Summer Term 2014 - BMS Advanced Course

Welcome to the website! The lecture is taught by Günter M. Ziegler. It is a continuation of Discrete Geometry I, which was taught last semester, but is not required as a prerequisite. If you have any questions, please ask them during class or send us an email!

# Course Description

This course is a continuation of Discrete Geometry I, taught last semester. We will refer to objects and topics from DG I (for example polytopes and convexity) every once in a while, but you will be able to follow DG II without having taken DG I. The aim of DG II is a skillful handling of discrete geometric structures with an emphasis on metric and convex geometric properties. The material will be a selection of the following topics:

Linear programming and some applications

- Linear programming and duality
- Pivot rules and the diameter of polytopes

Subdivisions and triangulations

- Delaunay and Voronoi
- Delaunay triangulations and inscribable polytopes
- Weighted Voronoi diagrams and optimal transport

Basic structures in convex geometry

- convexity and separation theorems
- convex bodies and polytopes/polyhedra
- polarity
- Mahler’s conjecture
- approximation by polytopes

Volumes and roundness

- Hilbert’s third problem
- volumes and mixed volumes
- volume computations and estimates
- Löwner-John ellipsoids and roundness
- valuations

Geometric inequalities

- Brunn-Minkowski and Alexandrov-Fenchel inequality
- isoperimetric inequalities
- measure concentration and phenomena in high-dimensions

Geometry of numbers

- lattices
- Minkowski's (first) theorem
- successive minima
- lattice points in convex bodies and Ehrhart's theorem
- Ehrhart-Macdonald reciprocity

Sphere packings

- lattice packings and coverings
- the Theorem of Minkowski-Hlawka
- analytic methods

Applications in optimization, number theory, algebra, algebraic geometry, and functional analysis

The course will use material from P. M. Gruber, " Convex and Discrete Geometry" (Springer 2007) and various other sources. There will be brief lecture notes available for course participants with detailed pointers to the literature.

# Contact

Contact | Office Hours: | ||

Lecture | Prof. Günter M. Ziegler | ziegler(at)math.fu-berlin.de | TBA |

Tutorial | Miriam Schlöter | schloeter(at)math.fu-berlin.de | THU 12 - 1 PM, RM K007 |

Tutorial | Albert Haase | a.haase(at)fu-berlin.de | TBA |

# Lectures

| ||

TUE | 10:15 - 11:45 | Arnimallee 6 Room 007/008 |

THU | 10:15 - 11:45 | Arnimallee 3 Room 119 |

- Lecture Notes Part 1
- Lecture Notes Parts 1-2
- Here's a link to Cynthia Vinzant's (U Mich) article on
*spectrahedra*that was mentioned in the lecture. - Lecture Notes Parts 1-3
- Lecture Notes Parts 1-4
- Lecture Notes Parts 1-5
- Lecture Notes Parts 1-6
- Lecture notes all

# Tutorials and Problems

Tutorials | ||

WED | 14:15 - 15:45 | Arnimallee 6, Room 007/008 |

In the tutorial, we will occasionally review topics from the lecture but mostly discuss problems and solutions to homework assignments. You are encouraged to pitch in by presenting a solution every once in a while.

Every week each student is required to solve homework assignments and hand them in. The problem sheets will be uploaded on Tuesdays and solutions should be turned in before the lecture on the following Tuesday. You will receive points for your solutions based on whether your solutions are correct and well-written.

Course requirements are the following: (1) You must score at least 50% of the total points of the problems assigned in each half of the semester. There will be 6 problem sheets in the first half of the semester. A sheet will have problems worth roughly 20 points. (2) You must pass an exam at the end of the semester which alone will determine your grade.