+49 331 977-1689
Current status and CV
I spend a year abroad at Harvard University from Feb - Dec 2017.
- Ricci curvature
- Heat equation
- Unbounded graph Laplacians
Ricci curvature on birth-death processes
(with Bobo Hua), 2017, submitted
Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds
(with Radoslaw K. Wojciechowski), 2017, submitted
Magnetic sparseness and Schrödinger operators on graphs
(with Michel Bonnefont, Sylvain Golénia, Matthias Keller, Shiping Liu), 2017, submitted
Relationships between cycles spaces, gain graphs, graph coverings, path homology, and graph curvature
(with Mark Kempton, Shing-Tung Yau), 2017, submitted
Distance bounds for graphs with some negative Bakry Emery curvature
(with Shiping Liu, Norbert Peyerimhoff, Christian Rose), 2017, submitted
Rigidity properties of the hypercube via Bakry-Emery curvature
(with Shiping Liu, Norbert Peyerimhoff), 2017, submitted
Ollivier-Ricci idleness functions of graphs
(with David Bourne, David Cushing, Shiping Liu, Norbert Peyerimhoff), 2017, submitted
Bakry-Emery curvature and diameter bounds on graphs
(with Shiping Liu, Norbert Peyerimhoff), 2016, submitted
Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds
(with Michela Egidi, Shiping Liu, Norbert Peyerimhoff), 2016, submitted
Curvature and higher order Buser inequalities for the graph connection Laplacian
(with Shiping Liu, Norbert Peyerimhoff), 2015, submitted.
Li-Yau inequality on finite graphs via non-linear curvature dimension conditions
2019 | A new discrete Hopf-Rinow theorem | Matthias Keller, Florentin Münch Zeitschrift: Discrete Mathematics Seiten: 2751-2757 Band: 342 Link zur Publikation, Link zum Preprint
A new discrete Hopf-Rinow theorem
Autoren: Matthias Keller, Florentin Münch (2019)
We prove a version of the Hopf–Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf–Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.
2017 | Remarks on curvature dimension conditions on graphs | Florentin Münch Zeitschrift: Calculus of Variations and Partial Differential Equations Reihe: 56 Verlag: Springer Seiten: 11 Link zum Preprint
Remarks on curvature dimension conditions on graphs
Autoren: Florentin Münch (2017)
We show a connection between the CDE′ inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the CDψ inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a CDφψ inequality as a slight generalization of CDψ which turns out to be equivalent to CDE′ with appropriate choices of φ and ψ. We use this to prove that the CDE′ inequality implies the classical CD inequality on graphs, and that the CDE′ inequality with curvature bound zero holds on Ricci-flat graphs.
Calculus of Variations and Partial Differential Equations
2016 | Note on short time behavior of semigroups associated to selfadjoint operators | Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, Andras Telcs Zeitschrift: Bullettin of the London Mathematical Society, to appear Link zum Preprint
Note on short time behavior of semigroups associated to selfadjoint operators
Autoren: Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, Andras Telcs (2016)
We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.
Bullettin of the London Mathematical Society, to appear
2015 | Geometry and spectrum of rapidly branching graphs | Matthias Keller, Felix Pogorzelski, Florentin Münch Zeitschrift: Mathematische Nachrichten Seiten: 1636–1647 Band: 289 Link zur Publikation, Link zum Preprint
Geometry and spectrum of rapidly branching graphs
Autoren: Matthias Keller, Felix Pogorzelski, Florentin Münch (2015)
We study graphs whose vertex degree tends and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness.
Li-Yau inequalities on finite graphs.
Ultrametrische Cantormengen und Ränder von Bäumen.