09:0009.30 
Registration

09:30–09:35 
opening words

09:40–10:30 
On the $K$theory of crossed products

10:40–11:30 
Coffee break

11:30–12:20 
Curved analogs of symmetry breaking differential operators and a singular Yamabe problem

12:3013:30 
Lunch

13:30–14:20 
Spectral projections on hyperbolic surfaces

14:30–15:20 
The cubic Dirac operator on compact quotients of the oscillator group

15:30–16:00 
Coffee break

16:00–16:50 
The renormalized volume of hyperbolic 3manifold

17:00–17:50 
Spectral theory on locally symmetric spaces and automorphic $L$functions

18:00–18:30 
Chat & coffee

19:00 
Conference dinner at restaurant Postkutsch
8 rue Xavier Brasseur, L4040 EschsurAlzette

09:30–10:20 
Geometric side of a local relative trace formula

10:30–11:00 
Coffee break

11:00–11:50 
Resonances on spaces with symmetry

12:00–12:50 
Vectorvalued Poisson Transforms in QuantumClassical Correspondences

13:00–13:05 
closing words

13:10 
Lunch

JeanPhilippe Anker (Université d'Orléans):
Spectral projections on hyperbolic surfaces
In an ongoing collaboration with Pierre Germain (Imperial College) and Tristan Léger (Princeton University), we study $L^2$ to $L^p$ estimates $(p>2)$ for spectral projections in a small window on (locally) symmetric spaces. For hyperbolic surfaces of infinite area and with no cusps, we have recently obtained almost optimal results [arXiv:2306.12827]. In this talk, I will give a brief survey of the problem, which goes back to the restriction theorem of SteinTomas in the Euclidean setting, comment on our result for hyperbolic surfaces and present the main steps of its proof.
Ulrich Bunke (Universität Regensburg):
On the $K$theory of crossed products
Given a $C^{*}$algebra A with a finite group action one can form the crossed product. In this talk I will discuss to which extend one can calculate the $K$theory of this crossed product in terms of the $K$theory spectrum of the algebra with its induced group action. Viewing the crossed product as a kind of homotopy orbits of the algebra this is an instance of the descent problem asking when the caninical map $K(A)_{hG} \rightarrow K(A_{hG})$ is an equivalence.
Colin Guillarmou (Université ParisSaclay):
Resonances on spaces with symmetry
I will review several situations where a notion of quantum or classical resonances can be defined and provide a notion of discrete spectrum, fitting in a general scattering setting. This includes for example certain locally symmetric spaces: in rank 1 the Laplacian on convex cocompact manifolds, the geodesic flow in similar settings, the Weyl chamber flow in higher rank, or even geodesic and pseudoRiemannian Laplacian on Lorentzian quasiFuchsian manifolds. In all these cases, there is a quantumclassical correspondence between the eigenstates. If time permits, I’ll mention also some infinite dimensional setting in conformal field theory where the underlying Lie algebra is infinite dimensional (Virasoro) and related questions appear and can be solved.
Joachim Hilgert (Universität Paderborn):
Vectorvalued Poisson Transforms in QuantumClassical Correspondences
Poisson transforms for symmetric spaces, respectively trees, play a decisive role in the quantumclassical correspondences (QCC) relating spectral data of Laplacians with spectral data of geodesic flows for compact quotients. For generic spectral parameters the scalar Poisson transforms are bijective and constitute the bridge to pass from one set of spectral data to the other. There is a known set of exceptional spectral parameters for which the scalar Poisson transforms are neither injective nor bijective. It turns out that for these spectral parameters one can still build QCCs using specific vectorvalued Poisson transforms of the type discussed in Martin Olbrich’s PhDthesis. In the case of compact rank 1 locally symmetric spaces the vector parts come from the minimal Ktypes of the socles of the (reducible) spherical principal series representations associated with the exceptional spectral parameter. In the case of finite graphs one can use the edge Poisson transform. The goal of this talk is to explain these exceptional QCCs and discuss topological interpretations of the exceptional spectral data.
Andreas Juhl (HumboldtUniversität Berlin):
Curved analogs of symmetry breaking differential operators and a singular Yamabe problem
A special case of symmetry breaking differential operators (SBDO) are intertwining operators for spherical principal series representations which map smooth functions on a sphere to smooth functions on an equatorial subsphere. Here the symmetry is broken since equivariance holds true only with respect to the subgroup of the conformal group of the big sphere which leaves the small sphere invariant. These operators interpolate between conformal powers of the Laplacian on both spheres. There are farreaching analogs of construction SBDO in the context of Riemannian manifolds $X$ with boundary $M$. We describe recent progress on such constructions in conformal differential geometry. The constructions rest on the solution of a singular version of the Yamabe problem. Among other things, this leads to the notions of extrinsic conformal Laplacians and extrinsic $Q$curvatures (generalizing Branson’s $Q$curvatures). The extrinsic conformal Laplacians are conformally covariant generalizations of socalled GJMSoperators (higherorder analogs of Yamabe and Paneitz operators). The extrinsic $Q$curvatures are linked to conformal invariants of hypersurfaces. We also describe loworder cases of the extrinsic conformal powers of the Laplacian and make explicit the related conformal invariants of hypersurfaces. The presentation partially rests on joint work with Bent Orsted developing an alternative perspective towards works of Gover and Waldron.
Ines Kath (Universität Greifswald):
The cubic Dirac operator on compact quotients of the oscillator group
The subject of the talk is a contribution to the analysis of Dirac operators on compact locally symmetric Lorentzian manifolds.
We will consider Kostant's cubic Dirac operator $D^{1/3}$ on fourdimensional compact homogeneous spaces $X=G/\Gamma$, where $G$ is a solvable Lie group endowed with a
biinvariant Lorentzian metric and $\Gamma$ is a (cocompact) lattice in $G$. More exactly, $G$ is the fourdimensional oscillator group, which is a
semidirect product of the threedimensional Heisenberg group and the real line. We use the canonical indefinite inner product on smooth sections of the spinor
bundle $\Sigma(X)$ to define a Krein space $L^2(\Sigma(X))$. Then the cubic Dirac operator is defined on $L^2(\Sigma(X))$ and $iD^{1/3}:L^2(\Sigma(X))\rightarrow L^2(\Sigma(X))$ is
essentially selfadjoint (as an operator on a Krein space). We determine the spectrum of $D^{1/3}$ on $X$ for each basic lattice $\Gamma\subset G$ by using an explicit decomposition
of the regular representation of $G$ on $L^2(\Sigma(X))$ into irreducible subrepresentations.
This is joint work with Margarita Kraus.
Werner Müller (Universität Bonn):
Spectral theory on locally symmetric spaces and automorphic $L$functions
Spectral theory on locally symmetric spaces is closely related to the theory of automorphic forms and has deep connections with number theory. Spectra of canonical differential operators on quotients of symmetric spaces by discrete groups, which are defined arithmetically, are expected to carry fundamental arithmetic information. In this talk I will give an introduction to the subject. Then I will discuss some aspects of spectral theory on locally symmetric spaces, especially the Weyl law, the limit multiplicity problem and approximation of $L^2$invariants. The continuous spectrum of canonical differential operators is governed by Eisenstein series. Through the constant terms of Eisenstein series, automorphic $L$functions come into play. The analytic properties of automorphic $L$functions have a significant impact on the structure of the spectrum. This is another issue which I will discuss in some detail.
JeanMarc Schlenker (Université du Luxembourg):
The renormalized volume of hyperbolic 3manifold
Convex cocompact hyperbolic manifolds have infinite volume. However, they have a welldefined "renormalized" volume, which has a number of interesting properties. We will outline the definition of this renormalized volume, and some of its key properties, including its relations to the volume of the convex core. We will then survey some recent results on the global behavior of the renormalized volume, considered as a function over the deformation space of convex cocompact hyperbolic metrics on a given manifold. If time allows, we will describe some open questions concerning the renormalized volume of convex cocompact hyperbolic manifolds having a given Riemann surface as asymptotic boundary.
Sofiane Souafi (Université de Strasbourg):
Geometric side of a local relative trace formula
In general, the trace formula is a way to express the trace of some operator into two different ways, one in terms of geometric objects, the other one in terms of spectral objects. In a joint work with P. Delorme and P. Harinck, we give an asymptotic description of the geometric side of a local trace formula for a reductive $p$adic group $G$ relative to some symmetric subgroup $H$. As a consequence of this result, we can express spherical characters of an $H$distinguished supercuspidal representation of $G$ using weighted orbital integrals.