Imagine drawing lines on a surface. Most of us are pretty lazy, so we most
likely manage only a small doodle. However, what if the drawing for the
rest of the surface can be filled in by invoking symmetries. If the
surface we are drawing on is arbitrary, what are all the ways we can
scribble such that this actually works? Is there a way to enumerate these
different ways? If the goal was to find molecular structures by drawing
them on surfaces, what surfaces would we start with and why?
The first and greater part of my talk will motivate and answer these
questions, while focusing on a new technique to explicitly enumerate and
construct all essentially different ways to decorate prominent examples of
triply periodic minimal surfaces.
The second part will focus on what we can say about the kinds of
structures that arise from this process and the kind of advantages this
new approach offers. This is a rather controversial topic, as most
chemists exclusively use crystallographic tables for the study of
symmetries in 3D structures.
There will be tie-ins to geometry, braid theory, combinatorial group and
tiling theory, physics, and even some chemistry.