Calcul de traces de projecteurs dans des C*-algèbres de groupoïdes

• Jean Renault

Room: JJ232 Je présenterai une construction géométrique du projecteur de Rieffel de la C*-algèbre de la rotation irrationnelle et le calcul de sa trace.

Harish-Chandra’s philosophy of cusp forms via Lie groupoids

• Robert Yuncken

Room: JJ232 Harish-Chandra spent his career understanding the unitary representations of real reductive Lie groups like SL(n,R). One of the crucial points in this theory is his "philosophy of cusp forms", which says that any tempered unitary representation of a real reductive group (with compact centre) is either discrete series, meaning it is a subrepresentation of the regular representation, or it is induced from a parabolic subgroup, such as the block upper-triangular subgroup in SL(n,R). This sets up an inductive argument over ever smaller subgroups. I will describe how Harish-Chandra’s principal follows from a Lie groupoid construction due to Omar Mohsen plus some C*-algebra theory. (Joint work with Jacob Bradd and Nigel Higson)

Generalised Kontsevich-Vishik trace associated with a graph

• Sylvie Paycha

Room: JJ232 I shall report on ongoing work with S. Scott and B. Zhang by which we generalise regularised spectral zeta functions to a generalised Kontsevich-Vishik trace associated with a Feynman graph. These in turn generalise Feynman amplitudes on a Riemannian manifold studied by Dang and Zhang [JEMS 2021] in two ways. Whereas they consider graphs decorated by a single Riemannian Laplacian on a Riemannian manifold, we consider a general closed manifold and decorate the edges of the graph with arbitrary classical pseudo-differential operators. Whereas Dang and Zhang use complex powers of the Laplacian to regularise, we consider general holomorphic perturbations of the operators decorating the edges. Similarly to their approach, our method involves several complex parameters in the spirit of analytic renormalisation by Speer. We claim that the resulting regularised Feynman amplitudes admit analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo, Paycha and Zhang. We give an explicit determination of the affine hyperplanes supporting the poles, which only depends on the Betti number of the graph and the orders of the operators. Neither the poles nor the method by which we determine them make use of the underlying geometry of the manifold.

The Mackey analogy as a stratified equivalence

• Pierre Clare

Room: JJ232 The Mackey analogy refers to a correspondence between the tempered representation theory of a real reductive group $G$ and that, much simpler, of its associated Cartan motion group $G_0$. It takes the form of a bijection, due to Higson in the complex case and Afgoustidis in the general case, between the tempered duals of these groups, which preserves certain invariants. With Nigel Higson and Angel Román, we constructed an embedding of C*-algebras $C^\ast(G_0)\longrightarrow C^\ast_r(G)$, which characterizes the bijection and induces the Connes-Kasparov isomorphism. After briefly reviewing the correspondence and its C*-algebraic aspects, I will report on joint work with Afgoustidis on the properties of the embedding. We will see that it preserves certain natural stratifications on the tempered duals of $G$ and $G_0$ respectively, shedding a new light on the topological properties of the Mackey bijection.

The trace theorem on Carnot manifolds

• Edward McDonald

Room: JJ232 The algebra of pseudodifferential operators affiliated to a Carnot manifold has a natural trace which is a generalisation of the Guillemin-Wodzicki residue constructed by Dave-Haller and Couchet-Yuncken. I will explain why this residue coincides with the Dixmier trace.

Factorization of the Dirac Operator on Foliations

• Lucas Lemoine

Room: JJ232 On a foliation induced by a Riemannian submersion, J. Kaad and W. D. van Suijlekom constructed two natural operators whose product is the Dirac operator, up to a bounded curvature term. This result gives a geometric outlook on the work of A. Connes, M. Hilsum, and G. Skandalis about wrong-way functoriality in bounded KK-theory, where this curvature term remained implicit. In this talk, using recent developments in unbounded KK-theory, we adapt their results to extend this factorization to foliations.

Sub riemannian tangent spaces and groupoids

• Paul Le Breton

Room: JJ232 A sub-riemannian structure on a manifold $M$ naturally produces a distance $d_{CC}$. The study of the metric space $(M,d_{CC})$ raises 2 natural questions: - For $x \in M$ is there a metric space that encodes the infinitesimal properties of the structure at $x$ as does the tangent space $(T_xM, g_x)$ in riemannian geometry ? - If such a space exists what kind of algebraic structure can it be endowed with ? This problem has been entirely solved by Mohsen in 2021 using the following very elegant and elementary fact: the quotient of any group $G$ by any subgroup $H$ always has a canonical groupoid structure which coincides with the classical quotient group structure as soon as $H$ is normal in $G$. My goal in this talk is to present Mohsen's construction.

Geometric obstructions to fully ellipticity for families with embedded corners

• Florian Thiry

Room: JJ232 First we present how K-theory can be used to express obstruction to fully ellipticity from ellipticity. Then we set a geometrical context, namely families of manifold with embedded corners and we introduce some tools related to it (as cononormal homology, Monthubert groupoid for families, ...). Finally we compute the fully ellipticity obstruction in this context in terms of smaller indices and conormal cycles.

Stratification of the Helffer-Nourrigat Cone

• Clément Cren

Room: JJ232 Given a sub-riemannian manifold (or singular filtrations of the tangent bundle), Androulidakis, Mohsen and Yuncken constructed a calculus adapted to the corresponding hypoelliptic problems arising from the filtration. The principal symbol of an operator then becomes a family of operators in representations of nilpotent groups, the osculating groups (there is one attached to each point of the manifold), called the Helffer-Nourrigat cone. The dimension and structure of these groups may vary from one point to another, and not all of them have to be taken into account in the cone. I will explain how to stratify this cone of representations such that each stratum becomes locally compact and Hausdorff, and how this stratification behaves when restricted to each of the osculating groups.

Stability by functional calculus of the algebra of smooth functions with compact support

• Georges Skandalis

Room: JJ232 For which Lie groupoid $G$ is the convolution algebra $C_c^\infty (G)$ stable by holomorphic functional calculus in $C^*(G)$? We will answer this question completely. In particular, we will show that this is the case if the groupoid $G$ is proper. Joint work with Claire Debord and Kévin Massard

Functional calculus

• Omar Mohsen

Room: JJ232

Tempiric representations and the Connes-Kasparov isomorphism

• Jacob Bradd

Room: JJ232 (With Nigel Higson and Robert Yuncken.) An important result of Vogan in representation theory for real reductive groups states that if $K$ is a maximal compact subgroup of a real reductive group $G$, then the tempered irreducible representations of $G$ with real infinitesimal character (the "tempiric" representations, as coined by Afgoustidis), up to equivalence, are in bijection with irreducible unitary representations of $K$ up to equivalence (given by taking the unique minimal $K$-type). This is central to the Mackey bijection for general real reductive groups, proved by Afgoustidis. In this talk we will concisely reprove the Mackey bijection and, going further, we will prove that the "linearized" version of this Vogan bijection is equivalent (in a simple way) to the Connes-Kasparov isomorphism, which has striking implications in both directions of the equivalence. In one direction, thanks to Lafforgue this gives an almost completely index-theoretic proof of Vogan's (linearized) bijection, a purely (and deep) representation theoretic result, and in the other direction it shows that Vogan practically proved the Connes-Kasparov isomorphism (before it was even conjectured).

Algebras and states in QFTs with timelike boundaries

• Alessandro Contini

Room: JJ232 Based on ongoing work with Alexander Strohmaier, we review the algebraic approach to quantum field theory on curved spacetimes in the presence of a timelike boundary. In particular, we look at quasi-free states which are determined by their action on pairs of observables, and introduce a notion of Hadamard states in this context: they are bisolutions to the mixed Dirichlet-Cauchy problem for a normally hyperbolic operator, satisfying a precise microlocal condition in addition. We will hint at how one can prove the existence of such states, using techniques from the b-calculus.

Towards a comparison of Geometric Quantization with Deformation Quantization

• Iakovos Androulidakis

Room: JJ232 E. Hawkins proposed a scheme for geometric quantization of a Poisson manifold by means of deformation to normal cone of a symplectic groupoid, where the C*-algebra involved is twisted by polarization. We report on work in progress with P. Antonini, F. Bonechi, N. Ciccoli and V. Zenobi, where we review these ideas using a natural filtration. We redefine the notion of polarization allowing to include singular examples. The upshot of this project is to eventualy provide a framework allowing to compare geometric quantization with deformation quantization.