See also the Topology seminar and Student seminar pages.
I will explain how to compute the homotopy of THH(Z/p) and its C_{p^n}-fixed points, for all n, and how to extract the homotopy of TC(Z/p; p) from this, using methods that generalize to other ring spectra. This recovers the result of Hesselholt-Madsen. We also discuss K(Z/p) at p.
I will explain how to compute the mod p homotopy of THH(Z) and its C_{p^n}-fixed points, for p>2 and all n, and how to extract the mod p homotopy of TC(Z; p) from this. This recovers the result of Bökstedt-Madsen. We also discuss K(Z_p) and K(Q_p) at p.
I will explain how to compute the mod p and v_1 homotopy of THH(l) and its C_{p^n}-fixed points, for p>3 and all n, and how to extract the mod p and v_1 homotopy of TC(l; p) from this. This recovers the result of Ausoni-Rognes. We also discuss K(l_p) and K(L_p) at p.
John Rognes proved that the sphere spectrum S has only split extensions: this is based on the algebraic fact that finite extensions of the integers Z must ramify. On the other hand, working Bousfield locally with respect to K(n), the n-th Morava K-theory at a prime p, it is know that the local sphere L_{K(n)}S has a large connected extension called the n-the Lubin-Tate spectrum, E_n; the group here is the Morava stabilizer group. I will describe work with Birgit Richter in which we show that every finite Galois extension of E_n splits, at least for odd primes. The proof involves three cases corresponding to the existence of a quotient of the Galois group which is either cyclic of order p, cyclic of prime order different from p, or non-abelian simple. To deal with the last case, we need to use the Feit-Thompson Theorem to deduce that the order is even.
I will explain how to compute the mod p and v_1 homotopy of THH(l/p) and its C_{p^n}-fixed points, for p>3 and all n, and how to extract the mod p and v_1 homotopy of TC(l/p; p) from this. This represents work done in 2005 by Ausoni-Rognes. We also discuss K(l/p), K(L/p) and the algebraic K-theory of the "fraction field" ff(l_p) = p^{-1} L_p.