Report on the work done in 2015. 1. Results What I did this year was a little bit of going off on a tangent, except that the tangent happens to be a rather old one -- in fact predating the current project by several years. Namely, it concerns a result from 2006 that was my first work on cyclic homology in positive characteristic -- a proof of Kontsevich-Soibelman Degeneration Conjecture. The conjecture asserts that for a smooth and proper DG algebra over a field of characteristic 0 , the non-commutative Hodge-tode Rham spectral sequence degenerates (if the DG algebra is Morita-equivalent to an algebraic variety, this is the standard Hodge-to-de Rham degeneration). Positive characteristic enters the picture via the method of the proof. It seems that classical analytic approach to Hodge-to-de Rham degeneration leads nowhere. What works is an algebraic approach suggested by Deligne and lllusie in 1987, based on reduction to positive characteristic and utilizing the properties of the Cartier map. What I did in 2006 was to generalize this whole story to the setting of general associative DG algebras. Unfortunately, parts of the argument were quite technical, and somewhat ad hoc. Moreover, at some point the technicalities became overwhelming, so I had to impose a simplifying assumption -- namely, that the DG algebra in question is concentrated in non-negative cohomological degrees. This assumption was needed because I used simplicial methods and Dold-Kan equivalence. It is pretty unnatural, and although it does cover some interesting case such as that of the usual commutative algebraic varieties, it was pretty irksome to have to impose it. For various reasons, I revisited the subject starting from the very end of last year, and by now I believe I'm almost finished with it. I like my new approach much better. First of all, I no longer need to use simplicial methods, so the irksome unnatural assumption vanishes without a trace. Moreover, I now feel that I am much closer to understanding how things really work in the story. Namely, the new approach mimics that of Deligne and Illusie more closely, in that it relies on two completely different spectral sequence with the same first page -- the Hodge-to-de Rham spectral sequence, on one hand, and the so-called "conjugate spectral sequence", on the other hand. In the noncommutative case, the first page for both is a periodization of Hochschild homology. The first spectral sequence converges to periodic cyclic homology. It took a while to realize that the second sequence does not. What it converges to, in fact, is a completely new additive invariant of DG algebras and DG categories that is obtained by totalizing the periodic cyclic bicomplex in the "wrong" way. I call this new invariant "co-periodic cyclic homology". The new invarint has been completely overlooked by the classics, probably because it is identically 0 over a field of characteristic 0 . It has been suggested by M. Kontsevich back in 2005 that in positive characteristic, it is an interesting thing to consider, but the suggestion has not been taken seriously at the time. A new perspective came from recent work of A. Beilinson and B. Bhatt, where a very similar phenomenon appears in the study of derived de Rham cohomology. Starting from there, it was a relatively straightforward matter to figure out the whole story. In the end, I proved that co-periodic cyclic homology is Morita-invariant, and moreover, it is an additive invariant in the sense of Keller (that is, it has a natural long exact sequence attached to a localization sequence of DG categories). It always has a conjugate spectral sequence converging to it. Then on one hand, there are many practically important examples when