Dresden 2017 – wissenschaftliches Programm
CPP 60: Focus: Topological Problems in the Physics of Polymers, Biopolymers and Fibers I (joint session BP/CPP, organized by CPP)
CPP 60.9: Vortrag
Donnerstag, 23. März 2017, 17:45–18:00, ZEU 260
Three-Dimensional Nets from Hyperbolic Tilings — •Benedikt Kolbe — Technische Universität Berlin
A main focus of modern crystallography is to explore the systematic enumeration and construction of nets in Euclidean space. The EPINET project attempts to enumerate crystalline frameworks that arise as structures derived from hyperbolic tilings. Since hyperbolic geometry represents the most prevalent class of geometry not only in nature as a model of the geometry of minimal surfaces, but also mathematically, the use of hyperbolic surfaces to construct possible structures in 3-space is very natural. Among many others, this recipe has also led to deep results in topology from Thurston and others and is also used in modern descriptions of conformal field theory.
We focus on the study of those networks that one encounters typically in the material sciences-periodic structures. Therefore, to classify the emerging structures we employ Delaney Dress combinatorial tiling theory.
We will explain some of the mathematics and intuition involved in a new approach to enumerating the 3-dimensional structures that arise through hyperbolic tilings and give some results on the knotted networks they represent. This work is aimed at generalizing Delaney-Dress tiling theory and to develop a complexity ordering of different tilings.
The most prominent triply periodic minimal surfaces are used to illustrate the approach and provide first examples. The goal of this project is to ultimately construct systematically and order by complexity all networks that arise by decorations of hyperbolic surfaces.