PROJECT 04M03F Combinatorial Topology and Applications in Discrete and Computational Geometry |
The project evolved from the former theme "Applied algebraic topology" which existed inside the general project "Geometry and topology". The creation of a separate subproject with this title reflects a growing interest in applications of topological ideas and methods in combinatorics, combinatorial geometry, theoretical computer science (discrete and computational geometry, complexity theory) etc.The project is closely linked with the Seminar for Geometry, Topology and Algebra (GTA seminar).
The objectives of the project are following.
1. Research on the highest level in the areas of combinatorial (algebraic) topology and geometry which have proven to be useful in other areas of mathematics, computer science and natural sciences. They include equivariant methods (Borsuk-Ulam theorems, index theory), applied homotopy and homology (arrangements of subspaces, Goresky-MacPherson type formulas, diagrams of spaces) special varieties (Grassmann, toric, algebraic curves), group actions (euclidean and hyperbolic tessellations, space forms, ornaments) etc.
2. Applications in combinatorial (discrete) geometry (ham sandwich and equipartition theorems, common transversals, Tverberg type theorems, convex polytopes, graph and knot theory).
3. Support of all other aspects of topological applications. Included are applications of existing general "computer algebra" packages (Mathematica, Macaulay, Singular etc.), analysis of algorithms in "discrete and computational geometry and topology", development of special purpose algorithms and computer programs in geometry (e.g. for classifying hyperbolic tessellations) etc.