**Programme - Schedule / Horaire**

**Monday / lundi:**

9:10-9:20: Welcome / Accueil

9:20-10:10: Giovanni Landi (pdf)

10:10-10:40: Coffee Break / Pause café

10:40-11:30: Heath Emerson

11:40-12:10: Yanli Song (pdf)

Lunch Break / Pause midi

2:00-2:50: Bahram Rangipour

3:00-3:50: Farzad Fathizadeh

3:50-4:20: Coffee Break / Pause café

4:20-5:10: Frédéric Latrémolière (pdf)

**Tuesday / mardi:**

9:20-10:20: Eckhard Meinrenken

10:10-10:40: Coffee Break

10:40-11:30: Jean-Marie Lescure (pdf)

11:40-12:10: Robin Deeley

Lunch Break / Pause midi

2:00-2:50: Bruno Iochum (pdf)

3:00-3:50: Hang Wang (pdf)

3:50-4:20: Coffee Break / Pause café

4:20-5:10: Paulo Carrillo Rouse

5:20-5:50: Nicolas Prudhon (pdf)

**Wednesday/ mercredi:**

9:20-10:10: Nigel Higson

10:10-10:40: Coffee Break / Pause café

10:40-11:30: Claire Debord (pdf)

11:40-12:30: Georges Skandalis (pdf)

19h00: Banquet at Chez Queux, 158 rue Saint-Paul est

Free Afternoon / Après-midi libre

**Thursday / jeudi:**

9:20-10:10: Paolo Piazza

10:10-10:40: Coffee Break / Pause café

10:40-11:20: Sasha Gorokhovsky

11:30-12:10: Branimir Ćaćić

Lunch Break / Pause midi

2:00-2:50: Adam Rennie

3:00-3:50: Hervé Oyono-Oyono (pdf)

3:50-4:20: Coffee Break / Pause café

4:20-5:10: Erik van Erp

**Friday / vendredi:**

9:20-10:10: Boris Tsygan

10:10-10:40: Coffee Break / Pause café

10:40-11:30: Piotr M. Hajac (pdf)

11:40-12:30: Alan Carey (pdf)

**Abstracts / Résumés**

**Branimir Ćaćić (Texas A&M University)**

*Title:* Principal bundles in unbounded KK-theory

*Abstract:* Ammann and Bär, in their work on Dirac spectra, obtained a useful decomposition of the Dirac operator on a circle bundle into a vertical term, a horizontal term, and a zero-order error term. More recently, Forsyth and Rennie have obtained analogous factorisations of suitable torus-equivariant spectral triples in unbounded KK-theory. In this talk, I will discuss, in terms of explicit factorisations in unbounded KK-theory, a direct generalisation of Ammann and Bär's geometric decomposition to the case of principal $G$-bundles for $G$ a compact connected Lie group; in particular, when $G = \mathbb{T}^n$, this recovers Forsyth and Rennie's factorisation as applied to the commutative spectral triple of a principal $\mathbb{T}^n$-bundle. I will then discuss applications to factorising suitable toric noncommutative manifolds qua noncommutative principal bundles, e.g., Brain, Mesland, and Van Suijlekom's factorisations of irrational noncommutative $2$-tori and $\theta$-deformed $3$-spheres. This is joint work with Bram Mesland.

**Alan Carey (Australian National University)**

*Title:* Spectral flow and the essential spectrum.

*Abstract:* Spectral flow is often used in a "bare hands" fashion in theoretical models in condensed matter physics. In this application the operators have non-empty essential spectrum. Nevertheless the question has been asked whether it is possible to compute spectral flow using the "suspension" trick and turning it into a problem of calculating the index of an operator on a manifold of one higher dimension. I will explain our answer to this question.

**Paulo Carrillo-Rouse (University of Toulouse)**

*Title:* An Atiyah-Singer type formula for manifolds with corners.

*Abstract:* Given a fully elliptic b-pseudodifferential operator on a manifold with corners (a la Melrose-Piazza for example)
we can associate a class in the K-theory of a vector bundle over a naturally associated open piecewise smooth manifold with corners which we could call the b-tangent bundle at infinity, once in this topological K-theory group we can take the Chern character of this class and integrate ( together with a cup with an appropriate Todd class). We prove that in this way we recover the Fredholm index of the operator.
In this talk I will explain the above paragraph and try to sketch the proof of the main theorem, this is based on joint work in progress with Jean-Marie Lescure. If time allows it I will discuss the local and non local components of our index formula and try to explain the link with the so called eta invariant at least for the manifold with boundary case.

**Claire Debord (University of Clermont-Ferrand)**

*Title:* Groupoids and pseudodifferential calculus

*Abstract:* In this talk, we will first recall how one can express order zero pseudodifferential operators on a groupoid G as integrals of kernels associate with the adiabatic deformation Gad of G. This will be the starting point to investigate pseudodifferential operators arising from the data of a groupoid G over a manifold M together with a codimension 1 submanifold V of M transverse to G and see how these are related to the Boutet de Monvel calculus.

**Robin Deeley (University of Clermont-Ferrand)**

*Title:* Comparing dynamical zeta functions

*Abstract:* Associated to an Axiom A diffeomorphism there are at least two natural zeta functions. One defined from the action on homology (based on the classical Lefschetz fixed point formula) and one defined directly from periodic point information. In general, these functions are not equal. I will introduce the definition of Axiom A, basic sets, and these zeta functions. The main goal of talk is to explore these zeta functions by relating Putnam's homology theory for basic sets with the homology of the manifold. Thus, the heart of the matter is the relationship between the classically defined homology of the manifold and Putnam's homology, which is "noncommutative" being defined via the K-theory of groupoid C*-algebras associated to shifts of finite type.

**Heath Emerson (University of Victoria)**

*Title:* Noncommutative Geometry, hyperbolic groups and their boundaries

*Abstract:* In this talk we describe some noncommutative geometric features of the action of a Gromov hyperbolic group on its boundary. The type of geometry suitable for this context is a certain natural Hölder geometry possessed by the boundary. This geometry is left invariant by the group action. We use it to construct canonical representatives of K-homology classes for the crossed-product and, by restriction, for the reduced C*-algebra of the group. One consequence is that for every Gromov hyperbolic group G there is a constant d, a geometric dimensional invariant of the group, for which every K-homology class over the reduced C*-algebra of G is represented by a Fredholm module which is d+-summable over the group ring.

**Farzad Fathizadeh (Caltech)**

*Title:* Scalar curvature for the noncommutative 4-torus and its functional relations

*Abstract:* I will explain how the scalar curvature of the conformally
perturbed noncommutative 4-torus can be computed by making use of a
noncommutative residue. This method justifies the remarkable cancellations
that occurred when the curvature was computed previously in a joint work
with M. Khalkhali, in which the rearrangement lemma was used. Furthermore,
this method readily allows to recover the 2-variable function in the
curvature formula as the sum of a finite difference and a finite product
of the 1-variable function. The simplification of the curvature formula
for the dilatons associated with an arbitrary projection and an explicit
computation of the gradient of the analog of the Einstein-Hilbert action
will be outlined.

**Alexander Gorokhovsky (University of Colorado at Boulder)**

*Title:* Localized analytic indices and cyclic cohomology

*Abstract:* In their work on Novikov conjecture, A . Connes and H. Moscovici
introduced a notion of localized analytic indices for an elliptic operator
and gave a topological formula for their computation.

The goal of my talk is to reinterpret the localized analytic indices
as pairing in cyclic (co)homology and to describe a theorem computing
them in topological terms. This is joint work with H. Moscovici.

**Piotr Hajac (IMPAN)**

*Title:* Non-contractibility of compact quantum groups and index pairings for their non-reduced suspensions

*Abstract:* Using the concept of an equivariant join G*G of a compact quantum group G with itself, we define the contractibility of G as the existence of a global section of the compact quantum principal bundle G*G over the non-reduced suspension SG. We unravel the pullback structure of finitely generated projective modules associated to G*G, and make it fit the Milnor connecting homomorphism formula in K-theory of unital C*-algebras. Then, taking advantage of the compatibility of the index pairing with the connecting homomorphisms of the Mayer-Vietoris six-term exact sequences for K-theory and K-homology (which is a manifestation of the associativity of the Kasparov product), we prove that SUq(2) is not contractible, i.e. that Pflaum's quantum instanton bundle SUq(2)*SUq(2) is not trivializable. Finally, we conjecture the non-contractibility of all non-trivial compact quantum groups, and explain how it fits the bigger picture of noncommutative Borsuk-Ulam-type conjectures. (Based on joint work with P. F. Baum, L. Dabrowski, T. Hadfield and E. Wagner.)

**Nigel Higson (Penn State University)**

*Title:* Oka principle: commutative and noncommutative

*Abstract:* Oka proved in 1938 that topological line bundles over closed, complex submanifolds of complex affine space admit unique holomorphic structures, and nearly twenty years later, Grauert proved the same thing for topological vector bundles of any rank. Oka's theorem is in some sense "commutative," since it concerns the abelian Lie group GL(1,C), whereas Grauert's theorem concerns the non-abelian groups GL(n,C). But there are further extensions of both theorems into the realm of the noncommutative, for instance the Oka principle of Bost. In this lecture I shall try to explain a new way of applying an Oka principle in an effort to better understand the Connes-Kasparov isomorphism and its meaning in representation theory.

**Bruno Iochum (CNRS-Aix-Marseille University)**

*Title:* Crossed product extensions of spectral triples

*Abstract:* Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{A},\mathcal{H},\mathcal{D})$ on the crossed product $G \ltimes_{\alpha, red} \mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^*$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold. This is joint work with Thierry Masson.

**Giovanni Landi (University of Trieste)**

*Title:* Sigma-model solitons on noncommutative spaces

*Abstract:* We use results from time-frequency analysis and Gabor analysis to construct new classes of sigma-model solitons
over the Moyal plane and over noncommutative tori, taken as source spaces, with a target space made of two points. A natural action functional leads to self-duality equations for projections in the source algebra. Solutions, having non-trivial topological content, are constructed via suitable Morita duality bimodules.

**Frédéric Latrémolière (University of Denver)**

*Title:* The Gromov-Hausdorff Propinquity

*Abstract:* The search for a noncommutative analogue of the Gromov-Hausdorff distance, motivated by questions in mathematical physics as well as the desire to extend techniques from metric geometry to noncommutative geometry, raises many interesting challenges and possibilities. In this presentation, we will introduce the Gromov-Hausdorff propinquity, a recently introduced metric which generalizes the Gromov-Hausdorff distance to quantum compact metric spaces, which are a form a noncommutative generalizations of Lipschitz algebras. We will present several examples of non-trivial convergence and approximation results within the framework of this new metric on classes of C*-algebras, and prove an analogue of Gromov's compactness theorem for our new metric based on a noncommutative analogue of the covering number for metric spaces.

**Jean-Marie Lescure (University of Clermont-Ferrand)**

*Title:* Convolution of distributions on Lie groupoids

*Abstract:* This is joint work with Dominique Manchon and Stéphane Vassout. We extend the convolution product on a Lie groupoid $G$ to a large class of distributions. We obtain a convolution algebra and show that $G$-operators are all convolution operators. We explain how the symplectic groupoid $T^*G$ of Costes-Dazord-Weinstein appears naturally when one analyses the wave front set of the convolution of two distributions on $G$. Following this idea, we apply the Hörmander's theory of Lagrangian distributions to develop a calculus of Fourier integral operators on Lie groupoids.

**Eckhard Meinrenken (University of Toronto)**

*Title:* Splitting theorems

*Abstract:* The Weinstein splitting theorem for Poisson manifolds says that locally, any Poisson manifold is a direct product of a symplectic leaf with a transverse Poisson structure. Zung's splitting theorem is a similar result for Lie algebroids, and the Stefan-Sussmann theorem may be viewed as a splitting theorem for generalized foliations. I will explain a simple approach to splitting theorems, which lends itself to various generalizations. Based on joint work with Henrique Bursztyn and Hudson Lima.

**Hervé Oyono-Oyono (University of Metz)**

*Title:* Quantitative indices and Novikov conjecture.

*Abstract:* We define for compact metric spaces quantitative indices that take into account propagation phenomena for pseudo-differential calculus. We relate these quantitative indices to the Novikov conjecture. As an application, we provide an elementary proof of Novikov conjecture for group with finite asymptotic dimension.

**Paolo Piazza (University of Rome La Sapienza)**

*Title:* Relative pairings in K-Theory and higher index theory

*Abstract:* This lecture will illustrate the use of relative K-theory groups, relative cyclic cohomology groups and their pairings in higher index theory. Starting from the study of the divisor flows of Lesch-Moscovici-Pflaum, I will then move on and survey recent results in higher Atiyah-Patodi-Singer index theory: first on Galois coverings (joint with Gorokhovsky and Moriyoshi) and then on spaces endowed with a proper cocompact G-action, with G a Lie group or a Lie groupoid (joint with Posthuma).

**Nicolas Prudhon (University of Metz)**

*Title:* Exhausting families of representations and spectra of pseudodifferential operators.

*Abstract:* A powerful tool in the spectral theory and the study of Fredholm conditions for (pseudo)differential operators is provided by families of representations of a naturally associated algebra of bounded operators. Motivated by this approach, we define the concept of an exhausting family of representations of a $C^*$-algebra $A$. Let $F$ be an exhausting family of representations of $A$.

We have then that an abstract differential operator $D$ affiliated to $A$ is invertible if, and only if, $\phi(D)$ is invertible for all $\phi\in F$. This property characterizes exhausting families of representations. We provide necessary and sufficient conditions for a family of representations to be exhausting. If $A$ is a separable $C^*$-algebra, we show that a family $F$ of representations is exhausting if, and only if, every irreducible representation of $A$ is (weakly) contained in a representation $\phi\in F$. However, this result is not true, in general, for non-separable $C^*$-algebras. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator $P$ is invertible if, and only if, its Mellin transform $\hat{P) (\tau)$ is invertible, for all $\tau\in\mathbb{R}^n$.

**Bahram Rangipour (University of New Brunswick)**

*Title:* Topological Hopf algebras and their Hopf cyclic cohomology.

*Abstract:* We show that Hopf cyclic cohomology is well understood in the realm of topological Hopf algebras. We bring several examples showing that the topological (contrary to algebraic) framework helps to have a full correspondence between classical symmetries (e.g., Lie groups) and their representations and cohomology, and non classical ones (e.g., Hopf algebras) and their SAYD modules and Hopf cyclic cohomology. We shall also give some applications to the study of characteristic classes of foliations represented by groupoid action algebras.

**Adam Rennie (University of Wollongong)**

*Title:* Representing the defining extension of a class of Cuntz-Pimsner algebras as a Kasparov module.

*Abstract:* We show how to construct a Kasparov module representing the class of the defining extension of
the Cuntz-Pimsner algebras of biHilbertian bimodules. We do this without the benefit of a completely positive splitting, which is typically difficult to obtain without changing coefficients. The bimodule structure instead provides us with an expectation with which we can construct the Kasparov module. We will give numerous examples, and an application to the bulk-edge correspondence for the quantum Hall effect. Joint work with Chris Bourne, Alan Carey, Dave Robertson, and Aidan Sims.

**Georges Skandalis (University of Paris 7)**

*Title:* A generalized Boutet de Monvel index theorem.

*Abstract:* Let $G$ be a smooth groupoid and $V$ a hypersurface in $M=G^{(0)}$ transverse to the groupoid.

In her talk, Claire Debord will explain the construction of a corresponding Boutet de Monvel calculus in terms of a deformation groupoid.
We will compute the $K$ theory of the total symbol algebra and the corresponding index morphism.

**Yanli Song (University of Toronto)**

*Title:* Equivariant indices of Spin-c Dirac operators for proper moment maps.

*Abstract:* Given a compact, connected Lie group acting on a possibly non-compact manifold, we can associate it with an equivariant map from the manifold to the Lie algebra, which generalizes the moment map introduced in the symplectic case. Under the assumption that the moment map is proper, we will explain how to define an equivariant index of Spin-c Dirac operators on the manifold and decompose the index into irreducible representations according to a quantization commutes with reduction principle. This joint work with Peter Hochs (University of Adelaide).

**Boris Tsygan (Northwestern University)**

*Title:* An extension of the Fedosov construction

*Abstract:* In the early nineties, Fedosov proposed a simple geometric construction of deformation quantization of symplectic manifolds. It turns out that a slight extension of this construction leads to new geometric structures related to deformation quantization and allows to define new invariants in symplectic geometry.

**Erik van Erp (Dartmouth College)**

*Title:* A Groupoid Approach to Pseudodifferential Calculi

*Abstract:*
The tangent groupoid was introduced by Alain Connes as a geometric device for glueing a pseudodifferential operator to its principal symbol. We carry this idea further and show that classical pseudodifferential operators have a simple (and coordinate-free) definition in terms of the geometry of the tangent groupoid. The same definition also works for the Heisenberg calculus and its generalizations, by appropriately adapting the construction of the tangent groupoid. (Joint work with Bob Yuncken)

**Hang Wang (University of Adelaide)**

*Title:* Noncommutative geometry, equivariant cohomology, and conformal invariants.

*Abstract:* We will explain how to apply the framework of noncommutative geometry in the setting of conformal geometry. We plan to describe three main results. The first result is a reformulation of the local index formula of Atiyah-Singer in conformal geometry, i.e., in the setting of the action of a group of conformal-diffeomorphisms. The second result is the construction of new conformal invariants out of equivariant characteristic classes. The third result is a version in conformal geometry of the Vafa-Witten inequality for eigenvalues of Dirac operators. This is joint work with Raphaël Ponge (Seoul National University and McGill University).