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E-raamat: Higher Index Theory

(University of Hawaii, Manoa), (Texas A & M University)
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Higher Index TheorySuurem pilt
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"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 Ktheory 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 operatoralgebras. 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"--

Arvustused

'This book is an exceptional blend of clear, concise, delightfully written exposition and thorough scholarship. The book also has much to offer more experienced researchers.' Peter Haskell, European Mathematical Society

Muu info

Friendly introduction to higher index theory, a growing subject at the intersection of geometry, topology and operator algebras.
Introduction 1(4)
PART ONE BACKGROUND
5(156)
1 C*-algebras
7(53)
1.1 Definition and Examples
8(7)
1.2 Invertible Elements and Spectrum
15(5)
1.3 Commutative C*-algebras
20(7)
1.4 Functional Calculus
27(4)
1.5 Ideals and Quotients
31(6)
1.6 Spatial Theory
37(4)
1.7 Multipliers and Corners
41(3)
1.8 Tensor Products
44(9)
1.9 Exercises
53(4)
1.10 Notes and References
57(3)
2 K-theory for C*-algebras
60(82)
2.1 Algebraic Ko
61(6)
2.2 Approximation and Homotopy in K0
67(5)
2.3 Unbounded Traces
72(14)
2.4 The Algebraic Index Map
86(3)
2.5 The Topological K1 Group
89(3)
2.6 Bott Periodicity and the Six-Term Exact Sequence
92(4)
2.7 Some Computational Tools
96(14)
2.8 Index Elements
110(5)
2.9 The Spectral Picture of K-theory
115(11)
2.10 The External Product in K-theory
126(4)
2.11 Exercises
130(10)
2.12 Notes and References
140(2)
3 Motivation: Positive Scalar Curvature on Tori
142(19)
3.1 Differential Geometry
142(4)
3.2 Hilbert Space Techniques
146(4)
3.3 K-theory Computations
150(3)
3.4 Some Historical Comments
153(4)
3.5 Content of This Book
157(2)
3.6 Exercises
159(2)
PART TWO ROE ALGEBRAS, LOCALISATION ALGEBRAS AND ASSEMBLY
161(158)
4 Geometric Modules
163(35)
4.1 Geometric Modules
164(7)
4.2 Covering Isometries
171(3)
4.3 Covering Isometries for Coarse Maps
174(3)
4.4 Covering Isometries for Continuous Maps
177(4)
4.5 Equivariant Covering Isometries
181(14)
4.6 Exercises
195(2)
4.7 Notes and References
197(1)
5 Roe Algebras
198(19)
5.1 Roe Algebras
198(8)
5.2 Equivariant Roe Algebras
206(3)
5.3 Relationship to Group C*-algebras
209(3)
5.4 Exercises
212(3)
5.5 Notes and References
215(2)
6 Localisation Algebras and K-homology
217(57)
6.1 Asymptotically Commuting Families
218(4)
6.2 Localisation Algebras
222(5)
6.3 K-homology
227(8)
6.4 General Functoriality
235(16)
6.5 Equivariant K-homology
251(8)
6.6 The Localised Roe Algebra
259(4)
6.7 Other Pictures of K-homology
263(5)
6.8 Exercises
268(4)
6.9 Notes and References
272(2)
7 Assembly Maps and the Baum--Connes Conjecture
274(45)
7.1 Assembly and the Baum--Connes Conjecture
275(8)
7.2 Rips Complexes
283(12)
7.3 Uniformly Contractible Spaces
295(3)
7.4 Classifying Spaces
298(10)
7.5 The Coarse Baum--Connes Conjecture for Euclidean Space
308(5)
7.6 Exercises
313(2)
7.7 Notes and References
315(4)
PART THREE DIFFERENTIAL OPERATORS
319(98)
8 Elliptic Operators and K-homology
321(32)
8.1 Differential Operators and Self-Adjointness
322(7)
8.2 Wave Operators and Multipliers of L*(M)
329(8)
8.3 Ellipticity and K-homology
337(11)
8.4 Schatten Classes
348(3)
8.5 Exercises
351(1)
8.6 Notes and References
352(1)
9 Products and Poincare Duality
353(40)
9.1 A Concrete Pairing between K-homology and K-theory
354(2)
9.2 General Pairings and Products
356(6)
9.3 The Dirac Operator on Rd and Bott Periodicity
362(6)
9.4 Representable K-homology
368(7)
9.5 The Cap Product
375(7)
9.6 The Dirac Operator on a Spinc Manifold and Poincare Duality
382(8)
9.7 Exercises
390(1)
9.8 Notes and References
391(2)
10 Applications to Algebra, Geometry and Topology
393(24)
10.1 The Kadison--Kaplansky Conjecture
393(7)
10.2 Positive Scalar Curvature and Secondary Invariants
400(4)
10.3 The Novikov Conjecture
404(6)
10.4 Exercises
410(1)
10.5 Notes and References
410(7)
PART FOUR HIGHER INDEX THEORY AND ASSEMBLY
417(90)
11 Almost Constant Bundles
419(10)
11.1 Pairings
419(4)
11.2 Non-positive Curvature
423(4)
11.3 Exercises
427(1)
11.4 Notes and References
427(2)
12 Higher Index Theory for Coarsely Embeddable Spaces
429(55)
12.1 The Bott--Dirac Operator
430(13)
12.2 Bounded Geometry Spaces
443(3)
12.3 Index Maps
446(15)
12.4 The Local Isomorphism
461(7)
12.5 Reduction to Coarse Disjoint Unions
468(5)
12.6 The Case of Coarse Disjoint Unions
473(4)
12.7 Exercises
477(2)
12.8 Notes and References
479(5)
13 Counterexamples
484(23)
13.1 Injectivity Counterexamples from Large Spheres
485(2)
13.2 Expanders and Property (τ)
487(6)
13.3 Surjectivity Counterexamples from Expanders
493(10)
13.4 Exercises
503(1)
13.5 Notes and References
504(3)
APPENDICES
507(2)
A Topological Spaces, Group Actions and Coarse Geometry
509(18)
A.1 Topological Spaces
509(4)
A.2 Group Actions on Topological Spaces
513(4)
A.3 Coarse Geometry
517(8)
A.4 Exercises
525(1)
A.5 Notes and References
526(1)
B Categories of Topological Spaces and Homology Theories
527(5)
B.1 Categories We Work With
527(1)
B.2 Homology Theories on C
528(2)
B.3 Exercises
530(1)
B.4 Notes and References
531(1)
C Unitary Representations
532(4)
C.1 Unitary Representations
532(2)
C.2 Fell's Trick
534(1)
C.3 Notes and References
535(1)
D Unbounded Operators
536(12)
D.1 Self-Adjointness and the Spectral Theorem
536(3)
D.2 Some Fourier Theory for Unbounded Operators
539(1)
D.3 The Harmonic Oscillator and Mehler's Formula
540(6)
D.4 Notes and References
546(2)
E Gradings
548(13)
E.1 Graded C*-algebras and Hilbert Spaces
548(3)
E.2 Graded Tensor Products
551(7)
E.3 Exercises
558(2)
E.4 Notes and References
560(1)
References 561(14)
Index of Symbols 575(2)
Index 577
Rufus Willett is Professor of Mathematics at the University of Hawaii, Manoa. He has interdisciplinary research interests across large-scale geometry, K-theory, index theory, manifold topology and geometry, and operator algebras. Guoliang Yu is the Powell Chair in Mathematics and University Distinguished Professor at Texas A & M University. He was an invited speaker at the International Congress of Mathematicians in 2006, is a Fellow of the American Mathematical Society and a Simons Fellow in Mathematics. His research interests include large-scale geometry, K-theory, index theory, manifold topology and geometry, and operator algebras.
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