Abstract: In this talk, I will present an idea for extending the action of $\mathbb{C}(z)$ from the Riemann sphere to the unit hyperbolic 3-ball. I will first discuss the arithmetic reasons for expecting such an action. After reviewing classical hyperbolic geometry on the unit 3-ball and the quaternionic upper half-space, I will give axioms which such a (putative) action should satisfy. I will discuss obstructions which make several natural candidates for such an action fail. Then, I will present a family of examples (which includes the monomials $z^n$), that do satisfy the axioms, and have many other desirable properties as well. While these examples fall short of a general theory (in particular, general existence and uniqueness theorems are needed), they suggest numerous questions, both theoretical and numerical, for further investigation.
To add/edit talks, please log in on the department web page, then return to Announce. Alternatively if you know the Announce
username/password, click the link below: