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Tuesday, December 1, 2015

**Abstract:** In this talk, I will report recent progress that the 61-sphere has a unique smooth structure. Following results of Moise, Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, we show that the only odd dimensional spheres with a unique smooth structure are in dimension 1, 3, 5 and 61. Following recent work of Isaksen, we also show that in dimensions from 5 through 61, the only spheres with a unique smooth structure are in dimension 5, 6, 12, 56 and 61. Recent work of Behrens-Hill-Hopkins-Mahowald shows that the next sphere with a unique smooth structure, if exists, is in dimension at least 126. The computation of the stable homotopy groups of spheres at the prime 2 is essential to this result. I will review classical techniques and explain our new technique. This work is joint with Guozhen Wang.