JOURNÉESANALYSE ETPHYSIQUEMATHÉMATIQUE

Karsten BOHLEN – Regensburg David DOS SANTOS FERREIRA – LorraineMatthias LESCH (Excusé) – BonnFrancis NIER – Paris 13Yu QIAO – Xi'an & LorraineAnton THALMAIER – LuxembourgStéphane VASSOUT – Paris Diderot

4 & 5 octobre 2018IECL - Metz

3 rue Augustin Fresnel

WWW.JOURNEES-APM.IECL.UNIV-LORRAINE.FR

sebastien.breteaux / victor.nistor / tilmann.wurzbacher @univ-lorraine.fr

Accès Access

France – Metz – Technopôle – 3 rue Augustin Fresnel – UFR MIM

Salle de séminaire . . . . . . . . . . . . . . . . . . . . . . . . . UM-ARC-052 . . . . . . . . . . . . . . . . . . . . . . . . . Seminar Room

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• En voiture GPS : 49°05’41.5”N 6°13’47.2”E By car •

• En transport en commun Public transportation •depuis la Gare de Metz-Ville from Metz-Ville Train Station

Bus: Mettis B – Direction: Hôpital Mercy – Stop: Grandes Écoles

1Photo de Raymond Mortini.

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Conférenciers Speakers

Karsten BOHLEN Universität Regensburg, GermanyDavid DOS SANTOS FERREIRA Université de Lorraine, FranceMatthias LESCH: EXCUSÉ Universität Bonn, GermanyFancis NIER Université de Paris 13, FranceYu QIAO Shaanxi Normal University, ChinaAnton THALMAIER Université du Luxembourg, LuxembourgStéphane VASSOUT Université Paris Diderot, France

Programme Program

Exposés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Talks

Jeudi 4 — Thursday 4 Vendredi 5 — Friday 5

14.30 – 15.30 9.00 – 10.00A. THALMAIER K. BOHLEN

K K16.15 – 17.15 10.30 – 11.30

F. NIER S. VASSOUT

17.30 – 18.30 11.30 – 12.30Y. QIAO D. DOS SANTOS FERREIRA

Repas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Social Event

Conference diner for speakers and organizers

Thursday evening at 8PM

Brasserie des Arts et Métiers: 2 Bis Rue Gambetta, 57000 Metz

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Résumés Abstracts

ANTON THALMAIER

University of Luxembourg, Luxembourg

Brownian motion, heat equation under geometric flows and entropy formulasWe introduce the notion of a canonical Brownian motion on manifolds evolving under a geometricflow. In this framework we characterize the Ricci flow and outline a probabilistic approach to entropyformulas. In particular, we define variants of Perelman’s entropy functionals, using the Wienermeasure as reference measure, and investigate their monotonicity along the flow. If time permits, wesketch possible applications.

BOHLEN KARSTEN

Universität Regensburg, Germany

Index theory and K-homology on manifolds with a Lie structure at infinityI consider so-called Lie manifolds, which can be viewed as an axiomatization of numerous differenttypes of compactifications of complete non-compact manifolds with bounded geometry and pre-scribed behavior "at infinity". On such manifolds there is a pseudodifferential calculus and one canconsider fully elliptic operators which give rise to Fredholm operators on appropriate Sobolev spaces.A problem, proposed by Victor Nistor, asks for a general index formula of Atiyah-Singer type, valid forFredholm pseudodifferential operators contained in the Lie calculus. In this talk, which is based onjoint work with Jean-Marie Lescure, I present a solution to the problem.

DAVID DOS SANTOS FERREIRA

Université de Lorraine, France

On the linearized anisotropic Calderón problemThe anisotropic Calderon problem is the inverse problem consisting in determining a metric on acompact Riemannian manifold with boundary from the Dirichlet-to-Neuman map. The resolution ofthe problem in a conformal class follows from a similar inverse problem on the Schrödinger equationand remains an open question in dimensions higher than 3. In previous works, we could solve thisinverse problem under structural assumptions on the known metric (namely that it is conformal to awarped product with an Euclidean factor) and additional geometric assumptions on the transversalmanifold. The proof of uniqueness relies on the high frequency limit in a Green identity involving pairsof complex geometrical optics solutions to the Schrödinger equation. This talk will be concernedwith a description of the resolution of this nonlinear inverse problem under strong assumptions onthe metric and our attempts to remove the additional transversal assumptions on the geometry byrefraining from passing to the limit in the linearised problem. Unfortunately, this path only leads topartial results on the linearised problem for the time being, that is recovery of singularities of thepotential in the transversal variables.

This a joint work with Yaroslav Kurylev, Matti Lassas, Tony Liimatainen and Mikko Salo.

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MATTHIAS LESCH

Universität Bonn, Germany

Sums of linear operators in Hilbert C*-modulesGiven two closed unbounded operators A, B in a Banach space. There is a rich literature on theproblem whether the sum A+B is closed and regular on the intersection of the domains DA ∩ DB .The seminal paper by da Prato and Grisvard (1975) and its successors are mostly motivated byapplications to PDE.Another completely different and quite recent motivation comes from the unbounded picture of KK-theory. Here, one typically encounters pairs of *weakly anticommuting* self-adjoint operators acingon a Hilbert C*-module.In my talk I will present a Hilbert C*-module version of a noncommutative Dore-Venni type Theoremfor noncommuting operators. This will generalize previous work on weakly anticommuting operatorsarising from Kasparov products.

FRANCIS NIER

Université de Paris 13, France

Boundary conditions for hypoelliptic LaplaciansI will explain what are the natural boundary conditions for Bismut hypoelliptic Laplacians correspond-ing to Dirichlet and Neumann boundary conditions for Witten Laplacian. The subelliptic estimateswill be specified for such boundary value problems. Following the use of boundary value problemfor Witten Laplacian in the study of their exponentially small eigenvalues in relation with persistenthomology, I will present how those boundary conditions for the hypoelliptic case should allow todevelop a similar analysis.

YU QIAO

Shaanxi Normal University, China

Neumann-Poincaré operators on three-dimensional wedges and pseudo-differentialoperators on Lie groupoidsLet W be a three-dimensional wedge, and K be the Neumann-Poincaré operator associated to Wand the Laplace operator. We show that 1

2 ± K are isomorphisms on suitable weighted Sobolevspaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem onW. Then we make a connection between the operator K and pseudo-differential operators oncertain Lie groupoid. More precisely, we show that the operator K is an element of C∗(G) ⊗M2(C),

where G is the action groupoid M o G, with G =

(1 0a b

): a ∈ R, b ∈ R+,

, and M is a sort of

compactification of G. This result could be used to prove the Fredholmness of 12 + KΩ, where Ω

is “a domain with edge singularities" and KΩ is the Neumann-Poincaré operator associated to theLaplacian and Ω.

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STÉPHANE VASSOUT

Université Paris Diderot, France

Fourier Integral Operators on Lie GroupoidsBased on our work with Jean-Marie Lescure, I will explain the construction of Fourier IntegralOperators recently developed in the framework of Lie groupoids. This calculus not only contains theoriginal calculus for smooth compact manifolds, but also fits with many different generalizations ofthis calculus developped in various geometric situations. Actually, once the geometric data of theparticular case considered is encoded by a Lie groupoid, then the general framework above allowsto give right away the FIO calculus adapted for this particular geometric situation.

Organisateurs Organizing Committee

Sébastien BRETEAUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sebastien.breteaux@univ-lorraine.frVictor NISTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .victor.nistor@univ-lorraine.frTilmann WURZBACHER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .tilmann.wurzbacher@univ-lorraine.fr

Financement Funded by

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