Heisenberg modules and Balian–Low theorems - Applications of operator algebras to Gabor analysis

Abstract

The main way to study a periodic signal is to decompose it into a sum of simple signals, namely sine waves. However, when a signal changes substantially over time, such as a piece of music, different methods are needed. One method is to use Gabor frames. A Gabor frame represents a given signal in a way that emphasizes the signal’s frequency content at each point in time. For instance, a Gabor frame will represent an audio signal analogously to how sheet music is written.

Constructing good Gabor frames is not an easy task, and this problem has connections to many other areas in mathematics. In my dissertation, I have connected this problem to an area called operator algebras. A basic theorem about Gabor frames is the Balian-Low theorem, which is rooted in the uncertainty principle from quantum mechanics. I have shown that this theorem has a conceptual interpretation in operator algebras. Moreover, one can generally talk about Gabor frames in an abstract setting, namely in the context of an abelian topological group. I have completely classified the groups to which the Balian-Low theorem extends. One of the groups to which it extends is the rational adele group from number theory.