Work of Adams and Quillen gives a precise connection between special values of the Riemann zeta function, v_1-periodic stable homotopy groups of spheres, and the algebraic K-theory of the integers. Behrens and Laures show that certain v_2-periodic homotopy groups correspond to certain integral modular forms. I show that this bridge can be extended to K-theory and this v_2-periodic family is also detected in iterated algebraic K-theory of the integers. This result fits in the red-shift philosophy of Ausoni-Rognes, which extends the Lichtenbaum-Quillen conjectures to chromatic heights.
In addition to discussing my result on iterated algebraic K-theory of the integers, I will discuss algebraic K-theory height shifting results from joint work with A. Salch and joint work with J.D. Quigley. If time permits, I will also mention work in progress with J. Hahn and D. Wilson on algebraic K-theory of Morava K-theory and work in progress with C. Ausoni, D. Culver, E. Höning, and J. Rognes on algebraic K-theory of the second truncated Brown-Peterson spectrum.