Arrangements of Geometric Objects and Topological Graphs

My aim in this project is to solve selected combinatorial problems aboutarrangements of basic geometric objects such as points, lines, curves,polygons, polyhedra, discs, convex sets etc. Closely related to this, I alsointend to work on problems in topological graph theory, which can be regarded as the study of arrangements of curves on a fixed set of endpoints. My project is divided into two parts.In the first part I want to address the three following types of problemsabout geometrically defined graphs and hypergraphs.1. I plan to work on cover-decomposability problems in which we ask for a given set of geometric shapes, if for a given L there exists a number K such that any K-fold cover of the space by given geometric shapes can be decomposed into L covers.2. By using methods of structural graph theory I plan to attack problemsrelated to the quasi-perfectness of intersection graphs of geometric objects. Thus, my main goal in this part of the project is to investigate the relation between the chromatic and clique number in intersection graphs of selected classes of geometric objects.3. I intend to prove some variants or generalizations of The CenterpointTheorem by employing methods of algebraic topology.The second part of my project deals with topological graphs, and it is centered around the following questions.1. What is the minimum number of pairs of crossing edges in a drawing of a graph in the plane? Is this number always the same as the minimum number of edge crossings in such a drawing?2. Can we test whether a clustered graph admits a clustered planar drawing in polynomial time?3. What is the maximal density of edges in a topological graph avoidinga given configuration?