PHDTE00012 Experimental Topology

In this course, we will explore surfaces, higher-dimensional manifolds and more general complexes from a triangulation point of view.

Our main focus will be on

- constructions (of extremal or otherwise interesting examples),

- tools (e.g., discrete Morse theory, homology computation, finite group presentations),

- combinatorial properties (such as face numbers, colorings and valence conditions),

- algorithmic aspects (of how to compose/decompose and to recognize triangulations),

- enumerative methods (in an isomorphism-free way)

- and applications (in material sciences).

We will see that almost all algorithmic problems in topology are hard to solve or even are unsolvable -- yet we will discuss random approaches that allow to settle some of these problems.

Our constructions and examples will cover, among other, equivelar surfaces, Seifert 3-manifolds, knotted triangulations, homotopy 4-spheres, non-PL spheres, random complexes, valence restricted triangulations and polycrystalline materials such as metals and steel.