Wednesday September 19, 2018 1:00 PM  2:00 PM Math Tower P131
 Jack Burkart, Stony Brook University
Improving Liouville's Theorem for Harmonic FunctionsLiouville's theorem says that the only harmonic functions on $R^n$ that are bounded above are actually constant. We will discuss the proof of this fact using Harnack's inequality, which is a primitive example of extremely useful "3Ball inequalities" that show up in harmonic analysis. We will compare this continuous version to the discrete version of the Liouville theorem after defining harmonic functions on $Z^2$ in terms of the mean value property. In particular, we will discuss a recent advancement from 2017 due to Buhovsky, Logunov, Malinnikova, and Sodin, which says that, in a way that we will make precise, a harmonic function bounded on 99.999% of $Z^2$ must actually be constant.
