Instructor: Dr. Gergely AMBRUS
Text: Classnotes by the lecturer
Prerequisite: basic notions of classical Euclidean geometry, combinatorics, and probability.
Course description: This course will concentrate on the combinatorial aspects of geometric structures: discrete sets of points, lines, and convex sets. The connection of geometry and combinatorics has been extremely fruitful in the last century, and the results obtained from this dual viewpoint had found applications in a broad range of subjects from number theory to analysis. We are going to start from the elementary notions of Euclidean geometry, therefore no special prerequisite is needed. Besides the classical theorems, we will also learn about recent developments of the last decade. A large part of discrete and combinatorial geometry has rooted in Budapest, making it a perfect location to study this beautiful subject!
- Points and lines on the plane, incidence problems, distance problems
- Erdos-Szekeres type questions
- Lattice points, Minkowski's theorem, applications in number theory
- Convexity, combinatorial aspects, Helly's theorem
- Packings and coverings, estimates also in high dimensions
- Transversals and epsilon nets
- Plank problems
- Appications in number theory and elementary information theory