Informal Pizza Seminar

The Isospectral Fruits of Representation Theory:
Quantum Graphs and Drums

Gilad Ben-Shach

McGill

In 1966, Marc Kac asked the famous question “Can one hear the shape of a drum?” In other words, is every spectrum of frequencies associated with a unique shape of a drum? This question was answered in 1992, when a pair of drums with different shapes, but the same spectrum was discovered. Since then, the question has been extended to other physical objects, including quantum graphs. A graph is a collection of vertices, connected by edges. If we apply an operator, and boundary conditions on these graphs, we can consider them to be quantum graphs, and can examine wavefunctions defined on the edges. The set of eigenvalues of the operator is called the spectrum. We present a theorem that can be used as a method to construct pairs of isospectral quantum graphs — two graphs that have different shapes, but have the same spectrum. The basis of the theorem lies in representations of Algebraic groups. Not only has this method yielded pairs of isospectral graphs, it has also been used to produce larger families of graphs, which had not previously been done. We also used the method to reproduce existing examples of isospectrality, in graphs, drums, and other geometric objects. In this presentation, we will examine the construction method, along with examples. The method may prove useful in many fields of research, ranging from string theory, to condensed matter, and even psychology.