Author: Anton Petrunin
Publisher: Springer Nature
ISBN: 3031391624
Category : Mathematics
Languages : en
Pages : 107
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Book Description
This book serves as an introductory asset for learning metric geometry by delivering an in-depth examination of key constructions and providing an analysis of universal spaces, injective spaces, the Gromov-Hausdorff convergence, and ultralimits. This book illustrates basic examples of domestic affairs of metric spaces, this includes Alexandrov geometry, geometric group theory, metric-measure spaces and optimal transport. Researchers in metric geometry will find this book appealing and helpful, in addition to graduate students in mathematics, and advanced undergraduate students in need of an introduction to metric geometry. Any previous knowledge of classical geometry, differential geometry, topology, and real analysis will be useful in understanding the presented topics.
Author: Dmitri Burago
Publisher: American Mathematical Soc.
ISBN: 0821821296
Category : Mathematics
Languages : en
Pages : 434
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Book Description
"Metric geometry" is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations. The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and Carnot-Caratheodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces).
Author: Motoko Kotani
Publisher: Advanced Studies in Pure Mathe
ISBN: 9784931469587
Category : Mathematics
Languages : en
Pages : 514
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Book Description
The first Seasonal Institute of the Mathematical Society of Japan (MSJ-SI) “Probabilistic Approach to Geometry” was held at Kyoto University, Japan, on 28th July 2008 - 8th August, 2008. The conference aimed to make interactions between Geometry and Probability Theory and seek for new directions of those research areas. This volume contains the proceedings, selected research articles based on the talks, including survey articles on random groups, rough paths, and heat kernels by the survey lecturers in the conference. The readers will benefit of exploring in this developing research area.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America
Author: Stephanie Alexander
Publisher: American Mathematical Society
ISBN: 1470475367
Category : Mathematics
Languages : en
Pages : 303
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Book Description
Alexandrov spaces are defined via axioms similar to those of the Euclid axioms but where certain equalities are replaced with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above (CBA) and curvature bounded below (CBB). Even though the definitions of the two classes of spaces are similar, their properties and known applications are quite different. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering Alexandrov spaces with CBA and with CBB simultaneously. The book also includes an extensive problem list with solutions indicated for every problem.
Author: Barrett O'Neill
Publisher: Academic Press
ISBN: 0080570577
Category : Mathematics
Languages : en
Pages : 483
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Book Description
This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.
Author: Stephanie Alexander
Publisher: Springer
ISBN: 3030053121
Category : Mathematics
Languages : en
Pages : 95
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Book Description
Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.
Author: Luigi Ambrosio
Publisher: Springer Science & Business Media
ISBN: 376438722X
Category : Mathematics
Languages : en
Pages : 333
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Book Description
The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.
Author: Ivan Cheltsov
Publisher: Springer Nature
ISBN: 3031178599
Category : Mathematics
Languages : en
Pages : 882
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Book Description
This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang The conferences were focused on the following two related problems: • existence of Kähler–Einstein metrics on Fano varieties • degenerations of Fano varieties on which two famous conjectures were recently proved. The first is the famous Borisov–Alexeev–Borisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) Tian–Yau–Donaldson Conjecture on the existence of Kähler–Einstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide. These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the Moscow–Shanghai–Pohang conferences, while the others helped to expand the research breadth of the volume—the diversity of their contributions reflects the vitality of modern Algebraic Geometry.
Author: Ana Cannas da Silva
Publisher: Springer
ISBN: 354045330X
Category : Mathematics
Languages : en
Pages : 240
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Book Description
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
Author: Dmitri Burago
Publisher: American Mathematical Society
ISBN: 1470468530
Category : Mathematics
Languages : en
Pages : 415
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Book Description
“Metric geometry” is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations. The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and Carnot-Carathéodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces). The authors tend to work with “easy-to-touch” mathematical objects using “easy-to-visualize” methods. The authors set a challenging goal of making the core parts of the book accessible to first-year graduate students. Most new concepts and methods are introduced and illustrated using simplest cases and avoiding technicalities. The book contains many exercises, which form a vital part of the exposition.
