# Mobius randomness and homogeneous dynamics

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Ryan Peckner, Princeton University
Fine Hall 314

Qualitative approaches to understanding the randomness of the primes offer a first step toward the extremely difficult quantitative challenge of sharply bounding sums involving the Mobius function. Recently, Sarnak has conjectured such a qualitative description that subsumes many previously known examples: any observable sequence of complex numbers from a zero-entropy topological dynamical system must fail to correlate with the Mobius function. I will outline this conjecture, go over some interesting known cases, and describe a bit of my thesis work, in which I aim to prove that the conjecture is true for all zero-entropy dynamics on homogeneous spaces of semisimple Lie groups.