## Talk by Dr. Elizabeth Gillaspy (Dept. of Mathematical Sciences, University of Montana)

### 10/24/2019  Wilson Hall 1-144  4:10-5:00pm

Abstract:

The space $E^\infty$ of infinite paths in a directed graph E is naturally a Cantor set; Julien and Savinien (building on work by Pearson and Bellissard) have studied spectral triples for such Cantor sets, and in particular shown that the Laplace-Beltrami operators ${\Delta_s}_{s \in \mathbb{R}}$ of these spectral triples all have the same eigenspaces. In joint work with Farsi, Julien, Kang, and Packer, we have discovered that this eigenspace decomposition agrees with a wavelet-type orthogonal decomposition of $L^2(E^\infty, M)$ first introduced by Marcolli and Paolucci. Moreover, the Dixmier trace measure associated to the Pearson-Bellissard spectral triple agrees with the Hausdorff measure on the Cantor set $E^\infty$.

In my talk, I will sketch the proofs of the above results; no prior knowledge of spectral triples, wavelets, or Hausdorff measure will be assumed