Smooth manifolds and observables [Documento eletrónico] / Jet Nestruev
Language: eng.Country: Switzerland, Swiss Confederation, Cham.Edition Statement: 2nd ed. Publication: Cham : Springer International Publishing, 2020Description: XVIII, 433 p. : il.ISBN: 978-3-030-45650-4.Series: Graduate Texts in Mathematics, vol. 220Subject - Topical Name: Manifolds (Mathematics) | Algebra | Quantum physics | Spintronics Online Resources:Click here to access onlineItem type | Current library | Collection | Call number | Copy number | Status | Date due | Barcode | |
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QA613.SPR FCT Manifolds, vector fields, and differential forms, an introduction to differential geometry | QA613.SPR FCT Mirzakhani's curve counting and geodesic currents | QA613.SPR FCT Submanifold theory , beyond an introduction | QA613.SPR FCT Smooth manifolds and observables | QA613.SPR FCT Topology of infinite-dimensional manifolds | QA613.SPR FCT 81047 The heat Kernel Lefschetz fixed point formula for the spin-c Dirac operator | QA613.SPR FCT 81214 An introduction to manifolds |
This textbook demonstrates how differential calculus, smooth manifolds, and commutative algebra constitute a unified whole, despite having arisen at different times and under different circumstances. Motivating this synthesis is the mathematical formalization of the process of observation from classical physics. A broad audience will appreciate this unique approach for the insight it gives into the underlying connections between geometry, physics, and commutative algebra. The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles. Smooth Manifolds and Observables is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics. This second edition adds ten new chapters to further develop the notion of differential calculus over commutative algebras, showing it to be a generalization of the differential calculus on smooth manifolds. Applications to diverse areas, such as symplectic manifolds, de Rham cohomology, and Poisson brackets are explored. Additional examples of the basic functors of the theory are presented alongside numerous new exercises, providing readers with many more opportunities to practice these concepts.
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