Abstract

The harmonic oscillator is a cornerstone of classical, statistical, and quantum mechanics, both from theoretical and computational perspectives. In classical mechanics, it admits a rich variety of geometric descriptions, offering a natural playground to explore how underlying structures can inform the design of numerical methods. In statistical mechanics, the harmonic oscillator connects intimately with the Gaussian distribution, providing insight into sampling from measures. By exploiting these geometric and probabilistic structures, one can develop efficient algorithms for sampling from probability distributions and tailored numerical schemes for the solution of stochastic differential equations. This talk takes a guided tour through these perspectives, highlighting how a simple system can illuminate deep ideas across mechanics, geometry, probability, and computation.