After successful completion of the course, students are able to analyze lattices in R^n and to determine, or estimate, their basic geometric quantities. The lecutres serves as a foundation for further specialization in the geometry of numbers.

The geometry of numbers was originally introduced by Hermann Minkowski in order to study question from number theory with geometric tools. Throughout the years, this theory has developed into an independent field of research that sees various applications both inside and outside of mathematics. Methods from the geometry of numbers are used, for instance, in integer optimization, coding theory and in the context of sphere packing problems.

The main subject of the geometry of numbers are lattices (integral spans of independent vectors) in R^n and their interplay with convex sets. The lecture gives an introduction to Minkowski geometry of numbers from a modern point of view. It covers the following subjects:

• Basic structural propoerties of lattices in R^n,
• Determinants of lattices and their sublattices,
• Successive minima and Minkowski's first and second theorem,
• Applications of Minkowski's theorem,
• Basis reduction,
• Packing problems.

Mathematical Defintions, theorems and proofs.

There will be 10 lectures á 90 minutes.

A preliminary meeting in which the dates of the lectures are decided will take place on March 5 at 10am in the room DA07 H16 ("Freihaus" building, 7th floor, green area).

oral exam

Lecture notes will be made available during the semester.

Linear Algebra and Analysis.