Georgia Institute of Technology College of Sciences

Eigenvalues of linear operators often correspond to physical observables; for example they determine the energy levels in quantum mechanics and the frequencies of vibration in acoustics. Properties such as the shape of a system are encoded in the the set of eigenvalues, known as the "spectrum," but in subtle ways. I'll talk about some classic theorems about how geometry and topology show up in the spectrum of differential operators, and then I'll present some recent work, with connections to physical models such as quantum waveguides, wires, and graphs.