To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. This title is also available as Open Access on Cambridge Core. With plentiful exercises, some with solutions, and worked examples, this will be indispensable for graduate students and researchers working at the intersection of functional analysis, non-linear differential equations and differential geometry. The geometric techniques developed are then showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids, and rough path theory. Numerous examples highlight both surprising connections between finite- and infinite-dimensional geometry, and challenges occurring solely in infinite dimensions. Topics covered include diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. The numerous exercises and examples make this book an excellent self-study resource or text for a one-semester course or seminar. The concluding section discusses current open problems and related topics. Highlights include a proof of boundary rigidity for simple surfaces as well as scattering rigidity for connections. An in-depth analysis of various geodesic X-ray transforms is carried out together with related uniqueness, stability, reconstruction and range characterization results. The required geometric background is developed in detail in the context of simple manifolds with boundary. The presentation is self-contained and begins with the Radon transform and radial sound speeds as motivating examples. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research.
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