The stable ${\displaystyle C^{*}}$-algebra of a Smale space is obtained from the étale groupoid associated to the space’s stable equivalence relation. When the original system is mixing, the stable ${\displaystyle C^{*}}$-algebra is simple, separable, nuclear and stably finite. I will outline the construction of an explicit inductive-limit decomposition of the stable ${\displaystyle C^{*}}$-algebra when the stable sets are totally disconnected. From this, one can often compute the ${\displaystyle K}$-theory of the stable ${\displaystyle C^{*}}$-algebra. The talk will be example-based. In particular, no knowledge of Smale spaces is required. This talk is based on joint work with Allan Yashinski.

(Joint with Anna Puskas) In this talk, we determine all of the imaginary ${\displaystyle n}$-quadratic fields with class number dividing 32, ${\displaystyle n>1}$; results for ${\displaystyle n=1}$ were previously computed by Watkins. Our results provide techniques to find complete lists of imaginary ${\displaystyle n}$-quadratic fields of class number ${\displaystyle 2^m}$ for nonnegative integers ${\displaystyle m}$. Further, given a fixed ${\displaystyle m>0}$ we find a bound ${\displaystyle B(m)}$ on ${\displaystyle n}$ for which there are no imaginary ${\displaystyle n}$-quadratic fields of class number ${\displaystyle 2^m}$ whenever ${\displaystyle n>B(m)}$.

I will describe results obtained in collaboration with the Blue Brain Project on the topological analysis of the structure and function of digitally reconstructed microcircuits of neurons in the rat cortex.

I will explain the construction of a new model for the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately.  The key to the construction is the lifted Boardman-Vogt tensor product of modules over operads, developed earlier in joint work with Dwyer.

(Joint work with Bill Dwyer and Ben Knudsen.)