Wolfgang Lück

L2-Invariants: Theory and Applications to Geometry and K-Theory

• Broschiertes Buch

Produktbeschreibung

In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. It is particularly these interactions with different fields that make L2-invariants very powerful and exciting. The book presents a comprehensive introduction to this area of research, as well as its most recent results and developments. It is written in a way which enables the reader to pick out a favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material.

Produktdetails

• Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-07810-1
• Seitenzahl: 612
• Erscheinungstermin: 16. November 2010
• Englisch
• Abmessung: 235mm x 155mm x 33mm
• Gewicht: 914g
• ISBN-13: 9783642078101
• ISBN-10: 3642078109
• Artikelnr.: 32056953

Autorenporträt

Wolfgang Lück has worked on topology, K-theory, and global analysis. He completed his PhD in 1984 under the supervision of Prof. Tammo tom Dieck at Göttingen, where he also obtained the venia legendi in 1989. He has held permanent positions at the universities at Lexington, Mainz and Münster, and is currently Professor at the University of Bonn. He was awarded the Max Planck Research Award in 2003, the Gottfried Wilhelm Leibniz Award in 2008 and the von Staudt Prize in 2025. His other honors include membership of the Leopoldina (since 2010) and of the Nordrhein-Westfälische Akademie der Wissenschaft und der Künste (since 2013), Fellowship of the American Mathematical Society (since 2013), and he was president of the Deutsche Mathematiker Vereinigung during 2009–2010. In addition, he was a Max Planck Fellow from 2013–2023 and obtained an ERC Advanced Grant in 2014. To date, he has directed the theses of 30 PhD students. In Bonn, he was the director of the Hausdorff Institute from 2011–2017 and the spokesperson of the Cluster of Excellence Hausdorff Center for Mathematics from 2019–2022. He has been married to Sibylle Lück since 1984 and has four children and four grandchildren.

Inhaltsangabe

0. Introduction.- 1. L2-Betti Numbers.- 2. Novikov-Shubin Invariants.- 3. L2-Torsion.- 4. L2-Invariants of 3-Manifolds.- 5. L2-Invariants of Symmetric Spaces.- 6. L2-Invariants for General Spaces with Group Action.- 7. Applications to Groups.- 8. The Algebra of Affiliated Operators.- 9. Middle Algebraic K-Theory and L-Theory of von Neumann Algebras.- 10. The Atiyah Conjecture.- 11. The Singer Conjecture.- 12. The Zero-in-the-Spectrum Conjecture.- 13. The Approximation Conjecture and the Determinant Conjecture.- 14. L2-Invariants and the Simplicial Volume.- 15. Survey on Other Topics Related to L2-Invariants.- 16. Solutions of the Exercises.- References.- Notation.

Rezensionen

From the reviews:

"The book under review represents a fundamental monograph on the theory of L2-invariants. ... To a great extent, it is self-contained. ... The book is very clearly written, it contains many examples and we can find exercises at the end of each chapter. ... At many places in the book, the reader will find hints for further research. ... The book will be of great interest to specialists but it can also be strongly recommended for postgraduate students."

(EMS Newsletter, March, 2005)

"L2-invariants were introduced into topology by Atiyah in 1976 ... . Since then, the theory has been developed successfully by many researchers, among them the author of this monograph ... . This book is an excellent survey of many up-to-date results ... . It could be used as a very good introduction to the subject of L2-invariants ... usable either for self-study or as a text for a graduate course. ... Lück's book will become the primary reference about L2-variants for the foreseeable future."

(Thomas Schick, Mathematical Reviews, 2003 m)

"L2-invariants were introduced into topology by Atiyah in the 1970's ... . The present book is the first substantial monograph on this topic. ... This is an impressive account of much of what is presently known about these invariants ... . It combines features of a text and a reference work; to a considerable degree the chapters can be read independently, and there are numerous nontrivial exercises, with nearly 50 pages of detailed hints at the end."

(Jonathan A. Hillman, Zentralblatt MATH, Vol. 1009, 2003)