Prof. Dr. Matthias Keller


+49 331 977-2259
+49 331 977-2899

Research interests

  • Dirichlet forms and unbounded operators on graphs
  • Spectral and geometric properties of non-positively curved graphs
  • Spectral theory of random Schrödinger operators

Scientific Output

@ googlescholar, mathscinet

  • Preprints
  • Publications
  • Theses
  • Notes and Extended Abstracts


General Cheeger inequalities for p-Laplacians on graphs
(with Delio Mugnolo), submitted.

Note on short time behavior of semigroups associated to selfadjoint operators
(with Daniel Lenz, Florentin Münch, Marcel Schmidt, Andras Telcs), submitted.

Sectional curvature of polygonal complexes with planar substructures
(with Norbert Peyerimhoff, Felix Pogorzelski), submitted.

Note on uniformly transient graphs
(with Matthias Keller, Daniel Lenz, Marcel Schmidt, Radosław K. Wojciechowski), submitted.

Global properties of Dirichlet forms in terms of Green's formula
(with Sebastian Haeseler,  Daniel Lenz, Jun Masamune, Marcel Schmidt), submitted.


Geometry and spectrum of rapidly branching graphs
(with Felix Pogorzelski, Florentin Münch), to appear in Mathematische Nachrichten.

A Feynman-Kac-Ito Formula for magnetic Schrödinger operators on graphs
(with Batu Güneysu, Marcel Schmidt), to appear in Probability Theory and Related Fields.

Eigenvalue asymptotics for Schrödinger operators on sparse graphs
(with Michel Bonnefont, Sylvain Golénia), to appear in Annales de l'Institut Fourier.

Diffussion determines the recurrent graph
(with Daniel Lenz, Marcel Schmidt, Melchior Wirth),  Advances in Mathematics, Volume 269 (2015), 364-398.

Graphs of finite measure
(with Agelos Georgakopoulos, Sebastian Haeseler, Daniel Lenz, Radosław Wojciechowski), to appear in Journal de Mathématiques Pures et Appliquées.

Intrinsic metrics on graphs - A survey
to appear in  Mathematical Technology of Networks (Proc. Bielefeld 2013), Proc. Math. & Stat. Springer-Verlag, New York.

An invitation to trees of finite cone type: random and deterministic operators
(with Daniel Lenz and Simone Warzel) to appear in Markov Processes and Related fields.

Harmonic functions of general graph Laplacians
(with Bobo Hua), Calculus of Variations and Partial Differential Equations  51 (2014), 343-362.

On the l^p spectrum of Laplacians on graphs
(with Frank Bauer, Bobo Hua), Advances in Mathematics, Volume 248, 25 November 2013, 717-735.

Note on basic features of large time behaviour of heat kernels
(with Daniel Lenz, Hendrik Vogt, Radoslaw Wojciechowski), to appear in Journal für die reine und angewandte Mathematik (Crelle's Journal).

Cheeger inequalities for unbounded graph Laplacians
(with Frank Bauer, Radoslaw Wojciechowski),  to appear in Journal of the European Mathematical Society.

Volume growth and bounds for the essential spectrum for Dirichlet forms
(with Sebastian Haeseler, Radoslaw Wojciechowski),  Journal of the London Mathematical Society,  Volume 88, Issue 3  (2013), 883--898.

Spectral Analysis of Certain Spherically Homogeneous Graphs
(with Jonathan Breuer), Operators and Matrices, Volume 7, Number 4 (2013), 825-847.

A note on self-adjoint extensions of the Laplacian on weighted graphs
(with Xueping Huang, Jun Masamune, Radoslaw Wojciechowski), Journal of Functional Analysis, Volume 265, Issue 8 (2013),  1556-1578.

Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs
(with Daniel Lenz, Radoslaw K. Wojciechowski), Mathematische Zeitschrift, Volume 274, Issue 3 (2013), 905-932.

On the spectral theory of trees with finite cone type
(with Daniel Lenz and Simone Warzel), Israel Journal of Mathematics, Volume 194, Issue 1,  (2013), 107-135.

Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions
(with Sebastian Haeseler, Daniel Lenz, Radoslaw Wojciechowski), Journal of Spectral theory, Volume 2, Issue 4  (2012), 397-432.

Absolutely continuous spectrum for multi-type Galton Watson trees
Annales Henri Poincaré, Volume 13, Issue 8 (2012), 1745-1766.

Absolutely continuous spectrum for random operators on trees of finite cone type
(with Daniel Lenz and Simone Warzel), Journal d'Analyse Mathematique, Volume 118, Issue 1 (2012), 363-396.

Dirichlet forms and stochastic completeness of graphs and subgraphs,
(with Daniel Lenz), Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume 2012, Issue 666 (2012),  189-223.

Curvature, geometry and spectral properties of planar graphs,
Discrete & Computational Geometry, Volume 46, Issue 3 (2011), Page 500-525.

Generalized solutions and spectrum for Dirichlet forms on graphs
(with Sebastian Haeseler),  Random Walks, Boundaries and Spectra, Progress in Probability, (2011) Birkhäuser, 181-201.

Cheeger constants, growth and spectrum of locally tessellating planar graphs
(with Norbert Peyerimhoff), Mathematische Zeitschrift, 268, Issue 3-4, (2011), 871-886.

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation
(with Daniel Lenz), Mathematical Modeling of Natural Phenomena "Spectral Problems" Vol. 5 No. 4 (2010),  198-204.

The essential spectrum of the Laplacian on rapidly branching tessellations,
Mathematische Annalen 346, Issue 1 (2010), 51-66.

Habilitation thesis

"On the geometry and analysis of graphs"

PhD thesis

"On the spectral theory of operators on trees"
or at here

Diploma thesis

"Produkte zufälliger Matrizen und der Lyapunov-Exponent"

Extended abstracts and lecture notes

On Cheeger's inequality for graphs, Oberwolfach Report No. 7/2015, February 2015.

An overview of curvature bounds and spectral theory of planar tessellations, Proceedings of the CIRM Meeting, 3 nr 1, Discrete Curvature; theory and applications, 2013.

Absolutely continuous spectrum on trees-random potentials, random hopping and Galton-Watson trees  Oberwolfach Report No. 50/2011, October 2011.

"Curvature and spectrum on graphs" Oberwolfach Report No. 02/2012, January 2012.

"Applications of operator theory - Discrete Operators" SS 2012.

"A minicourse on the L^p spectrum of graphs", Bizerte, Tunesia, March 2014.

CV in english, CV in deutsch


University of Potsdam since 10/2015 Professor

Technion Haifa 02-03/2015 Visiting Assistant Professor

Hebrew University Jerusalem  10/2012-09/2013 and 02-06/2011 Post-Doc

Friedrich Schiller University Jena 10/2008-09/2015 Post-Doc and PhD Student

Princeton University 10/2007-05/2008 Visiting Student/Research Collaborator

TU Chemnitz 07/2006-09/2008 PhD Student



Habilitation 15/2015, Friedrich Schiller University Jena

PhD 12/2010, Friedrich Schiller University Jena

Diploma 06/2006, TU Chemnitz


Grants and Projects

Programme to support junior researchers in obtaining third-party funding, Line A  (Advanced), 09/2014-09/2015

Golda Meir Fellowship 10/2012 - 09/2013

Short visit grant ESF 07/2012

DFG Project "Geometry of discrete spaces and spectral theory of non-local operators" (joint with D. Lenz) since 07/2012

PhD Fellowship Klaus Murmann Foundation (sdw)  07/2007 - 06/2010



Florentin Münch, Topic on Concepts of Curvature on Graphs and Manifolds, PhD

Michael Schwarz, Topic on Boundary Theory of Graphs, PhD


Florentin Münch, Li-Yau inequalities on finite graphs, Master Jena 2014

Melchior Wirth, Does diusion determine the graph structure?, Bachelor Jena 2013.

Oliver Siebert, Spectra of lamplighter random walks associated with percolation, Bachelor Jena 2013

Florentin Münch, Ultrametrische Cantormengen und Ränder von Bäumen (Ultra metric Cantor sets and boundaries of trees), Bachelor Jena 2012

Ricardo Kalke, Ricci curvature on graphs, Staatsexamen Jena 2012


(Co-)Organized Scientific meetings

One Day Workshop 2015 "New Directions in Mathematical Physics and beyond", Jena January 14th 2015 (organized by Gerhard Bräunlich, Matthias Keller, Markus Lange, Marcel Schmidt)

International Conference Fractal Geometry and Stochastics V, in Tabarz, local organizing committee

Geometric aspects in probability and analysis September 14th 2013 in Jena (organized by Matthias Keller, Daniel Lenz and Marcel Schmidt)

One Day Workshop "Schrödingeroperators - December 8th 2011, Friedrich Schiller Universität Jena

Graduate student symposium within the Summer school on "Graphs and spectra" held at the TU Chemnitz in the week 18--23 July 2011, for more information see here

Graduate student symposium September 15-16th 2010, Friedrich Schiller University Jena

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