Daniel Prosianik: Diskret geometri:Incidensgeometri och alternativt synsätt på Kakeya-problemet

Abstract.

This thesis provides insights into discrete geometry by analyzing two interesting problems: the minimum number of crossings in a graph draw in the plane and application of the polynomial method on finite field Kakeya problem. This thesis begins by presenting the crossing number inequality - an important theorem in the incidence geometry field, that determines the least amount of crossings a graph can have when drawn in the plane. We then apply this inequality to the Szemerédi-Trotter theorem, which is about the possible number of incidences between lines and points. Finally, we explore the polynomial method in the context of the finite field Kakeya problem, as well as a background to the original Kakeya problem. By applying the polynomial method, we demonstrate its potential in future usage and provide an alternative perspective on the Kakeya problem.