Title: What does the average eigenvalue know about geometry, physics, or the complexity of a graph?
Speaker: Evans M. Harrell (Professor Emeritus, School of Mathematics, Georgia Institute of Technology)
Abstract: Inverse spectral theory refers to the use of eigenvalues and related information to understand a linear operator. Linear operators and matrices are used, for example, to describe such things as vibrating membranes, quantum phenomena, and graphs (in the sense of networks). The eigenvalues respond to the shape of the membrane, the form of the interaction potential, or respectively the connectedness of the graph, but not in a formulaic way. The effort to tease out these details from the knowledge of the eigenvalue spectrum was memorably described al little over 50 years ago by Mark Kac in an article entitled “Can one hear the shape of a drum?” (Incidentally, the answer is “Often, but not always.”)
In this lecture I will give some perspectives on inverse spectral problems and then concentrate on what can be learned from the statistical distribution of the eigenvalues, such as means, deviations, and partition functions. Much of this work is joint with J. Stubbe of EPFL.
Notes from this talk can be viewed here.