Speaker: Jason Mitrovich
University of Notre Dame
Will give a Graduate Student Geometry Seminar entitled:
An Introduction to Ergodic Theory
Abstract: The word ergodic is an example of a greek portmanteau, combining the words $\epsilon \rho \gamma o \nu$ (ergon $=$ work) and $o \delta o\sigma$ (odos $=$ way). Its inventor, Boltzmann (often called the father of statistical mechanics), tried to explain the behavior of gases by looking at the statistical combinations of all its molecules. To place this on mathematical grounds, Boltzmann in 1871 was forced to make an assumption that the volume of a region $E$ in state space is proportional to the time the system spends living in $E$. This assumption became known as Boltzmann's \emph{ergodic hypothesis}. The need to validate (or invalidate) this hypothesis became the bedrock for ergodic theory by introducing a mathematical formulation of the problem as well as a proof (or counterexample). A precise mathematical description and proof of validity was provided by Birkhoff in 1931 and is titled the mean ergodic theorem. In this seminar, I build up to the statement of the mean ergodic theorem and some applications if time permits.
Date: 03-28-2025
Time: 4:00 pm
Location: 258 Hurley Bldg