Program

Monday, October 2, 2023

Leibenson's equation on Riemannian manifolds

Alexander Grigor'yan
Bielefeld University

We consider on arbitrary Riemannian manifold the Leibenson equation $$\partial _{t}u=\Delta _{p}u^{q}.$$
This equation comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium.
Our main result is that if $p>1$ and $q>1/(p-1)$  then bounded solutions of this equation have finite propagation speed.
In $R^n$ this result has been long known due to explicit Barenblatt solutions.
The proof on manifolds is based on a certain mean value inequality for subsolutions.
This work is joint with Philipp S\"urig.

Boundedness of Schrödinger operator in energy space

Gilles Carron

University of Nantes

It is a joint work with Maël Lansade (Marseille). In a very nice paper, Maz’ya and Verbitsky have obtained and if and only if criterium for a Schrödinger operator $\Delta-V$ to be bounded between the homogeneous space $\overset{o}W^{1,}$ into its dual. We obtain a similar criterium on complete Riemannian manifolds satisfying the Poincaré inequalities and the volume doubling condition.  We will explain a relationship of this problem with the boundedness of the Hodge projector on some weighted L^2 space. 

Dynamic Isoperimetry and Operator Splitting for Detection of Coherent sets

Kathrin Völkner
FU Berlin

Coherent sets are regions of the phase space in a dynamical system which resist mixing with the surrounding space. Typical applications come from complex physical systems such as atmospheric flows and ocean dynamics. The theory of dynamic isoperimetry developed over the past decade is a geometric approach to characterise and detect coherent sets. Recently, G. Froyland and P. Koltai introduced the so-called inflated dynamic Laplacian which detects emergence and decay of finite-time coherent sets via eigenfunctions of a Laplace-Beltrami operator in a time-expanded manifold. In this talk I will discuss spectral properties of this operator and propose an approximation of its eigenvalues and eigenspaces via a Trotter product formula for its semigroup.

Tuesday, October 3, 2023

Excursion to Klaffenbach Castle

Dinner at Klaffenbach Castle

Abbildung 1 https://www.schloesserland-sachsen.de/typo3temp/assets/images/csm_Wasserschloss-Klaffenbach-Ansicht-Sebastian-Rose-Schloesserland-Sachsen-800x600px_852fcf67dd_log_abc1dc9ab7.jpg

Wednesday, October 4, 2023

Johnson-Schwartzman gap labelling and applications

David Damanik
Rice University

We give an introduction to Johnson's gap labelling based on the Schwartzman homomorphism and describe several recent applications of this theory, including a positive answer to a question of Bellissard about the structure of the almost sure spectra of random Schr\"odinger operators.

Hess-Schrader-Uhlenbrock inequality for the heat semigroup on differential forms over Dirichlet spaces tamed by distributional curvature lower bounds

Kazuhiro Kuwae
Fukuoka University

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and Tamanini (’22) as a Dirichlet space having a weak form of Bakry-\'Emery curvature lower bounds in distribution sense. After their work, Braun ('22+) established a vector calculus for it, in particular, the space of $L^2$-normed $L^{\infty}$-module describing vector fields, $1$-forms, Hessian in $L^2$-sense. In this framework, we establish the Hess-Schrader-Uhlenbrock inequality for $1$-forms as an element of $L^2$-cotangent bundles, (an $L^2$-normed $L^{\infty}$-module), which extends the result on the Hess-Schrader-Uhlenbrock inequality under an additional condition by Braun (‘22+).

Curvature and Analysis on Path Space

Eva Kopfer
University of Bonn

n this talk we discuss the interplay between geometry and stochastic analysis. This manifests in functional inequalities on path space such as gradient estimates or Harnack inequalities.

Non-local differential complexes and localization maps

Michael Hinz
Bielefeld University

We will first give an introduction to differential complexes of Kolmogorov-Alexander-Spanier type on metric measure spaces associated with unbounded non-local Dirichlet forms and mention some results. We will then comment on the approximation of local differential complexes by non-local ones. Such approximations produce natural cochain maps taking multivariate functions into differential forms. These maps may be seen as an energy based localization that can replace traditional localization by algebraic means or in terms of smooth curves. The results are joint with Jörn Kommer.

Thursday, October 5, 2023

Nonlinear Fokker-Planck-Kolmogorov Equations and Nonlinear Markov Processes

Michael Röckner
Bielefeld University

Since the middle of last century a substantial part of stochastic analysis has been devoted to the relationship between (parabolic) linear partial differential equations (PDEs), more precisely, linear Fokker-Planck-Kolmogorov equations (FPKEs), and stochastic differential equations (SDEs), or more generally Markov processes. Its most prominent example is the classical heat equation on one side and the Markov process given by Brownian motion on the other. This talk is about the nonlinear analogue, i.e., the relationship between nonlinear FPKEs on the analytic side and McKean-Vlasov SDEs (of Nemytskii-type), or more generally, nonlinear Markov processes in the sense of McKean on the probabilistic side. This program has been initiated by McKean already in his seminal PNAS-paper from 1966 and this talk is about recent developments in this field. Topics will include existence and uniqueness results for distributional solutions of the nonlinear FPKEs on the analytic side and equivalently existence and uniqueness results for weak solutions of the McKean-Vlasov SDEs on the probabilistic side. Furthermore, criteria for the corresponding path laws to form a nonlinear Markov process will be presented. Among the applications are e.g. porous media equations (including such with nonlocal operators replacing the Laplacian and posssibly being perturbed by a transport term) and their associated nonlinear Markov processes. But also the 2D Naiver-Stokes equation in vorticity form and its associated nonlinear Markov process will be discussed.

Nonlinear Dirichlet forms associated with quasiregular mappings

Lucian Beznea
IMAR Bucharest

We present a general procedure of constructing nonlinear  Dirichlet forms in the sense introduced by Petra van Beusekom,  starting from a strongly local, regular, Dirichlet form, admitting a  carré du champ operator. As a particular case, we shall describe the  nonlinear form associated with a quasiregular mapping.

(Title)

Max von Renesse
Leipzig University

(Abstract)

An almost rigidity on the positive Green function with non-negative Ricci curvature

Shouhei Honda
Tohoku University

Colding established the sharp gradient estimate on the smoothed distance function by the positive Green function on a non-parabolic manifold with nonnegative Ricci curvature. He also proved the rigidity result. In this talk, we prove an almost rigidity result of it via a non-smooth geometric analysis. This is a joint work with Yuanlin Peng (Tohoku University).

Friday, October 6, 2023

Geometric scattering theory for Dirac operators

Sebastian Boldt
Leipzig University

Given a noncompact spin manifold with a fixed topological spin structure and two complete Riemannian metrics on it, we prove a criterion for the existence and completeness of the wave operators associated with the corresponding Dirac operators. This result does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow. This is joint work with Batu Güneysu.

Generalised norm resolvent convergence: different approaches  and examples

Olaf Post
University of Trier

In this talk I will present mainly two different approaches  of operator norm convergence for operators acting in different Hilbert  spaces. One uses so-called identification operators, another one  introduced by Weidmann uses a common Hilbert space in which all spaces  are embedded. We discuss both concepts and show their equivalence. We  illustrate both concepts by examples and discuss also how it can be  specialised to Dirichlet forms.