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| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 114 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 0821804138 |
Lecture notes from the conference held Aug. 1995 in Boulder, Colo.
| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 106 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821825593 |
"July 1993, volume 104, number 497 (fourth of 6 numbers)."
| Author | : Both Professors of Maths John Roe |
| Publisher | : Oxford University Press, USA |
| Total Pages | : 106 |
| Release | : 2014-08-31 |
| Genre | : MATHEMATICS |
| ISBN | : 9781470400743 |
Coarse geometry'' is the study of metric spaces from the asymptotic point of view: two metric spaces (such as the integers and the real numbers) which look the same from a great distance'' are considered to be equivalent. This book develops a cohomology theory appropriate to coarse geometry. The theory is then used to construct higher indices'' for elliptic operators on noncompact complete Riemannian manifolds. Such an elliptic operator has an index in the $K$-theory of a certain operator algebra naturally associated to the coarse structure, and this $K$-theory then pairs with the coarse cohomology. The higher indices can be calculated in topological terms thanks to the work of Connes and Moscovici. They can also be interpreted in terms of the $K$-homology of an ideal boundary naturally associated to the coarse structure. Applications to geometry are given, and the book concludes with a discussion of the coarse analog of the Novikov conjecture.
| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 184 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 0821833324 |
Coarse geometry is the study of spaces (particularly metric spaces) from a 'large scale' point of view, so that two spaces that look the same from a great distance are actually equivalent. This book provides a general perspective on coarse structures. It discusses results on asymptotic dimension and uniform embeddings into Hilbert space.
| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 175 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 9781470421762 |
Coarse geometry is the study of spaces (particularly metric spaces) from a 'large scale' point of view, so that two spaces that look the same from a great distance are actually equivalent. This point of view is effective because it is often true that the relevant geometric properties of metric spaces are determined by their coarse geometry: two examples of important uses of coarse geometry are Gromov's beautiful notion of a hyperbolic group and Mostow's proof of his famous rigidity theorem. The first few chapters of the book provide a general perspective on coarse structures. Even when only metric coarse structures are in view, the abstract framework brings the same simplification as does the passage from epsilons and deltas to open sets when speaking of continuity. The middle section of the book reviews notions of negative curvature and rigidity.Modern interest in large scale geometry derives in large part from Mostow's rigidity theorem and from Gromov's subsequent 'large scale' rendition of the crucial properties of negatively curved spaces. The final chapters discuss recent results on asymptotic dimension and uniform embeddings into Hilbert space. John Roe is known for his work on index theory, coarse geometry, and topology. His exposition is clear and direct, bringing insight to this modern field of mathematics. Students and researchers who wish to learn about contemporary methods of understanding the geometry and topology of manifolds will be well served by reading this book. Also available from the AMS by John Roe is ""Index Theory, Coarse Geometry, and Topology of Manifolds"".
| Author | : Jurgen Eichhorn |
| Publisher | : World Scientific |
| Total Pages | : 353 |
| Release | : 2009-05-29 |
| Genre | : Mathematics |
| ISBN | : 9814474223 |
For closed manifolds, there is a highly elaborated theory of number-valued invariants, attached to the underlying manifold, structures and differential operators. On open manifolds, nearly all of this fails, with the exception of some special classes. The goal of this monograph is to establish for open manifolds, structures and differential operators an applicable theory of number-valued relative invariants. This is of great use in the theory of moduli spaces for nonlinear partial differential equations and mathematical physics. The book is self-contained: in particular, it contains an outline of the necessary tools from nonlinear Sobolev analysis.
| Author | : Christian Rosendal |
| Publisher | : Cambridge University Press |
| Total Pages | : 309 |
| Release | : 2021-12-16 |
| Genre | : Mathematics |
| ISBN | : 110884247X |
Provides a general framework for doing geometric group theory for non-locally-compact topological groups arising in mathematical practice.
| Author | : Rufus Willett |
| Publisher | : Cambridge University Press |
| Total Pages | : 595 |
| Release | : 2020-07-02 |
| Genre | : Mathematics |
| ISBN | : 1108853110 |
Index theory studies the solutions to differential equations on geometric spaces, their relation to the underlying geometry and topology, and applications to physics. If the space of solutions is infinite dimensional, it becomes necessary to generalise the classical Fredholm index using tools from the K-theory of operator algebras. This leads to higher index theory, a rapidly developing subject with connections to noncommutative geometry, large-scale geometry, manifold topology and geometry, and operator algebras. Aimed at geometers, topologists and operator algebraists, this book takes a friendly and concrete approach to this exciting theory, focusing on the main conjectures in the area and their applications outside of it. A well-balanced combination of detailed introductory material (with exercises), cutting-edge developments and references to the wider literature make this a valuable guide to this active area for graduate students and experts alike.
| Author | : C.E. Aull |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 438 |
| Release | : 2001-12-31 |
| Genre | : History |
| ISBN | : 9780792369707 |
This volume mainly focuses on various comprehensive topological theories, with the exception of a paper on combinatorial topology versus point-set topology by I.M. James and a paper on the history of the normal Moore space problem by P. Nyikos. The history of the following theories is given: pointfree topology, locale and frame theory (P. Johnstone), non-symmetric distances in topology (H.-P. Künzi), categorical topology and topological constructs (E. Lowen-Colebunders and B. Lowen), topological groups (M. G. Tkacenko) and finally shape theory (S. Mardesic and J. Segal). Together with the first two volumes, this work focuses on the history of topology, in all its aspects. It is unique and presents important views and insights into the problems and development of topological theories and applications of topological concepts, and into the life and work of topologists. As such, it will encourage not only further study in the history of the subject, but also further mathematical research in the field. It is an invaluable tool for topology researchers and topology teachers throughout the mathematical world.
| Author | : Liviu I. Nicolaescu |
| Publisher | : World Scientific |
| Total Pages | : 606 |
| Release | : 2007 |
| Genre | : Mathematics |
| ISBN | : 9812708537 |
The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that ?in learning the sciences examples are of more use than precepts?. We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a ?global and analytical bias?. We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincar duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-;Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand Hlder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.
