Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Geometry

Code

kokong1u0um17em

Title

Combinatorial convex geometry (l)

Usual semester

Autumn

Published semester

2025/26/2

ECTS

3

Language

hu

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

Convex polyhedra in the d-dimensional Euclidean space, Euler and Dehn–Sommerville equations, Upper Bound Theorem, relations for the face numbers of simplicial polyhedra. Lattices in ℝᵈ, successive minima and covering radius, the Minkowski–Hlawka theorem, Mahler’s compactness theorem, Swinnerton-Dyer’s theorem on K-critical lattices. Finiteness theorems, Ehrhart’s theorem on the number of lattice points in a lattice polytope. Flatness, Hermite-, Minkowski-, and Lovász-reduced bases. Prerequisite knowledge: linear algebra

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

B. Grünbaum: Convex polytopes, 2nd edition, Springer-Verlag, 2003. P.M. Gruber: Convex and Discrete Geometry, Springer-Verlag, 2006. P.M. Gruber, C.G. Lekkerkerker: Geometry of numbers, North-Holland, 1987.

Programmes of the course