Title: Almost commuting matrices

Abstract: A deceptively simple question is the following. If d commuting

Hermitian matrices can be diagonalized by a single unitary matrix,

does it follow that d almost commuting matrices can be simultaneously

approximately diagonalized by a single unitary? The answer depends

on how we interpret 'almost' and 'approximately' so we will restrict

to measuring with the spectral norm. We will cover Lin's theorem that

says this result is true for d=2. We will also show how K-theory

can be used to easily show the the same result if false for d=3.

If time permits, we will explore the connection with the existence

of generalized Wannier functions in condensed matter physics.