Sfb 288 Differential Geometry and Quantum Physics

Differential-Geometric and Topological Methods in Discrete Geometry and Combinatorics

Leader(s):

Prof. Günter M. Ziegler Priv.-Doz. Dr. Michael Joswig

This project concerns bridges between differential-geometric concepts on the one side and the discrete structures on the other side. We are trying to create, reinforce and extend such bridges.

A major tool and starting point in these investigations is R. Forman's "Discrete Morse Theory." Carsten Lange has found Discrete Weitzenböck formulas for this set-up, which connects the different discrete Laplace operators that occur, for example, on a cellular manifold.

A key example and source of motivation is the boundary complex of a convex polytope, viewed as a discrete, but "round", object "of positive curvature." Via our study of this example we are trying bound diameters of polytopes. This might in the long range lead to applications to central problems of discrete geometry such as the Hirsch conjecture and the analysis of the simplex method for Linear Programming.

Similarly, we hope for a geometrization of triangulated manifolds and symmetric complexes; the connection between curvature, extremal properties and symmetries is not yet clear in a discrete context. In particular in the context of triangulated spheres we hope for substantial new extremal and complexity results.