MATH 382
Fourier Analysis Spring 2025
Division III Quantitative/Formal Reasoning

Fourier analysis is the study of waves and frequencies. More precisely, the goal of Fourier analysis is to decompose a complicated function into a simple combination of pure waves, thereby gleaning insight into the behavior of the function itself. It’s difficult to overstate the impact of this branch of mathematics; it is foundational throughout theoretical mathematics (e.g., to study the distribution of prime numbers), applied mathematics (e.g., to solve differential equations), physics (e.g., to study properties of light and sound), computer science (e.g., to compute with large integers and matrices), audio engineering (e.g., to pitch-correcting algorithms), medical science (e.g., throughout radiology), etc. The goal of this course is to cover the basic theory (fourier series, the fourier transform, the fast fourier transform) and explore a number of applications, including Dirichlet’s theorem on primes in arithmetic progressions, the isoperimetric inequality, the heat equation, and Heisenberg’s uncertainty principle.