Cindy Zhang will present her General Exam "Dual Graphs of Crystallographic Fourier Series" on Thursday, May 16, 2024 at 3:30 PM in CS 402.

Committee Members: Ryan Adams (advisor), Bernard Chazelle, Benjamin Eysenbach

Abstract:

Many physical theories are grounded in mathematical symmetries, making it important to incorporate these symmetries into machine learning models of physical systems in order to obtain accurate and physically meaningful outputs. In materials science, the symmetries of interest are described by crystallographic groups. In physical systems where the particles satisfy the symmetries of a crystallographic group, many of the systems’ properties are described by functions invariant to these symmetries.

In this work, we present a dual graph construction for the Fourier representation of functions invariant to crystallographic group symmetries. Using these graphs, we derive for each crystallographic group: (i) the Fourier bases for group-invariant functions, (ii) the irreducible representations corresponding to energy levels of a quantum-mechanical system, and (iii) the eigenvalue multiplicities of the Laplacian as a function of the sum-of-squares. The Fourier bases provide a method of embedding exact group invariances into machine learning models.

Reading List:

https://docs.google.com/document/d/1TyRMtOigYf2aqO5cvcV7n00l5BLIm92xH5p8xKGjI58/edit

Everyone is invited to attend the talk, and those faculty wishing to remain for the oral exam following are welcome to do so.