Addressing the active and challenging field of spectral theory, this book develops the general theory of spectra of discrete structures, on graphs, simplicial complexes, and hypergraphs. In fact, hypergraphs have long been neglected in mathematical research, but because of the discovery of Laplace operators that can probe their structure, and their manifold applications from chemical reaction networks to social interactions, they have now become one of the most active areas of interdisciplinary research. The authors' analysis of spectra of discrete structures embeds intuitive and easily visualized examples, which are often quite subtle, within a general mathematical framework. They highlight novel research on Cheeger-type inequalities that connect spectral estimates with the geometry, more precisely the cohesion, of the underlying structure. Establishing mathematical foundations and demonstrating applications, this book will be of interest to graduate students and researchers in mathematics working on the spectral theory of operators on discrete structures.
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Develops the general theory of spectra of discrete structures and demonstrates applications with a particular emphasis on hypergraphs.
Preface; Acknowledgments; Organization of the book; Part I. Basics:
Foundational Material, Elementary Aspects, Examples:
1. Introduction;
2. The
abstract setting;
3. The classical case;
4. First properties of the spectrum;
5. Floer theory; Part II. Eigenvalues and Eigenfunctions on Simplicial
Complexes and Hypergraphs:
6. Lovász extensions;
7. Discrete p-Laplacians;
8.
Cheeger inequalities;
9. Nodal domains; Part III. Additional topics:
Interlacing, Tensors, Non-backtracking Laplacians, and Applications:
10.
Interlacing and spectral classes;
11. Spectral theory of hypergraphs via
tensors;
12. The non-backtracking Laplacian;
13. Applications; Bibliography;
Index.
Jürgen Jost is a founding Director and Scientific Member at the Max Planck Institute for Mathematics in the Sciences. He is a recipient of the Gottfried-Wilhelm-Leibniz award of the DFG (1993) and is a member of the German National Academy Leopoldina, the Academy of Science and Literature at Mainz, and the Saxonian Academy of Science at Leipzig. Raffaella Mulas is a tenured Assistant Professor at VU Amsterdam, where her research is supported by NWO VENI and VU Startpremie grants. Previously, she was a Group Leader at the Max Planck Institute for Mathematics in the Sciences, where her research group was supported by a Minerva Fast Track Fellowship from the Max Planck Society, and a postdoc at the Alan Turing Institute in London. Dong Zhang is an Assistant Professor at Peking University, and was previously a postdoctoral researcher at the Max Planck Institute for Mathematics in the Sciences. His research area is discrete analysis, especially nonlinear spectral graph theory and discrete-to-continuous extensions.
