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Wednesday, September 26, 2018

**Abstract:** We will discuss A. Tserunyan's proof of the classical pointwise ergodic theorem for pmp actions of $\mathbb{Z}$ given in [arXiv link], which reduces proving this analytic theorem to solving a finitary tiling problem on the Cayley graph of $\mathbb{Z}$. We hope to use this method in the future to give streamlined proofs of known instances of the pointwise ergodic theorem for other groups, as well as obtain new ones.