Frank H. Lutz: Combinatorial Roundness of Grains in Cellular Microstructures and Complicatedness Measurements for Cell Complexes

Polycrystalline materials are composed of crystal grains of varying size and shape, with some of the occurring grain types being more frequent than others. We will observe that almost all corresponding dual 2-dimensional spheres are `combinatorially round', in the following sense: there are no `short' separating cycles that partition the triangulations into two parts of similar sizes. In other words, almost all the examples are stacked with few vertices, flag, or close to flag.

    For more general complexes from material sciences or other, even with a billion of faces, homological information can be obtained with computational homology packages such as CHomP or RedHom.
These packages extensively use (NP-hard) discrete Morse theory as a (fast) preprocessing step to avoid (slow, polynomial time) Smith Normal Form computations. In fact, it is surprisingly hard to construct ``complicated'' examples on which homology calculations perform poorly. We will obtain an infinite series of such examples based on the Akbulut-Kirby potential counterexamples to the smooth 4-dimensional Poincaré conjecture and the Andrews-Curtis conjecture.