Leírás
Simple harmonics, such as monochromatic light waves or heart rhythms or standing patterns of a vibrating string, are basic building blocks of analysis: a compound signal like sunlight or the sound of your favorite instrument is composed of (many) single-color bands or single-pitch tones.
On more general spaces, flat ones or those with some curvature, the role of simple harmonics is played by eigenfunctions, objects central in contexts ranging from spectral geometry, a field whose spirit was captured by Mark Kac's famous question "Can you hear the shape of a drum?", to quantum mechanics, where they represent "pure quantum states" and where their concentration of mass is closely related to geometry and dynamics.
After describing some of these fundamental modes and what they can tell us about the underlying spaces, we will discuss what eigenfunctions have to do with number theory (things like primes, or divisors, or Fermat's Last Theorem) and how additional symmetries of arithmetic or geometric nature can drive their exceptional behavior not generically observed or predicted by physical models.