Bjørn Dundas, University of Bergen
Detecting periodic classes
Whereas the rational primes determine the arithmetic of the integers,
the sphere spectrum S gives rise to a tower of "periodic classes".
Just as Witt vectors gauge the difference between finite and infinite
characteristics, a variant of the Red-shift conjecture predicts that
the fixed points of the topological Hochschild homology described in
Hesselholt's talk does a similar job for the higher periodic classes.
In particular, starting with e.g., the prime field F_p one should be
able to detect any periodic class by just iterating the construction.
In this talk I will try to explain why this is indeed true, although
at present we are only able to "detect" (just like 2 is nonzero in
Z/4Z, but is still a zero divisor).
We will see that the iterations give rise to torus actions, and the
level of detection is intimately connected with the rank of the groups
acting. At the algebraic level this corresponds to Dress and
Siebeneicher's of the Witt vectors and the Burnside ring.
The talk builds on work by many people, in particular Waldhausen,
Connes, Bokstedt, Hsiang, Madsen, Hesselholt, Rognes, Ausoni, Veen,
Richter and Lindenstrauss.
Lars Hesselholt, University of Copenhagen & Nagoya University
Topological Hochschild homology and periodicity
In this talk, I will first review the topological or non-archimedean version of Connes' cyclic homology introduced by Bökstedt-Hsiang-Madsen in the late eighties. The basic novelty in the construction is to replace tensor products over the initial ring Z by tensor products over the initial ring spectrum S. Surprisingly, this change gives rise to a natural (inverse) Frobenius operator, whence the non-archimedean nature of the construction. I will next explain the structure of this theory in the case of the valuation ring in the perfectoid field of p-adic complex numbers, an important feature of which is a periodicity operator similar to Connes' S-operator in cyclic homology.
André Joyal, Université du Québec à Montréal
Witt vectors and the James construction (joint work with Pierre Cartier)
The Witt vectors construction is a comonad on the category of commutative rings.
We show that the comonad is cofreely cogenerated by a pointed endo-functor.
The proof depends on an abstract version of the James construction in topology.