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

• Jean Renault

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

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

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.

Sessions de travail -- Working sessions

The Mackey analogy as a stratified equivalence

• Pierre Clare

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.

Exposé -- Talk

• Edward McDonald

Exposé -- Talk

• Clément Cren

Exposé -- Talk

• Paul Le Breton

Exposé -- Talk

• Lucas Lemoine

Exposé -- Talk

• Florian Thiry

Sessions de travail -- Working sessions

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

• Georges Skandalis

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

Exposé -- Talk

• Omar Mohsen

Exposé -- Talk

• Jacob Bradd

Exposé -- Talk

• Alessandro Contini

Exposé -- Talk

• Iakovos Androulidakis