From Ptolemy to Fourier

Functional analysis is a powerful, modern branch of mathematical analysis where topics that are familiar from the theory of real numbers (sequences, series, convergence, etc) are generalized to the infinite dimensional world of functions. This talk provides an introduction to this field for students who have mastered elementary real analysis, tying together both historical motivations and the modern viewpoint. The first half of the talk focuses on the pre-history of the subject, looking at the mathematics of the geocentric model of the heavens developed by the Ptolemies and Ibn al-Shatir, among others. While the notion of epicycles very naturally leads to precise mathematical questions about sequences of functions and convergence, it took us many further centuries to fully understand what lies just beneath the surface of this idea. In the second half we visit Fourier and his study of heat, formalizing the questions these historical figures may have asked, and laying out the roadmap leading from here to the modern theory of Functional Analysis.