\(K\)-theory of the \(K(1)\)-local sphere via \(TC\)

An expository talk given for the Harvard Thursday Seminar in Spring 2024, which was on the topic of the recent paper “K-theoretic counterexamples to Ravenel’s telescope conjecture” by Burklund, Hahn, Levy, and Schlank. In this talk, I stated and gave proofs of the key theorems from “The algebraic \(K\)-theory of the \(K(1)\)-local sphere via \(TC\)” by Levy, including computation of the \(K\)-theory of the \(K(1)\)-local sphere in terms of \(TC\) of the Adams summand, and an analogue to the Dundas-Goodwillie-McCarthy theorem for certain \((-1)\)-connective rings. I also explained the Land-Tamme $\odot$ construction, which, given the input of a pullback square of \(\mathbb{E}_1\)-rings, allows comparison with a closely related commuting square which becomes a pullback under \(K\)-theory or another localizing invariant. The relevance of this talk to the seminar is that the counterexample to the telescope conjecture at height 2 is \(K(L_{K(1)}\mathbb{S})\). Notes are available.