Seminar

A classical question in geometry is whether surfaces with given geometric features can be realized as surfaces in Euclidean space. We will investigate surfaces with cone metrics and find concrete examples of triply periodic polyhedra that have identifiable conformal structures.

Related questions are

1. How do we come up with examples?

2. Are there (if so, how many) saddle connections on infinite polyhedra with their natural polyhedral metric? What are their Veech groups?

3. Furthermore, these examples connect to Novikov’s problem on hyperplane sections. The intersection of a triply periodic polyhedral surface $P$ and a hyperplane $H$ yields one-dimensional curves which are level sets of a 1-form on the surface. Then what can we say about the set $\{\omega H\}$ as we vary $H,$ thinking of it as a subset of the vector space of holomorphic 1-forms on the intersection. In other words, what does the family of translation surfaces look like?

These questions lie in the intersection of flat geometry and dynamics. It is also known that the motivation for Novikov’s problem comes from physics where periodic surfaces in $\mathbb{R}^3$ are understood as a Fermi surface of some metal, and the plane sections can be explained as a hyperplane orthogonal to the magnetic field.

In this talk, instead of answering these questions, we will discuss very specific examples of polyhedral surfaces and describe the questions that are asked above.