Joint work with G. Angelini-Knoll, Ch. Ausoni, D.L. Culver and E. H{\"o}ning. We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown-Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free \bF_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner.

Click here to watch the seminar