Dr. Martin Kell

Kontakt / Contact information

Fachbereich Mathematik
Universität Tübingen
Auf der Morgenstelle 10
72076 Tübingen
Germany

Tel: +49-(0)7071/29-7 62 18
[cryptmail:pdlowr-pduwlq1nhooCpdwk1xql0wxhelqjhq1gh|Hpdlo]
Raum: C 5 P 08

 

Vorlesung / Lecture - Linear Partial Differential Equations (in English)

Sprechstunde / Office hours - open door policy or via appointment

 

Forschungsinteressen / Research interests

  • Geometry and Analysis of Metric Spaces
  • Generalized Curvature
  • Flows on Metric Spaces

Aktuelle Forschung / Current research

I'm interested in metric spaces whose Sobolev spaces are not Hilbert spaces. Those spaces can be seen as generalizations of Finsler manifolds. One distinguished property is that the natural heat flow and the Laplacian are non-linear.  Whereas many results are known for Riemannian-like metric spaces, called infinitesimally Hilbertian, very few results are known for spaces with non-linear infinitesimal structure. A first approach is to find a sectional curvature analogue similar to Alexandrov curvature. Convexity of the distance can be seen as a non-positive curvature. Currently, I'm trying to obtain a meaningful notion of non-negative curvature which hopefully gives a Finslerian version of Alexandrov spaces with curvature bounded from below.

Wissenschaftliche Veröffentlichungen / Publications

  • (with Galaz-García, F., Mondino, A., Sosa, G.) On quotients of spaces with Ricci curvature bounded below (preprint 2017 arXiv)
  • Transport maps, non-branching sets of geodesics and measure rigidity, Advances in Mathematics 320, November 2017 (journal link, free until 2017-10-31, preprint arXiv)
  • (with Mondino, A.) On the volume measure of non-smooth spaces with Ricci curvature bounded below, accepted Annali della Scuola Normale Superiore di Pisa DOI 10.2422/2036-2145.201608_007 (preprint at arXiv)
  • Symmetric orthogonality and contractive projections in metric spaces, submitted (preprint 2016 arXiv)
  • Sectional curvature-type conditions on metric spaces, submitted (preprint 2016 arXiv)
  • On Cheeger and Sobolev differentials in metric measure spaces, submitted (preprint 2015 arXiv)
  • A note on non-negatively curved Berwald spaces, submitted (preprint 2015 arXiv)
  • Uniformly convex metric spaces Analysis and Geometry of Metric Spaces 2(1), December 2014. (preprint arXiv)
  • (with Bačák, M., Hua, B., Jost, J., Schikorra, A.) A notion of nonpositive curvature for general metric spaces Differential Geometry and its Applications 38, November 2015. (preprint arXiv)
  • q-heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space, Journal of Functional Analysis 271(8) October 2016 (preprint 2014 arXiv)
  • On Interpolation and Curvature via Wasserstein Geodesics, Advances in Calculus of Variations 10(2), April 2017. (preprint arXiv)
  • (with Hua, B., Xia, C.) Harmonic functions on metric measure spaces, submitted (preprint 2013 arXiv)
  • (with Jost, J., Rodrigues, Chr. S.) Representation of Markov chains by random maps: existence and regularity conditions, Calculus of Variations and Partial Differential Equations 54(3), November 2015 (preprint arXiv)
  • Conley index of isolated equilibria Topological Methods In Nonlinear Analysis 38(2), December 2011.

Andere Veröffentlichungen / Not intended for publication

  • A Note on Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces, preprint 2013 (at arXiv)
  • Local attractor continuation of non-autonomously perturbed systems, preprint 2011 (at arXiv)
  • Stability of the global attractor under Markov-Wasserstein noise, preprint 2011 (at arXiv)
  • Dissertation - On curvature conditions using Wasserstein spaces - University of Leipzig August 2014 (at Qucosa)
  • Diplomarbeit - Conley index of isolated equilibria - University of Rostock August 2010