Title: Topological Insulators and Joint Approximate Diagonalization of Matrices

Only commuting families of self-adjoint matrices can be jointly diagonalized, but that does not stop physicists and electrical engineers from trying. Settling for a unitary change of basis that brings all the matrices “close” to diagonal is often good enough. In signal processing, joint approximate diagonalization has led to effective algorithms for blind source separation. In quantum chemistry, joint approximate diagonalization can find a basis of low-energy space consisting of localized electron states.

There are limits to joint approximate diagonalization, some of which can be explained by K-theory. The choice of scalar field is surprisingly important. Theorems regarding almost commuting unitary matrices cannot be translated properly to results on almost commuting orthogonal matrices unless one takes care with the distinction between KO and KU groups.

Much of the talk will be on joint work with Matt Hastings on topological insulators. These are intriguing states of matter where a sample is conducting on the boundary, yet insulating in the interior. Numerical models of topological insulators suggest that fast algorithms to distinguish topological insulators from ordinary insulators can be found based on the K-theory obstructions to joint approximate diagonalization.

Local Host: David Pitts

This colloquium is funded by the UNL Research Council.