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Variations on a Theme of Borel

Author: Shmuel Weinberger

Publisher: Cambridge University Press

Published: 2022-12-08

Total Pages: 366

ISBN-13: 1108916848

In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture - some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology.

Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory

Author: Chris Wendl

Publisher: Cambridge University Press

Published: 2020-03-26

Total Pages: 198

ISBN-13: 1108759580

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Intersection theory has played a prominent role in the study of closed symplectic 4-manifolds since Gromov's famous 1985 paper on pseudoholomorphic curves, leading to myriad beautiful rigidity results that are either inaccessible or not true in higher dimensions. Siefring's recent extension of the theory to punctured holomorphic curves allowed similarly important results for contact 3-manifolds and their symplectic fillings. Based on a series of lectures for graduate students in topology, this book begins with an overview of the closed case, and then proceeds to explain the essentials of Siefring's intersection theory and how to use it, and gives some sample applications in low-dimensional symplectic and contact topology. The appendices provide valuable information for researchers, including a concise reference guide on Siefring's theory and a self-contained proof of a weak version of the Micallef–White theorem.

Advances in Noncommutative Geometry

Author: Ali Chamseddine

Publisher: Springer Nature

Published: 2020-01-13

Total Pages: 753

ISBN-13: 3030295974

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This authoritative volume in honor of Alain Connes, the foremost architect of Noncommutative Geometry, presents the state-of-the art in the subject. The book features an amalgam of invited survey and research papers that will no doubt be accessed, read, and referred to, for several decades to come. The pertinence and potency of new concepts and methods are concretely illustrated in each contribution. Much of the content is a direct outgrowth of the Noncommutative Geometry conference, held March 23–April 7, 2017, in Shanghai, China. The conference covered the latest research and future areas of potential exploration surrounding topology and physics, number theory, as well as index theory and its ramifications in geometry.

The Random Matrix Theory of the Classical Compact Groups

Author: Elizabeth S. Meckes

Publisher: Cambridge University Press

Published: 2019-08

Total Pages: 225

ISBN-13: 1108419526

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Provides a comprehensive introduction to the theory of random orthogonal, unitary, and symplectic matrices.

Coarse Geometry of Topological Groups

Author: Christian Rosendal

Publisher: Cambridge University Press

Published: 2021-12-16

Total Pages: 309

ISBN-13: 110884247X

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Provides a general framework for doing geometric group theory for non-locally-compact topological groups arising in mathematical practice.

Slenderness

Author: Radoslav Milan Dimitric

Publisher: Cambridge University Press

Published: 2019

Total Pages: 330

ISBN-13: 110847442X

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A leading expert presents a unified concept of slenderness in Abelian categories, with numerous open problems and exercises.

Defocusing Nonlinear Schrödinger Equations

Author: Benjamin Dodson

Publisher: Cambridge University Press

Published: 2019-03-28

Total Pages: 256

ISBN-13: 1108681670

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This study of Schrödinger equations with power-type nonlinearity provides a great deal of insight into other dispersive partial differential equations and geometric partial differential equations. It presents important proofs, using tools from harmonic analysis, microlocal analysis, functional analysis, and topology. This includes a new proof of Keel–Tao endpoint Strichartz estimates, and a new proof of Bourgain's result for radial, energy-critical NLS. It also provides a detailed presentation of scattering results for energy-critical and mass-critical equations. This book is suitable as the basis for a one-semester course, and serves as a useful introduction to nonlinear Schrödinger equations for those with a background in harmonic analysis, functional analysis, and partial differential equations.

Attractors of Hamiltonian Nonlinear Partial Differential Equations

Author: Alexander Komech

Publisher: Cambridge University Press

Published: 2021-09-30

Total Pages:

ISBN-13: 100903605X

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This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.