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Book DetailsProceedings of Symposia in Pure MathematicsVolume: 105; 2023; 579 ppMSC: 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 Ktheory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory.
The reader will find survey articles providing a userfriendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de RhamWitt 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 RhamWitt complex, Fontaine's theory and the FarguesFontaine curve.ReadershipGraduate students and researchers interested in cyclic cohomology and applications to arithmetic geometry, global analysis, topological and algebraic \(K\)theory, and physics.

Table of Contents

Articles

Alexandre Baldare, Moulay Benameur and Victor Nistor  ChernConnesKaroubi 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 ChernBaum 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 FarguesFontaine 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 oneloop corrections in the spectral action

Jinmin Wang, Zhizhang Xie and Guoliang Yu  $\ell ^1$higher index, $\ell ^1$higher rho invariant and cyclic cohomology


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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 Ktheory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory.
The reader will find survey articles providing a userfriendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de RhamWitt 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 RhamWitt complex, Fontaine's theory and the FarguesFontaine 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  ChernConnesKaroubi 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 ChernBaum 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 FarguesFontaine 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 oneloop corrections in the spectral action

Jinmin Wang, Zhizhang Xie and Guoliang Yu  $\ell ^1$higher index, $\ell ^1$higher rho invariant and cyclic cohomology