Potential Theory and Geometry on Lie Groups
Part of New Mathematical Monographs
 Author: N. Th. Varopoulos, Université de Paris VI (Pierre et Marie Curie)
 Publication planned for: May 2020
 availability: Not yet published  available from May 2020
 format: Hardback
 isbn: 9781107036499
Hardback
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This book provides a complete and reasonably selfcontained account of a new classification of connected Lie groups into two classes. The first part describes the use of tools from potential theory to establish the classification and to show that the analytic and algebraic approaches to the classification are equivalent. Part II covers geometric theory of the same classification and a proof that it is equivalent to the algebraic approach. Part III is a new approach to the geometric classification that requires more advanced geometric technology, namely homotopy, homology and the theory of currents. Using these methods, a more direct, but also more sophisticated, approach to the equivalence of the geometric and algebraic classification is made. Background material is introduced gradually to familiarise readers with ideas from areas such as Lie groups, differential topology and probability, in particular, random walks on groups. Numerous open problems inspire students to explore further.
Read more Introduces a new approach to the classification of Lie groups
 Provides the necessary background material on theory of currents, random walk theory, etc. for readers with no prior knowledge of these areas
 Contains numerous open problems to inspire further development
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×Product details
 Publication planned for: May 2020
 format: Hardback
 isbn: 9781107036499
 dimensions: 228 x 152 mm
 contains: 20 b/w illus. 130 exercises
 availability: Not yet published  available from May 2020
Table of Contents
Preface
1. Introduction
Part I. The Analytic and Algebraic Classification:
2. The classification and the first main theorem
3. NCgroups
4. The B–NB classification
5. NBgroups
6. Other classes of locally compact groups
Appendix A. Semisimple groups and the Iwasawa decomposition
Appendix B. The characterisation of NBalgebras
Appendix C. The structure of NBgroups
Appendix D. Invariant differential operators and their diffusion kernels
Appendix E. Additional results. Alternative proofs and prospects
Part II. The Geometric Theory:
7. The geometric theory. An introduction
8. The geometric NCtheorem
9. Algebra and geometries on Cgroups
10. The end game in the Ctheorem
11. The metric classification
Appendix F. Retracts on general NBgroups (not necessarily simply connected)
Part III. Homology Theory:
12. The homotopy and homology classification of connected Lie groups
13. The polynomial homology for simply connected soluble groups
14. Cohomology on Lie groups
Appendix G. Discrete groups
Epilogue
References
Index.
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