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Full Description
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.
Contents
A. Baldare, M. Benameur, and V. Nistor, Chern-Connes-Karoubi character isomorphisms and algebras of symbols of pseudodifferential operators
J. Block, N. Higson, and J. Sanchez Jr., On Perrot's index cocycles
P. Carrillo Rouse, The Chern-Baum Connes assembly map for Lie groupoids
A. Connes and C. Consani, Hochschild homology, trace map and $\zeta$-cycles
A. Connes and C. Consani, Cyclic theory and the pericyclic category
J. Cuntz, The image of Bott peridocity in cyclic homology
B. I. Dundas, Applications of topological cyclic homology to algebraic $K$-theory
D. Gepner, Algebraic $K$-theory and generalized stable homotopy theory
A. Gorokhovsky and E. van Erp, Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrose
L. Hesselholt, Topological cyclic homology and the Fargues-Fontaine curve
M. Khalkhali and I. Shapiro, Hopf cyclic cohomology and beyond
M. Lorentz, The Hochschild cohomology of uniform Roe algebras
E. McDonald, F. Sukochev, and X. Xiong, Quantum differntiability-The analytical perspective
R. Meyer and D. Mukherjee, Local cyclic homology for nonarchimedean Banach algebras
H. Moscovici, On the van Est analogy in Hopf cyclic cohomology
P. Piazza and X. Tang, Primary and secondary invariants of Dirac operators on $G$-proper manifolds
M. J. Pflaum, Localization in Hochschild homology
R. Ponge, Cyclic homology and group actions
E. Prodan, Cyclic cocycles and quantized pairings in materials science
M. Puschnigg, Periodic cyclic homology of crossed products
A. Savin and E. Schrohe, Trace expansions and equivariant traces on an algebra of Fourier integral operators on $\mathbb{R}^n$
Y. Song and X. Tang, Carton motion group and orbital integrals
B. Tsygan, On noncommutative crystalline cohomology
T. D. H. Van Nuland and W. D. van Suijlekom, Cyclic cocycles and one-loop corrections in the spectral action
J. Wang, Z. Xie, and G. Yu, $\ell^1$-higher index, $\ell^1$-higher rho invariant and cyclic cohomology.
