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Tuesday, April 16, 2019

**Abstract:** The Hurewicz image of the alpha family in the algebraic K-theory of the integers is know to correspond to special values of the Riemann zeta function, by work of Adams and Quillen. Lichtenbaum and Quillen conjectured that, more generally, there should be a relationship between special values of Dedekind zeta functions and algebraic K-theory. These conjectures have now largely been proven by work of Voevodsky and Rost. The red-shift conjectures of Ausoni-Rognes generalize the Lichtenbaum-Quillen conjecture to higher chromatic heights in a precise sense. In that same spirit, I conjecture that the n-th Greek letter family is detected in the Hurewicz image of the n-th iteration of algebraic K-theory of the integers. In my talk, I will sketch a proof of this conjecture in the case n=2 using the theory of trace methods. Specifically, I prove that the beta family is detected in the Hurewicz image of iterated algebraic K-theory of the integers. This is a higher chromatic height analogue of the result of Adams and Quillen. Consequently, by work of Behrens, Laures, and Larson iterated algebraic K-theory of the integers detects explicit information about certain modular forms.