Abstract

Be it in math classes or donut shops, all of us have seen many different shaped tori throughout our lives.
Mathematically, we know that the conformal structures on the torus are parameterized by a dimensional space, and so it’s natural to ask which tori have we really seen? What do the conformal structures look like, which don’t arise from identifying opposing sides of a Euclidean rectangle?

In this talk I will illustrate a beautiful result of Ulrich Pinkall, who showed that all possible conformal structures on the torus are realizable as embedded surfaces in the three sphere using the geometry of the Hopf fibration. We will interact with these tori in three ways: by conformally projecting into $\mathbb{R}^3$, by viewing them projectively from a vantage point in $\mathbb{R}^4$, and finally intrinsically as sub manifolds of $S^3$, rendered geometrically correct (so light follows great circles).

Slides

(The original slide deck had many videos and animations in it: I need to find a way to usefully display this without uploading too much data!)