Metric embeddings

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Embeddings of discrete metric spaces into Banach spaces have become crucial in computer science and topology. This book presents key techniques and results, focusing on bilipschitz and coarse embeddings. Topics covered include: (1) The finite determination of the embeddability of locally finite metric spaces into Banach spaces; (2) Methods for constructing embeddings; (3) Distortion related to Poincaré inequalities; (4) Families of expanders and graphs with unbounded girth and lower average degrees; (5) Banach spaces that do not allow coarse embeddings of expanders; (6) The structure of metric spaces that cannot be coarsely embedded into Hilbert spaces; (7) Applications of Markov chains to embeddability challenges; (8) Metric characterizations of Banach space properties; (9) Lipschitz free spaces. A significant portion is dedicated to presenting relevant results from Banach space theory and graph theory. The final chapter lists open problems, and each chapter, apart from the one on open problems, includes exercises and a notes section with references, discussions of related results, and further reading suggestions. This book serves as a valuable resource for those entering a rapidly evolving field with important connections to various areas of mathematics and computer science.