Title: Non-linear cyclic homology and a conjecture of J.-L. Loday, I, II
Speaker: Crichton Ogle (OSU)
Abstract: The starting point for the connection between cyclic homology and algebraic K-theory is a conjecture posed by J.-L. Loday in 1981, the complete answer to which is still unknown. These two talks will present some recent progress on determining the extent to which Loday's original conjecture can be proven, and what questions remain.
Talk I will define the "nilpotent" part of Loday's original non-linear cyclic complex, which (up to formal deformation), coincides with its linear analogue in ordinary cyclic homology. Following the work of Goodwillie and Cortinas, the connection of the homology of this complex to rationalized algebraic K-theory is now completely describable and yields the expected relation with the Goodwillie-Jones-Weibel Chern character to negative cyclic homology.
Talk II will describe the construction of Loday's conjectured map on the level of simplices, which is sufficient to provide a map to rational K-theory from a monomial covering of non-linear cyclic homology, and which affirmatively answers a question recently posed by B. Tsygan. The exact image of this map, and whether it factors as Loday originally conjectured, is much less clear. We will discuss some open questions related to this construction and how they potentially connect to questions in algebraic K-theory.
Note: Part I of this seminar will take place on February 15, 2018 while Part II will take place on February 22, 2018 at the same place and time.