Graphs and Discrete Dirichlet Spaces
Springer International Publishing (Verlag)
978-3-030-81458-8 (ISBN)
The book gives a complete account of key topics of infinite graphs, such as essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. A major feature of the book is the use of intrinsic metrics to capture the geometry of graphs. As for manifolds, Dirichlet forms in the graph setting offer a structural understanding of the interaction between spectral theory, geometry and probability. For graphs, however, the presentation is much more accessible and inviting thanks to the discreteness of the underlying space, laying bare the main concepts while preserving the deep insights ofthe manifold case.
Graphs and Discrete Dirichlet Spaces offers a comprehensive treatment of the spectral geometry of graphs, from the very basics to deep and thorough explorations of advanced topics. With modest prerequisites, the book can serve as a basis for a number of topics courses, starting at the undergraduate level.
lt;b>Matthias Keller studied in Chemnitz and obtained his PhD in Jena. He held positions in Princeton, Jerusalem and Haifa before becoming a professor at the University of Potsdam.
Daniel Lenz obtained his PhD in Frankfurt am Main. After prolonged stays in Jerusalem, Chemnitz and Houston, he is now a professor at the Friedrich Schiller University in Jena.
Radoslaw Wojciechowski got his PhD at the Graduate Center of the City University of New York following his undergraduate studies at Indiana University Bloomington. After a postdoc period in Lisbon he is now a professor at York College and the Graduate Center in New York City.
Part 0 Prelude.- Chapter 0 Finite Graphs.- Part 1 Foundations and Fundamental Topics.- Chapter 1 Infinite Graphs - Key Concepts.- Chapter 2 Infinite Graphs - Toolbox.- Chapter 3 Markov Uniqueness and Essential Self-Adjointness.- Chapter 4 Agmon-Allegretto-Piepenbrink and Persson Theorems.- Chapter 5 Large Time Behavior of the Heat Kernel.- Chapter 6 Recurrence.- Chapter 7 Stochastic Completeness.- Part 2 Classes of Graphs.- Chapter 8 Uniformly Positive Measure.- Chapter 9 Weak Spherical Symmetry.- Chapter 10 Sparseness and Isoperimetric Inequalities.- Part 3 Geometry and Intrinsic Metrics.- Chapter 11 Intrinsic Metrics: Definition and Basic Facts.- Chapter 12 Harmonic Functions and Caccioppoli Theory.- Chapter 13 Spectral Bounds.- Chapter 14 Volume Growth Criterion for Stochastic Completeness and Uniqueness Class.- Appendix A The Spectral Theorem.- Appendix B Closed Forms on Hilbert Spaces.- Appendix C Dirichlet Forms and Beurling-Deny Criteria.- Appendix D Semigroups, Resolvents and their Generators.- Appendix E Aspects of Operator Theory.- References.- Index.- Notation Index.
"This is an extremely well-written book, with motivations sprinkled throughout. One useful pedagogic device is the slow beginning with a chapter 0 giving the finite case separately, even though this is technically superfluous. The value of the book is enhanced by its many exercises. Undergraduate students and researchers should find it an extremely useful introduction to a beautiful theory." (Bhaskar Bagchi, zbMATH 1487.05003, 2022)
“This is an extremely well-written book, with motivations sprinkled throughout. One useful pedagogic device is the slow beginning with a chapter 0 giving the finite case separately, even though this is technically superfluous. The value of the book is enhanced by its many exercises. Undergraduate students and researchers should find it an extremely useful introduction to a beautiful theory.” (Bhaskar Bagchi, zbMATH 1487.05003, 2022)
| Erscheinungsdatum | 24.10.2021 |
|---|---|
| Reihe/Serie | Grundlehren der mathematischen Wissenschaften |
| Zusatzinfo | XV, 668 p. 4 illus. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 1184 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Graphentheorie |
| Schlagworte | Dirichlet form • Graph Laplacian • Heat equation on graphs • intrinsic metrics • markov semigroups • Schrödinger Operators • Spectral Graph theory • Stochastic Completeness |
| ISBN-10 | 3-030-81458-0 / 3030814580 |
| ISBN-13 | 978-3-030-81458-8 / 9783030814588 |
| Zustand | Neuware |
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