We all know how to do Fourier Analysis on an interval or on the entire real line. But what if our functions live on another subset of Euclidean space, let’s say on a regular hexagon in the plane? Can we use our beloved exponentials, functions of the form $e_\lambda(x) = {\rm exp}(2\pi i \lambda \cdot x)$ to analyze the functions defined on our domain? In other words, can we select a set of frequencies such that the corresponding exponentials form an *orthogonal basis* for L^2 of our domain? It turns out that the existence of such an orthogonal basis depends heavily on the domain. So the answer is yes, we *can* find an orthogonal basis of exponentials for the hexagon, but if we ask the same question for a disk, the answer turns out to be no. B. Fuglede conjectured in the 1970s that the existence of such an exponential basis is *equivalent* to the domain being able to *tile* space by translations (the hexagon, that we mentioned, indeed can tile, while the disk cannot). In this talk we will track this conjecture and the mathematics created by the attempts to settle it and its variants. We will see some of its rich connections to geometry, number theory and harmonic analysis and some of the spectacular recent successes in our efforts to understand exponential bases. Tea at 3:33 pm in Stevenson 1425. (Host: Akram Aldroubi)