Alexander Strang is an assistant teaching professor at UC Berkeley.  He earned a PhD in applied mathematics from Case Western Reserve University in 2020, and served as a Kruskal Instructor at the University of Chicago from 2020 - 2023. He teaches courses in probability, Bayesian statistics, and information theory. He works on empirical validation of hierarchical prior models, statistical testing in cryo-electron microscopy, and empirical game theory.

Abstract: Most numerical problems involve many related variables coupled through multiple equations. Most multivariate problems are intractable unless they are linear. Linear problems are tractable in high dimensions because linearity allows decomposition; one n-dimensional problem can be broken into n one-dimensional problems that can be solved independently or in sequence. Among the available decompositions, the most powerful are frequently diagonalization methods. Unfortunately, diagonalization is expensive so is impractical unless the user has access to a fast method. Remarkably, if the linear problem of interest admits a symmetry, then that symmetry can be used to constrain the space of available eigenvectors.

We will discuss methods for quickly identifying eigenvectors from symmetries of linear problems. In particular, we will recall classical results relating the eigenvectors of an operator to the eigenvectors of an operator it commutes with to show that, if an operator admits a symmetry of action, then its eigenfunctions can be expressed as products of Fourier basis functions with an amplitude function. The Fourier term is entirely specified by the structure of the symmetry, while the amplitude function must be computed from the operator.

We will emphasize this perspective as a unifying justification for the ubiquity of the Fourier Transform and show a practical application for quickly detecting ribosomes in Cryo-Electron-Microscopy images of cells.