The wave equation, scattering theory and the Bernstein transform

• Nigel Higson

Room: Hôtel des Planchettes I shall describe work in progress with Bob Yuncken and Tyrone Crisp. The Bernstein transform is a map from one function space to another, like the Fourier transform or the Radon transform. It is part of Bernstein’s theory of smooth representations of p-adic groups, but the version I’m going to discuss comes from the tempered representation theory of real groups such as SL(2,R). The main problem in either context is to prove that it exists! I’ll explain what the Bernstein transform is designed to do, how to characterize it, at least for SL(2,R), using scattering theory for a wave equation (this involves a geometric space that is very similar to the tangent groupoid) and why I think that eventually it will be best understood from the point of view of C*-algebra theory (for the moment we rely on constructions that only work for smooth subalgebras).

à préciser

• Pierre Julg

Room: Hôtel des Planchettes

Bounded $H^\infty$-calculus for Cone Differential Operators

• Elmar Schrohe

Room: Hôtel des Planchettes We model a conic manifold by a manifold $\mathbb B$ with boundary $\partial \mathbb B=:X$. In a collar neighborhood we introduce coordinates $(t,x)$, where $t$ is the distance to $\partial \mathbb B$ and $x$ the variable in $X$. A cone differential operator of order $\mu$ is an operator $A: C^\infty_c(\mathbb B^\circ)\to C^\infty_c(\mathbb B$ that can be written near $\partial \mathbb B$ in the form $$A=t^{-\mu}\sum_{k=0}^\mu a_k(t)(-t\partial_t)^k\quad \text{with }a_k\in C^\infty([0,1)\mathrm{Diff}^{\mu-k}(X)). $$ We consider an extension of $A$ in a weighted cone Sobolev space $\mathcal H^{s,\gamma}_p(\mathbb B)$ with domain $\mathcal D(A) = \mathcal H^{s+\mu,\gamma+\mu}_p(\mathbb B)\oplus \mathcal E$, where $\mathcal E$ is a space of asymptotics functions. Given a sector $$\Lambda_\theta= \{re^{i\phi}: r\ge0, \theta\le\phi\le2\pi-\theta\}, \quad 0<\theta<\pi,$$ we show that any extension of $A$ which is parameter-elliptic with respect to $\Lambda_\theta$ has a bounded $H^\infty$-calculus on $\mathbb C\setminus \Lambda_\theta.$ Parameter-ellipticity with respect to $\Lambda_\theta$ here requires the following \begin{enumerate} \item Denote by $\sigma_\psi^\mu(A)$ the principal symbol of $A$. Then $\sigma_\psi^\mu(A)-\lambda$ is invertible for $\lambda \in \Lambda_\theta$, even up to the boundary, if one takes into account the degeneracy. \item The principal conormal symbol $\sigma_M^\mu(A)(z)$ is invertible for all $z\in \mathbb C$ with ${\rm Re}\, z= \frac{n+1}2-\gamma -\mu $ or ${\rm Re}\, z= \frac{n+1}2-\gamma$. \item $\Lambda_\theta$ is a sector of minimal growth for the model cone operator $$\widehat A= t^{-\mu}\sum_{k=0}^\mu a_k(0)(-t\partial_t)^k$$ acting on cone Sobolev spaces over he infinite cone. \end{enumerate} Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities. (Joint work with J\"org Seiler (Torino))

Noncommutative completion and universality of Boutet de Monvel's algebra

• Karsten Bohlen

Room: Hôtel des Planchettes Given as data an embedding of certain singular manifolds, I will describe a procedure which associates to this data a $C^{\ast}$-algebra correspondence. Such a noncommutative completion of an embedding is functorial and universal. As a particular instance of this construction I will discuss Boutet de Monvel's algebra in the setting of singular manifolds.

The injectivity radius of Lie manifolds

• Paolo Antonini

Room: Hôtel des Planchettes Lie manifolds were introduced by Ammann, Lauter and Nistor. These are a large class of noncompact complete (Riemannian) manifolds well behaved for the study of index theory on noncompact spaces. Their geometric structure is described by a Lie algebra of vector fields on a suitable compactification with corners. Equivalently by a Lie algebroid over the compactification.In this talk we will present the main geometrical features of Lie manifolds. In particular we will explain how the theory of connections and their associated geodesic flow on Lie algebroids leads to the proof that every Lie manifold has uniformly positive injectivity radius, a result recently obtained in collaboration with G. De Philippis and N. Gigli.

Smooth $K$-theory

• Alain Berthomieu

Room: Hôtel des Planchettes Smooth K-theory is defined on a smooth manifold as an extension of topological K-theory by differential forms. In the case of a proper submersion, direct image for smooth K-theory is defines using Bismut and al.'s eta-forms, which are differential forms entering in the transgression of the local families index theorem. The construction of a direct image in the case of a closed immersion should use corresponding objects which are currents instead of smooth differential forms.

à préciser

• Omar Moshen

Room: Hôtel des Planchettes

Revisiting the $K$-theory of $CP^n$ from a (singular) foliation viewpoint

• Iakovos Androulidakis

Room: Hôtel des Planchettes This is report on work in progress with Nigel Higson. We are exploring an idea which comes from a very simple observation: The Bruhat cells of various flag manifolds are exactly the orbits of the action by a nilpotent matrix group. So one might try to use the apparatus developed for singular foliations in order to address representation theory problems. Making a start with this, we look at the case of $CP^n$ and the action by triangular matrices. It turns out that the nilpotency of this group allows us to shed some geometric light in the well-known K-theory group of $CP^n$, using index theory and techniques developed with Georges Skandalis to split singularities. Using these techniques we also construct interesting $K$-theory elements.

The $K$-theory of operator algebras on $L^p$ spaces

• Yeong Chyuan Chung

Room: Hôtel des Planchettes I will recall the notion of quantitative $K$-theory for filtered $C^\ast$-algebras developed by Oyono-Oyono and Yu, and outline how it can be extended to a larger class of Banach algebras, including operator algebras on $L^p$ spaces. This then provides a tool for investigating the $K$-theory of $L^p$ analogs of crossed products or uniform Roe algebras in some work in progress that I will briefly describe. If time permits, I will also list a few general questions about the $K$-theory of operator algebras on $L^p$ spaces.

à préciser Room: Hôtel des Planchettes