Institut für Angewandte Mathematik
Abt. Stochastische Analysis
Endenicher Allee 60
D-53115 Bonn
| Tel: | +49 228 73 62274 |
| Room: | 3.031 |
| E-mail: | fischer[at]iam.uni-bonn.de |
Teaching
Winter term 25/26
- V5F2 Selected Topics in Probability Theory: Dirichlet forms on graphs
- time: Tue 10-12, starting October 14.
- room: N0.007
- goals: The course aims to give an introduction to analysis on locally summable graphs and discrete Dirichlet spaces. We will focus on intrinsic metrics and spectral estimates (incl. Hopf-Rinow theorem, Cheeger and Brooks inequ.), the type problem, and stochastic completeness. Depending on the wishes of the participants, one of the following topics will be covered also: general theory of regular Dirichlet forms; Royden and Martin boundary theory; much more about recurrence/transience; basics of combinatorial, Bakry-Émery, and Ollivier’s Ricci curvature; p-potential theory on infinite graphs; Feynman–Kac formula on infinite graphs; or heat kernel bounds.
- Scripts: ISEM26 Lecture Notes (lectures 1-9: ~ first 100 pages, and lectures 13-14), Kamtue's Thesis (lectures 10-11, Chapters 3 and 4), research article by Münch-Wojciechowski (lectures 11-12, Sections 1 and 4.3)
- eCampus
- EXAM: Feb 19, 2026. The exam takes 30 min and is oral. The room is 3.031. Preferred time slots: 10:00-10:30, 10:45-11:15, 11:30-12:00 , …
- Preparation tasks:
- for lecture 2 (21.10.25): read p. 4-7 in the script (resolvents and semigroups)
- for lecture 3 (28.10.25): read p. 24-27 (Section 1.7 on the Dirichlet problem), and p. 29-32 (until the proof of Prop. 2.4). In lecture 3, we discussed the linked exercises.
- for lecture 4 (4.11.25): read p. 35-36 (until the minimum principle), and solve the exercises. The solution can be found here.
- for lecture 5 (11.11.25): read p. 47-56 (Sections 3.1 - 3.3).
- for lecture 6 (18.11.25): read p. 61-65. In lecture 6, we discussed the linked exercises.
- for lecture 7 (25.11.25): solve the exercises. On the same day and in the same room, I will give a research seminar talk at 4:15 pm about the Liouville property on graphs. Feel free to join.
- for lecture 8 (2.12.25): solve the exercises.
- for lecture 9 (9.12.25): in lecture 8 we concentrated on resolvents. Read the associated proofs for the semigroups (p. 85 - 89). Additionally, read Section 6.2 (p. 97-98).
- for lecture 10 (16.12.25): read Chapter 1 in the second script. We calculated some specific curvature bounds in the sense of Bakry-Émery during the lecture. For getting a better feeling for the curvature have a look here (normalized curvature with/without dimension). More about Gamma-calculus can be found e.g. here.
- there are NO lectures on Dec 23 and Jan 6. There are no preparation tasks for the remaining lectures.
Research Interests
- Theory of linear and quasi-linear Schrödinger operators on infinite graphs, especially
- Potential theory
- Martin boundary theory
- Spectral theory
Publications
Published Papers
- Quasi-Linear Criticality Theory and Green's Functions on Graphs (Journal d'Analyse Mathematique, 2025)
- Optimal Poincaré-Hardy-type Inequalities on Manifolds and Graphs (with Christian Rose, Indagationes Mathematicae, 2025)
- Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces (with Norbert Peyerimhoff, Israel Journal of Mathematics, 2025)
- Capacity of infinite graphs over non-Archimedean ordered fields (with Matthias Keller, Anna Muranova and Noema Nicolussi, Journal of Mathematical Analysis and Applications, 2025)
- On the Optimality and Decay of p-Hardy Weights on Graphs (Calculus of Variations and Partial Differential Equations, 2024)
- An Improved Discrete p-Hardy Inequality (with Matthias Keller and Felix Pogorzelski, Integral Equations and Operator Theory, 2023)
- A Non-Local Quasi-Linear Ground State Representation and Criticality Theory (Calculus of Variations and Partial Differential Equations, 2023)
- Riesz Decompositions for Schrödinger Operators on Graphs (with Matthias Keller, Journal of Mathematical Analysis and Applications, 2021)
Preprints
- Characterizations of
p -Parabolicity on Graphs (with Andrea Adriani and Alberto G. Setti, submitted)
Reports
- Optimal Poincaré-Hardy Inequalities on Manifolds and Graphs (OWR, 2025)
- A Fatou-Naim-Doob-Theorem for Schrödinger Operators on Graphs (OWR, 2021)
Doctoral Thesis
- Hardy Inequalities on Graphs (Potsdam, 2024)
Conference Organisation
- September 14-18, 2026: Geometric and Analytic functional inequalities in Discrete spaces, TSIMF, Sanya, China (together with Shubham Gupta, Bobo Hua and Matthias Keller)
Seminars, Conferences and Workshops
| 2026 | May | Conference (upcoming) | International Conference on Elliptic and Parabolic Problems Talk in MS-12: Characterizations of p-Parabolicity on Graphs | Gaeta, Italy |
| Apr. | Conference (upcoming) | Mathematical aspects of the physics with non-self-adjoint operators Poster: Characterizations of p-Parabolicity on Graphs | CIRM, France | |
| 2025 | Nov | Seminar | Talk: Characterizations of p-Parabolicity on Graphs | Potsdam, Germany |
| Nov | Seminar | Talk: On the equivalence of the Liouville property and the Khas'minskiı criterion | Bonn, Germany | |
| July | Workshop | Workshop on spectral geometry, PDEs and mathematical physics Talk: Optimal Poincaré-Hardy inequalities on graphs | Hagen, Germany | |
| July | Seminar | Talk: Optimal Poincaré-Hardy inequalities on graphs | Milan, Italy | |
| May | Seminar | Talk: Optimal Poincaré-Hardy inequalities on graphs | Chemnitz, Germany | |
| May | Seminar | Talk: Optimal Poincaré-Hardy inequalities on graphs | Bielefeld, Germany | |
| Mar. | Conference | Differential geometry beyond Riemannian manifolds | Bonn, Germany | |
| Mar. | Workshop | DOGS 25 Talk: p-Rellich Inequalities on Graphs | Jena, Germany | |
| Mar. | Workshop | Hardy Inequalities in Discrete and Continuum Settings Talk: Optimal Poincaré-Hardy inequalities on manifolds and graphs | Oberwolfach, Germany | |
| Feb. | Workshop | N^4 DF Talk: Optimal Poincaré-Hardy inequalities on manifolds and graphs | Jena, Germany | |
| 2024 | Sep. | Conference | Metric Measure Spaces, Ricci Curvature, and Optimal Transport Talk: Sharp Hardy-type inequalities on non-compact harmonic manifolds | Varenna, Italy |
| Sep. | Workshop | Calculus of Variations: a New Generation | Bonn, Germany | |
| June | Workshop | Walkshop on Mathematical Physics Talk: p-Parabolic Graphs | Hagen, Germany | |
| May | Seminar | Talk: Hardy-type inequalities for non-compact harmonic manifolds and graphs | Como, Italy | |
| Mar. | Conference | Analysis on fractals and networks, and applications Talk: Hardy-type inequalities for non-compact harmonic manifolds and graphs | CIRM, France | |
| 2023 | Nov. | Seminar | Talk: On the Optimality and Decay of p-Hardy Weights on Locally Finite Graphs | Wrocław, Poland |
| Nov. | PhD Seminar | Talk: p-Parabolic Graphs | Wrocław, Poland | |
| Oct. | Seminar | Talk: On the Optimality and Decay of p-Hardy Weights on Graphs | Dortmund, Germany | |
| Oct. | Conference | Panorama of Mathematics II | Bonn, Germany | |
| June | Conference | ILAS2023 Talk: A Riesz Decomposition for Schrödinger Operators on Graphs | Madrid, Spain | |
| May | Seminar | Talk: Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces | Potsdam, Germany | |
| Apr. | Workshop | Geometric Aspects of Evolution and Control Talk: Optimal p-Hardy Weights on Locally Finite Graphs | Hagen, Germany | |
| 2022 | Dec. | Seminar | Talk: Optimal p-Hardy Weights on Locally Finite Graphs | Bonn, Germany |
| Dec. | Seminar | Talk: Optimal p-Hardy Weights on Locally Finite Graphs | Vienna, Austria | |
| Nov. | Seminar | Talk: Optimal Hardy Weights on Locally Finite Graphs | Como, Italy | |
| Okt. | Seminar | SIAM Chapter Potsdam Talk: Optimal p-Hardy Weights on Graphs | Potsdam, Germany | |
| Jun. | Conference (online) | OTAMP Talk: Characterisations of Criticality for Quasi-Linear Schrödinger Operators on Graphs | Stockholm, Sweden | |
| Apr. | Seminar | Talk (Analysis and PDE Seminar): Criticality Theory for Quasi-Linear Schrödinger Operators on Graphs | Durham, UK | |
| 2021 | Dec. | Conference | QGraph 2021 Talk: A Non-Linear Ground State Representation On Graphs | Stockholm, Sweden |
| Sep. | Conference (online) | DMV-ÖMG Jahrestagung 2021 Talk: Hardy Inequalities on Graphs | Passau, Germany | |
| Jan. | Conference (online) | Geometry, Dynamics and Spectrum of Operators on Discrete Spaces Talk: A Fatou-Naim-Doob-type Theorem for Schrödinger Operators on Graphs | Oberwolfach, Germany | |
| 2020 | Dec. | Seminar (online) | Talk: Reduced Functions, Minimally Thin Sets and a Fatou-Naim-Doob-type Theorem for Schrödinger Operators on Graphs | Leipzig, Germany |
| Dec. | Seminar (online) | Ph.D. Seminar Talk: What are Optimal Hardy Inequalities? | Potsdam, Germany | |
| Oct. | Workshop | Analysis on graphs Talk: Markov Semigroups associated with Graphs on l^p | Jena, Germany | |
| Sep. | Conference (online) | DMV Jahrestagung Talk: An Improved Discrete p-Hardy Inequality | Chemnitz, Germany | |
| 2019 | Sep. | Conference | DMV Jahrestagung Talk: A Riesz Decomposition for Schrödinger Operators on Graphs | Karlsruhe, Germany |
| July | Seminar | Ph.D. day Talk: An Improved Discrete p-Hardy Inequality | Potsdam, Germany | |
| June | Conference | Geometric aspects of harmonic analysis and spectral theory Talk: A Riesz Decomposition for Schrödinger Operators on Graphs | Haifa (Technion), Israel | |
| Apr. | Seminar | Talk: A Riesz Decomposition for Schrödinger Operators on Graphs | Hamburg (TUHH), Germany | |
| Apr. | Conference | Geometry at Infinity | Münster, Germany | |
| 2018 | Sep. | Summer School | Generalized Curvatures GenCurv2018 | Lausanne, Switzerland |
| July | Seminar | Ph.D. day Talk: A Riesz Decomposition for Schrödinger Operators on Graphs | Potsdam, Germany | |
| June | Workshop | Dirichlet forms on graphs, Talk: Infinite Graphs: The Setting in a Nutshell | Jena, Germany | |
| 2017 | Nov. | Conference | Discrete And Continuous Models in the Theory of Networks | Bielefeld, Germany |
| Oct. | Conference | Analysis and Geometry on Graphs and Manifolds | Potsdam, Germany | |
| 2016 | Dec. | Workshop | Dirichlet Forms, Operator Theory and Mathematical Physics Talk: Dirichlet Forms and Stochastic Completeness on Graphs | Hammamet, Tunisia |