19214901 Vorlesung

SoSe 16: BasisM Diskrete Geometrie II

Günter Ziegler

Hinweise für Studierende

Zielgruppe:

The target audience are students with an interest in discrete mathematics and (convex) geometry. The course is a good entry point for a specialization in discrete geometry. The topics addressed in the course supplement and deepen the understanding for discrete-geometric structures appearing in differential geometry, topology, combinatorics, and algebraic geometry.

Schließen

Zusätzl. Angaben / Voraussetzungen

Solid background in linear algebra and some analysis. Basic knowledge and experience with polytopes and/or convexity (as from the course "Discrete Geometry I") will be helpful. .

Kommentar

Inhalt:

This is the second in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures with an emphasis on metric and convex geometric properties. In the course we will develop central themes in metric and convex geometry including proof techniques and applications to other areas in mathematics.

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 discrete geometr

  • point configurations and arrangements
  • incidence problems
  • geometric selection theorems
    • epsilon-nets

    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

    Schließen

Literaturhinweise

The course will use material from P. M. Gruber, " Convex and Discrete Geometry" (Springer 2007) and various other sources.

27 Termine

Regelmäßige Termine der Lehrveranstaltung

Di, 19.04.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 26.04.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 03.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 10.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 17.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 24.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 31.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 07.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 14.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 21.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 28.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 05.07.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 12.07.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Di, 19.07.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A6/SR 009 Seminarraum (Arnimallee 6)

Do, 21.04.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 28.04.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 12.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 19.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 26.05.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 02.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 09.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 16.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 23.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 30.06.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 07.07.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 14.07.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Do, 21.07.2016 10:00 - 12:00

Dozenten:
Univ.-Prof. Dr. Günter Ziegler

Räume:
A3/SR 119 Seminarraum (Arnimallee 3-5)

Studienfächer A-Z