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Cyclic Cohomology at 40: Achievements and Future Prospects

Edited by: A. Connes Collège de France, Paris, France and Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
C. Consani Johns Hopkins University, Baltimore, MD
B. I. Dundas University of Bergen, Bergen, Norway
M. Khalkhali Western University, London, ON, Canada
H. Moscovici The Ohio State University, Columbus, OH
Available Formats:
Softcover ISBN: 978-1-4704-6977-1
Product Code: PSPUM/105
List Price: $139.00 MAA Member Price:$125.10
AMS Member Price: $111.20 Electronic ISBN: 978-1-4704-7263-4 Product Code: PSPUM/105.E List Price:$137.00
MAA Member Price: $123.30 AMS Member Price:$109.60
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $208.50 MAA Member Price:$187.65
AMS Member Price: $166.80 Click above image for expanded view Cyclic Cohomology at 40: Achievements and Future Prospects Edited by: A. Connes Collège de France, Paris, France and Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France C. Consani Johns Hopkins University, Baltimore, MD B. I. Dundas University of Bergen, Bergen, Norway M. Khalkhali Western University, London, ON, Canada H. Moscovici The Ohio State University, Columbus, OH Available Formats:  Softcover ISBN: 978-1-4704-6977-1 Product Code: PSPUM/105  List Price:$139.00 MAA Member Price: $125.10 AMS Member Price:$111.20
 Electronic ISBN: 978-1-4704-7263-4 Product Code: PSPUM/105.E
 List Price: $137.00 MAA Member Price:$123.30 AMS Member Price: $109.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$208.50 MAA Member Price: $187.65 AMS Member Price:$166.80
• Book Details

Proceedings of Symposia in Pure Mathematics
Volume: 1052023; 579 pp
MSC: Primary 11; 14; 16; 18; 19; 46; 58; 81;

This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada.

Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory.

The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc., in which cyclic homology plays the role of a unifying theme.

The researcher will find frontier research articles in which the cyclic theory provides a computational tool of great relevance. In particular, in analysis cyclic cohomology index formulas capture the higher invariants of manifolds, where the group symmetries are extended to Hopf algebra actions, and where Lie algebra cohomology is greatly extended to the cyclic cohomology of Hopf algebras which becomes the natural receptacle for characteristic classes. In algebraic topology the cyclotomic structure obtained using the cyclic subgroups of the circle action on topological Hochschild homology gives rise to remarkably significant arithmetic structures intimately related to crystalline cohomology through the de Rham-Witt complex, Fontaine's theory and the Fargues-Fontaine curve.

Graduate students and researchers interested in cyclic cohomology and applications to arithmetic geometry, global analysis, topological and algebraic $K$-theory, and physics.

• Articles
• Alexandre Baldare, Moulay Benameur and Victor Nistor - Chern-Connes-Karoubi character isomorphisms and algebras of symbols of pseudodifferential operators
• Jonathan Block, Nigel Higson and Jesus Sanchez, Jr - On Perrot’s index cocycles
• P. Carrillo Rouse - The Chern-Baum Connes assembly map for Lie groupoids
• Alain Connes and Caterina Consani - Hochschild homology, trace map and $\zeta$-cycles
• Alain Connes and Caterina Consani - Cyclic theory and the pericyclic category
• Joachim Cuntz - The image of Bott peridocity in cyclic homology
• Bjørn Dundas - Applications of topological cyclic homology to algebraic $K$-theory
• David Gepner - Algebraic $K$-theory and generalized stable homotopy theory
• Alexander Gorokhovsky and Erik van Erp - Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrose
• Lars Hesselholt - Topological cyclic homology and the Fargues-Fontaine curve
• Masoud Khalkhali and Ilya Shapiro - Hopf cyclic cohomology and beyond
• Matthew Lorentz - The Hochschild cohomology of uniform Roe algebras
• E. McDonald, F. Sukochev and X. Xiong - Quantum differntiability–The analytical perspective
• Ralf Meyer and Devarshi Mukherjee - Local cyclic homology for nonarchimedean Banach algebras
• Henri Moscovici - On the van Est analogy in Hopf cyclic cohomology
• Paolo Piazza and Xiang Tang - Primary and secondary invariants of Dirac operators on $G$-proper manifolds
• Markus Pflaum - Localization in Hochschild homology
• Raphaël Ponge - Cyclic homology and group actions
• Emil Prodan - Cyclic cocycles and quantized pairings in materials science
• Michael Puschnigg - Periodic cyclic homology of crossed products
• Anton Savin and Elmar Schrohe - Trace expansions and equivariant traces on an algebra of Fourier integral operators on $\mathbb {R}^n$
• Yanli Song and Xiang Tang - Carton motion group and orbital integrals
• Boris Tsygan - On noncommutative crystalline cohomology
• Teun van Nuland and Walter van Suijlekom - Cyclic cocycles and one-loop corrections in the spectral action
• Jinmin Wang, Zhizhang Xie and Guoliang Yu - $\ell ^1$-higher index, $\ell ^1$-higher rho invariant and cyclic cohomology

• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 1052023; 579 pp
MSC: Primary 11; 14; 16; 18; 19; 46; 58; 81;

This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada.

Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory.

The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc., in which cyclic homology plays the role of a unifying theme.

The researcher will find frontier research articles in which the cyclic theory provides a computational tool of great relevance. In particular, in analysis cyclic cohomology index formulas capture the higher invariants of manifolds, where the group symmetries are extended to Hopf algebra actions, and where Lie algebra cohomology is greatly extended to the cyclic cohomology of Hopf algebras which becomes the natural receptacle for characteristic classes. In algebraic topology the cyclotomic structure obtained using the cyclic subgroups of the circle action on topological Hochschild homology gives rise to remarkably significant arithmetic structures intimately related to crystalline cohomology through the de Rham-Witt complex, Fontaine's theory and the Fargues-Fontaine curve.

Graduate students and researchers interested in cyclic cohomology and applications to arithmetic geometry, global analysis, topological and algebraic $K$-theory, and physics.

• Articles
• Alexandre Baldare, Moulay Benameur and Victor Nistor - Chern-Connes-Karoubi character isomorphisms and algebras of symbols of pseudodifferential operators
• Jonathan Block, Nigel Higson and Jesus Sanchez, Jr - On Perrot’s index cocycles
• P. Carrillo Rouse - The Chern-Baum Connes assembly map for Lie groupoids
• Alain Connes and Caterina Consani - Hochschild homology, trace map and $\zeta$-cycles
• Alain Connes and Caterina Consani - Cyclic theory and the pericyclic category
• Joachim Cuntz - The image of Bott peridocity in cyclic homology
• Bjørn Dundas - Applications of topological cyclic homology to algebraic $K$-theory
• David Gepner - Algebraic $K$-theory and generalized stable homotopy theory
• Alexander Gorokhovsky and Erik van Erp - Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrose
• Lars Hesselholt - Topological cyclic homology and the Fargues-Fontaine curve
• Masoud Khalkhali and Ilya Shapiro - Hopf cyclic cohomology and beyond
• Matthew Lorentz - The Hochschild cohomology of uniform Roe algebras
• E. McDonald, F. Sukochev and X. Xiong - Quantum differntiability–The analytical perspective
• Ralf Meyer and Devarshi Mukherjee - Local cyclic homology for nonarchimedean Banach algebras
• Henri Moscovici - On the van Est analogy in Hopf cyclic cohomology
• Paolo Piazza and Xiang Tang - Primary and secondary invariants of Dirac operators on $G$-proper manifolds
• Markus Pflaum - Localization in Hochschild homology
• Raphaël Ponge - Cyclic homology and group actions
• Emil Prodan - Cyclic cocycles and quantized pairings in materials science
• Michael Puschnigg - Periodic cyclic homology of crossed products
• Anton Savin and Elmar Schrohe - Trace expansions and equivariant traces on an algebra of Fourier integral operators on $\mathbb {R}^n$
• Yanli Song and Xiang Tang - Carton motion group and orbital integrals
• Boris Tsygan - On noncommutative crystalline cohomology
• Teun van Nuland and Walter van Suijlekom - Cyclic cocycles and one-loop corrections in the spectral action
• Jinmin Wang, Zhizhang Xie and Guoliang Yu - $\ell ^1$-higher index, $\ell ^1$-higher rho invariant and cyclic cohomology
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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