Higher Structures in Topology, Geometry, and Physics

Higher Structures in Topology, Geometry, and Physics

Author: Ralph M. Kaufmann

Publisher: American Mathematical Society

Published: 2024-07-03

Total Pages: 332

ISBN-13: 1470471426

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This volume contains the proceedings of the AMS Special Session on Higher Structures in Topology, Geometry, and Physics, held virtually on March 26–27, 2022. The articles give a snapshot survey of the current topics surrounding the mathematical formulation of field theories. There is an intricate interplay between geometry, topology, and algebra which captures these theories. The hallmark are higher structures, which one can consider as the secondary algebraic or geometric background on which the theories are formulated. The higher structures considered in the volume are generalizations of operads, models for conformal field theories, string topology, open/closed field theories, BF/BV formalism, actions on Hochschild complexes and related complexes, and their geometric and topological aspects.


New Foundations for Physical Geometry

New Foundations for Physical Geometry

Author: Tim Maudlin

Publisher:

Published: 2014-02

Total Pages: 374

ISBN-13: 0198701306

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Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original Theory of Linear Structures.


Topology and Geometry for Physics

Topology and Geometry for Physics

Author: Helmut Eschrig

Publisher: Springer

Published: 2011-01-26

Total Pages: 397

ISBN-13: 3642147003

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A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.


Higher Structures in Geometry and Physics

Higher Structures in Geometry and Physics

Author: Alberto S. Cattaneo

Publisher: Springer Science & Business Media

Published: 2010-11-25

Total Pages: 371

ISBN-13: 081764735X

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This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics— such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics— and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more. Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. The volume is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures.


A Course in Modern Mathematical Physics

A Course in Modern Mathematical Physics

Author: Peter Szekeres

Publisher: Cambridge University Press

Published: 2004-12-16

Total Pages: 620

ISBN-13: 9780521829601

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This textbook, first published in 2004, provides an introduction to the major mathematical structures used in physics today.


Geometry, Topology and Physics

Geometry, Topology and Physics

Author: Mikio Nakahara

Publisher: Taylor & Francis

Published: 2018-10-03

Total Pages: 596

ISBN-13: 1420056948

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Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.


Geometry and Physics

Geometry and Physics

Author: Jürgen Jost

Publisher: Springer Science & Business Media

Published: 2009-08-17

Total Pages: 226

ISBN-13: 3642005411

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"Geometry and Physics" addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics. Jürgen Jost, a well-known research mathematician and advanced textbook author, also develops important geometric concepts and methods that can be used for the structures of physics. In particular, he discusses the Lagrangians of the standard model and its supersymmetric extensions from a geometric perspective.


Differential Topology and Geometry with Applications to Physics

Differential Topology and Geometry with Applications to Physics

Author: Eduardo Nahmad-Achar

Publisher:

Published: 2018

Total Pages: 0

ISBN-13: 9780750320726

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"Differential geometry has encountered numerous applications in physics. More and more physical concepts can be understood as a direct consequence of geometric principles. The mathematical structure of Maxwell's electrodynamics, of the general theory of relativity, of string theory, and of gauge theories, to name but a few, are of a geometric nature. All of these disciplines require a curved space for the description of a system, and we require a mathematical formalism that can handle the dynamics in such spaces if we wish to go beyond a simple and superficial discussion of physical relationships. This formalism is precisely differential geometry. Even areas like thermodynamics and fluid mechanics greatly benefit from a differential geometric treatment. Not only in physics, but in important branches of mathematics has differential geometry effected important changes. Aimed at graduate students and requiring only linear algebra and differential and integral calculus, this book presents, in a concise and direct manner, the appropriate mathematical formalism and fundamentals of differential topology and differential geometry together with essential applications in many branches of physics." -- Prové de l'editor.


Topology of Gauge Fields and Condensed Matter

Topology of Gauge Fields and Condensed Matter

Author: M. Monastyrsky

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 372

ISBN-13: 1489924035

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''Intended mainly for physicists and mathematicians...its high quality will definitely attract a wider audience.'' ---Computational Mathematics and Mathematical Physics This work acquaints the physicist with the mathematical principles of algebraic topology, group theory, and differential geometry, as applicable to research in field theory and the theory of condensed matter. Emphasis is placed on the topological structure of monopole and instanton solution to the Yang-Mills equations, the description of phases in superfluid 3He, and the topology of singular solutions in 3He and liquid crystals.