The mapping class group, hyperbolic tilings, and structures in three-dimensional Euclidean space

Abstract: We discuss some recent breakthroughs concerning an inherently interdisciplinary project between mathematicians, physicists, chemists, and computer scientists that attempts to produce structures in three-dimensional Euclidean space from graph embeddings on triply-periodic minimal surfaces. The mapping class group (MCG) of a surface is the group of homeomorphisms of the surface modulo isotopies of the surface. It has a long history in topology and represents an active area of research. We present in this talk a recent new application of MCGs relevant for crystallography, materials science, structure formation, and knot theory. We first explain the necessary set-up for the construction of candidates for new crystalline structures from graph embeddings on surfaces, where intrinsically hyperbolic triply-periodic minimal surfaces in three-dimensional Euclidean space are used as a scaffold for promising three-periodic structures. We then give an overview of new results on MCGs that facilitates an enumeration of isotopy classes of graph embeddings with a given group of symmetries. Lastly, we present a catalogue of three-dimensional structures that have resulted from this project and explain some of the difficulties involved as well as future directions.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry