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Tuesday, September 15, 2015

**Abstract:** Computing the stable homotopy groups of spheres is a major focus of modern algebraic topology. One way of finding infinite families of nontrivial elements involves finite CW-complexes with non-nilpotent self-maps. I will show computationally (joint with Bhattacharya and Mahowald) that the spectra $A_1$ whose cohomology is $A(1)$ admits a self-map detected in Morava K-theory by $v_2^{32}$. Time permitting, I will describe Mahowald's proof of the $K(1)$-local telescope conjecture and introduce a finite complex $Z$ that I hope will shed light on the $K(2)$-local telescope conjecture.