Picard curves are genus 3 curves of the form y^3=f(x), where f(x) is a polynomial of degree 4. They are the simplest non-hyperelliptic curves. This talk will discuss work with Chris Rasmussen, in which we found all Picard curves defined over the rationals with good reduction at all primes except p=3. Our work was inspired by Nigel Smart's enumeration of genus 2 curves with good reduction at all primes except p=2. As part of this project, we wrote code in SageMath to solve the equation x+y=1 over the S-units of some number fields. In collaboration with Alejandra Alvarado, Angelos Koutsianas, Christelle Vincent, and Mckenzie West, we recently generalized the implementation and have submitted our functions for release in upcoming releases of Sage.
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Blackadar and Kirchberg introduced the notion of a matricial field (MF) $C}^{*$-algebra; a $C}^{*$-algebra is MF if it can be embedded into a corona of matrix algebras. They asked whether every stably finite $C}^{*$-algebra satisfies the MF property; this is the $C}^{*$-analogue of the Connes Embedding Problem. I will discuss this problem in the context of $C}^{*$-dynamical systems and group $C}^{*$-algebras. We show that both stable finiteness and the MF property admit equivalent $K$-theoretic properties. We use deep results from the classification literature to lift $K$-theoretic dynamical information and thus answer the BK question in the affirmative for crossed products of certain classifiable $C}^{*$-algebras by free groups. We will also see that semi-direct product groups of the form $G\u22ca{\mathbb{F}}_{r}$, where $G$ is amenable, admit MF reduced group $C}^{*$-algebras. This is joint work with Chris Schafhauser.