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Tuesday, February 24, 2015

**Abstract:** Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP-homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and while $\eta:S^{1,1,}\to S^{0,0}$ induces zero on BP-homology, it is non-nilpotent. Work with Haynes Miller has led to a calculation of $\eta^{-1}\pi_{*,*}(S^{0,0})$, proving a conjecture of Guillou and Isaksen. I’ll introduce the motivic homotopy category and the motivic Adams-Novikov spectral sequence before describing this theorem. Then I’ll show that there are more periodicity operators in chromatic motivic homotopy theory than in the classical story. In particular, I will describe a new non-nilpotent self map.