Groupoid Methods in Wavelet Analysis, Redux

Paul Muhly, professor emeritus of Mathematics; University of Iowa

This series of two talks is based on joint work with Marius Ionescu. I will describe how we use groupoid methods to give new proofs of the main theorems in Stephane Mallat's famous Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315 (1989). Our techniques extend to settings much more general than those considered by Mallat.  One of them involves composition operators on the unit circle that are generated by finite Blaschke products whose zeros are suitably constrained.  My goal is to convince the audience of the utility of groupoids and to promote the mantra: Groups act on sets, Groupoids act on fibred sets.