Official Oberwolfach pages


Send your talk proposals to konstantin.pankrashkin@math.u-psud.fr

Schedule (very preliminary)




Sunday

  • 17:00-19:00: Arrivals, Check-in
  • 18:30-20:00, Dinner
Sunday 20:00 - Monday 8:00: Free discussion or sleep

 

Monday

  • 8:00-9:00, Breakfast
  • 9:00-10:30, Luca Rizzi: An introduction to sub-Laplacians and their self-adjointness properties
  • 10:45-12:15, Maryna Kachanovska: Transparent boundary conditions for wave propagation in fractal trees
    In this talk we will consider the problem of numerical modelling of wave propagation in lungs, described by a weighted wave equation on a 1D fractal tree (which, in turn, models a bronchiolar network). We will define the Dirichlet and Neumann problems on the tree and discuss the underlying variational framework, as well as some properties of the associated spaces (density of compactly supported functions, compact embeddings). The principal difficulty for numerical computations stems from the fact that the tree has an infinite number of edges. In order to cope with this problem, we truncate the computational domain with the help of transparent boundary conditions (Dirichlet-to-Neumann operator). We will present several characterizations of the symbol of the DtN operator (solution of a non-linear equation, meromorphic series etc.) and will show how they can be used for approximating the DtN operator by local operators. Time permits, we will discuss the error induced by this approximation (whose estimate relies on the asymptotic behaviour of eigenvalues of the weighted laplacian on fractal trees), and present some numerical experiments.This talk is based on joint works with Patrick Joly (INRIA) and Adrien Semin (TU Darmstadt).
  • 12:30-14:00, Lunch
  • 14:00-16:45, Free discussion
  • 16:45-18:15, Daniel Lenz: Dirichlet forms and boundaries of graphs. Part 1
  • 18:30-20:00 Dinner
Monday 20:00 - Tuesday 8:00: Free discussion or sleep

 

Tuesday

  • 8:00-9:00, Breakfast
  • 9:00-10:30, Matthias Keller: Dirichlet forms and boundaries of graphs. Part 2
  • 10:45-12:15, Wolfgang Woess:  An introduction to Martin boundary theory for Markov
    transition operators
    The theoretical construction of the Martin compactification, its basic properties,  and what is known about "geometric" realisations. In particular: under quite general assumptions the Martin compactification has a natural surjection onto the end compactification. Sources: my Markov chain book, some old papers, and a hint at recent developments.
  • 12:30-14:00, Lunch
  • 14:00-16:00, Free discussion
  • 16:00-17:00, Robert Neel: Extending Brownian motion to a family of Grushin-type singularities
    Abstract: We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. Stochastic analysis gives a quick proof and clear intuition for the fact that heat (or a diffusion that extends Brownian motion) can only cross the singularity for an intermediate range of the parameter. When crossing is possible, one must specify the behavior at the singularity. In the case when the singularity consists of one point, we give a complete description of these diffusions, and we describe a “best” extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the sense of Chen-Fukushima. This extension, however, does not correspond to the bridging extension, which was introduced by Boscain-Prandi, when they previously considered self-adjoint extensions of the Laplace-Beltrami operator on the Riemannian part for these surfaces. We clarify that several of the extensions they considered induce diffusions that are carried by the Martin compactification at the singularity, which is much larger than the (one-point) metric completion. In the case when the singularity is more than one-point, a complete classification of diffusions extending Brownian motion would be unwieldy. Nonetheless, we again describe a “best” extension which respects the isometry group, and in this case, the diffusion corresponds to the bridging extension. From the probabilistic perspective, one nice feature of this work is that a prominent role is played by Bessel processes (of every real dimension) and the classical theory of one-dimensional diffusions and their boundary conditions. Time permitting, we will also indicate some open questions relating random walks to the determination of a canonical diffusion extending Brownian motion (and thus a canonical extension of the Laplacian). This is joint work with Ugo Boscain.
  • 17:15-18:15, Adrien Semin: Density and trace results in generalized fractal networks
  • 18:30-20:00 Dinner
Tuesday 20:00 - Wednesday 8:00: Free discussion or sleep


To be continued...


Further talks to be scheduled (working titles):


  • Valentina Franceschi & Dario Prandi: On the essential self-adjointness of singular sub-Laplacians
    Given a smooth manifold $M$, a sub-Laplacian is a hypoelliptic operator $H$, naturally associated to a sub-Riemannian geometric structure and to a volume measure on it. If the structure is Riemannian and complete, the associated Laplace-Beltrami operator (in this case the volume is the intrinsic Riemannian measure) is essentially self-adjoint (Gaffney). This amounts to say that the solutions to the Schr\"odinger equation on $M$ are well defined without imposing any boundary conditions. If the structure is sub-Riemannian, sub-Laplacians are also essentially self-adjoint, assuming completeness of the metric structure and smoothnessn of the volume measure (Strichartz). In this seminar, we address the case where the structure is sub-Riemannian and (1) either the measure (chosen to be intrinsic) is non-smooth, (2) or the metric structure is non-complete. 

    Regarding (1), we present results obtained in collaboration with Dario Prandi and Luca Rizzi, concerning sub-Riemannian structures endowed with singular measures. A standing conjecture, formulated by Boscain and Laurent, asserts that singular sub-Laplacians are essentially self-adjoint out of the singularity. We will explain our results supporting the conjecture and underline the cases that are not included in our analysis. 

    Regarding (2), we present recent results obtained in a joint work with R. Adami, U. Boscain and D. Prandi, on 3D sub-Laplacians defined on non-complete sub-Riemannian manifolds, obtained by removing a point from a complete one. We show that, unlike the 3D Euclidean case, essential self-adjointness holds in this setting. 

  • Valentina Franceschi & Dario Prandi: Hardy inequalities in subriemannian settings

  • Noema Nicolussi: Self-adjoint extensions of infinite quantum graphs

Participants


  1. Ugo Boscain (organizer) http://www.cmapx.polytechnique.fr/~boscain
  2. Valentina Franceschi https://sites.google.com/site/franceschivale/
  3. Maryna Kachanovska https://sites.google.com/site/mkachanovska/Home
  4. Matthias Keller https://www.math.uni-potsdam.de/professuren/graphentheorie/team/prof-dr-matthias-keller/ 
  5. Aleksey Kostenko (organizer) https://www.mat.univie.ac.at/~kostenko/
  6. Daniel Lenz http://www.analysis-lenz.uni-jena.de/
  7. Mark Malamud https://scholar.google.com/citations?user=8TjZGSgAAAAJ
  8. Robert Neel https://www.lehigh.edu/~rwn209/
  9. Noema Nicolussi https://arxiv.org/search/math?searchtype=author&query=Nicolussi%2C+N
  10. Konstantin Pankrashkin (organizer) https://www.math.u-psud.fr/~pankrashkin/
  11. Dario Prandi https://www.darioprandi.com/
  12. Luca Rizzi https://sites.google.com/view/rizziluca/home
  13. Adrien Semin https://scholar.google.de/citations?user=7K_88igAAAAJ
  14. Igor Sheipak  https://scholar.google.fr/scholar?hl=fr&as_sdt=0%2C5&q=igor+sheipak
  15. Anton Thalmaier http://math.uni.lu/thalmaier/
  16. Wolfgang Woess https://www.math.tugraz.at/~woess/