. Pinamonti)

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Jonathan Rosenberg

T-duality for nonprincipal circle bundles and noncommutative geometry:

"Topological T-duality", which was first discovered by physicists, gives an involution on the set of pairs (p: X --> Z, H), where p: X --> Z is a principal S^1 bundle and H is a 3-dimensional cohomology class on X. Recently David Baraglia pointed out that this theory can be extended to cover the case of nonprincipal circle bundles as well. In recent work with Varghese Mathai, we show that Baraglia's results can be reproduced in two ways: a) via a homotopy-theoretic approach à la Bunke-Schick, and b) via noncommutative geometry and crossed products. The latter method leads to the general analysis of K-theory for crossed products by the semidirect product of Z/2 acting on the reals.

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Jean-Luc Sauvageot

Variations in non commutative potential theory: finite energy states, potentials, multipliers; and application to the construction of Fredholm modules:

We develop, in a non commutative workframe, the usual tools of Potential Theory à la Beurling-Deny, following the ideas of Gabriel Mokobodski. We show how those tools can help to associate Fredholm modules to very general Dirichlet forms on C*-algebras. This is joint work with Fabio Cipriani.

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Mauro Spera

The Riemann zeta function as an equivariant Dirac index

In this talk an interpretation of Riemann's zeta function is presented in terms of an ${\Bbb R}$-equivariant $L^2$-index of a Dirac-Ramond type operator, akin to the one on (mean zero) loops in flat space constructed in S.-Wurzbacher, JFA (2003). We build on the formal similarity between Euler's partitio numerorum function (the $S^1$-equivariant $L^2$-index of the loop space Dirac-Ramond operator) and Riemann's zeta function. Also, a Lefschetz-Atiyah-Bott interpretation of the result is given. Generalisations to Lapidus' fractal membranes are also discussed. A fermionic Bost-Connes type statistical mechanical model is presented as well, exhibiting a "phase transition" at (inverse) temperature $\beta = 1$", which also holds for some "well-behaved" g-prime systems in the sense of Hilberdink-Lapidus.

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Boris Tsygan

Microlocal determinant formulas

In addition to index and Riemann-Roch theorems that provide a topological formula for the Euler characteristic of a complex of geometric origin, there are analogous results that compute the determinant line of the cohomology of this complex. They include: a) Beilinson's microlocal formula for the determinant of cohomology of a constructible sheaf; b) Patel's formula for the determinant of cohomology of a D-module; c) a conjectural microlocal formula for the determinant of cohomology of an elliptic pair. The compatibility between a) and b), proved by Beilinson, implies product formulas for determinants of period matrices (the simplest case being the expression of the Beta function as a product of Gamma functions). The conjectural c) would be a multiplicative analog of the index theorem for elliptic pairs (Schapira-Schneiders; Bressler-Nest-Tsygan) and could be related to regularized determinants of elliptic differential operators on real analytic manifolds. We will review the subject and formulate the conjecture.

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Mathai Varghese

Index type invariants for twisted signature complexes:

Atiyah-Patodi-Singer proved an index theorem for non-local boundary conditions in the 1970's that have been widely used in mathematics and mathematical physics. A key application of their theory gives the index theorem for signature operators on oriented manifolds with boundary. As a consequence, they defined certain secondary invariants that were metric independent. I will discuss some recent work with Benameur where we extend the APS theory to signature operators twisted by an odd degree closed differential form satisfying some reality conditions, and study the corresponding secondary invariants. Several interesting new features arise, making the study worthwhile.

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Dan-Virgil Voiculescu

Is there a mod Hilbert-Schmidt BDF-type theorem for operators with trace-class self-commutator?

We revisit the question in the title. We show that part of the problem is about the K-theory of certain Banach algebras.

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