# 104.650 AKGEO AKDIS Geometry of numbers This course is in all assigned curricula part of the STEOP.\$(function(){PrimeFaces.cw("Tooltip","widget_j_id_21",{id:"j_id_21",showEffect:"fade",hideEffect:"fade",target:"isAllSteop"});});This course is in at least 1 assigned curriculum part of the STEOP.\$(function(){PrimeFaces.cw("Tooltip","widget_j_id_23",{id:"j_id_23",showEffect:"fade",hideEffect:"fade",target:"isAnySteop"});}); 2024S

2024S, VO, 2.0h, 3.0EC

## Properties

• Semester hours: 2.0
• Credits: 3.0
• Type: VO Lecture
• Format: Presence

## Learning outcomes

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.

## Subject of course

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.

## Teaching methods

Mathematical Defintions, theorems and proofs.

## Mode of examination

Oral

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).

## Course dates

DayTimeDateLocationDescription
Tue10:00 - 10:4505.03.2024 DA07 H16 ("Freihaus" building, 7th floor, green area)Preliminary meeting
Wed12:00 - 14:0010.04.2024 - 19.06.2024 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Mon10:00 - 13:0029.04.2024 DA07 H16 (Freihaus, 7. Stock, grüner Bereich)Replacement for April 24 and May 01
AKGEO AKDIS Geometry of numbers - Single appointments
DayDateTimeLocationDescription
Tue05.03.202410:00 - 10:45 DA07 H16 ("Freihaus" building, 7th floor, green area)Preliminary meeting
Wed10.04.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed17.04.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed24.04.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Mon29.04.202410:00 - 13:00 DA07 H16 (Freihaus, 7. Stock, grüner Bereich)Replacement for April 24 and May 01
Wed08.05.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed15.05.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed22.05.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed29.05.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed05.06.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed12.06.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture
Wed19.06.202412:00 - 14:00 DA07 H16 ("Freihaus" building, 7th floor, green area)Lecture

oral exam

Not necessary

## Curricula

Study CodeObligationSemesterPrecon.Info
860 GW Optional Courses - Technical Mathematics Mandatory elective

## Literature

Lecture notes will be made available during the semester.

## Previous knowledge

Linear Algebra and Analysis.

## Language

if required in English