Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Geometry

Code

digeop1u0um17gm

Title

Problems in discrete geometry (p)

Usual semester

Autumn

ECTS

2

Language

en

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyze mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Placements and coverings in the Euclidean plane. Dowker’s theorems; the theorems of László Fejes Tóth and Rogers on the densest translations and centrally symmetric arrangements, and on coverings. Questions of homogeneity. Lattice-like arrangements. Homogeneous placements (with group action). Space requirements, separability. Illumination, antipodality, equilateral sets. Transversality, epsilon-nets, Vapnik–Chervonenkis dimension. Problems concerning families of sets with a common transversal. Prerequisite knowledge: basic linear algebra; basic knowledge of affine and convex geometry

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

Fejes Tóth László: Regular figures, Pergamon Press, Oxford–London–New York–Paris, 1964. Fejes Tóth László: Lagerungen in der Ebene auf der Kugel und im Raum, Springer-Verlag, Berlin–Heidelberg–New York, 1972. Rogers, C. A.: Packing and covering, Cambridge University Press, 1964. Böröczky, K. Jr.: Finite packing and covering, Cambridge University Press, 2004. Jiri Matousek: Lectures on Discrete Geometry, Springer-Verlag, Berlin–Heidelberg–New York, 2002. Károly Bezdek, Classical Topics in Discrete Geometry (CMS Books in Mathematics), Springer-Verlag, Berlin–Heidelberg–New York, 2010.

Programmes of the course