19061
Vorlesung
SoSe 13: (V) BasisM Diskrete Geometrie II
Raman Sanyal
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 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 convex geometric properties. In the course we will develop central themes in discrete and convex geometry including proof techniques and applications to other areas in mathematics.
The material will be a selection of the following topics:
Basic structures in discrete and convex geometry
- convexity and separation theorems
- convex bodies and polytopes/polyhedra
- faces and boundary structures
- polarity
- approximation by polytopes
- subdivisions and triangulations (including Delaunay and Voronoii)
Notions and tools
- face numbers and invariants
- volumes and mixed volumes
- volume computations and estimates
- Löwner-John ellipsoids and roundness
- valuations
Geometric inequalities
- Brunn-Minkowski and Alexsandrov-Fenchel inequality
- isoperimetric inequalities
- measure concentration and phenomena in high-dimensions
- Minkowski's representation theorem for polytopes
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
- Theorem of Minkowski-Hlawka
- analytic methods
Applications in optimization, number theory, algebra, algebraic geometry, and functional analysis
Schließen
Basic structures in discrete and convex geometry
- convexity and separation theorems
- convex bodies and polytopes/polyhedra
- faces and boundary structures
- polarity
- approximation by polytopes
- subdivisions and triangulations (including Delaunay and Voronoii) Notions and tools
- face numbers and invariants
- volumes and mixed volumes
- volume computations and estimates
- Löwner-John ellipsoids and roundness
- valuations Geometric inequalities
- Brunn-Minkowski and Alexsandrov-Fenchel inequality
- isoperimetric inequalities
- measure concentration and phenomena in high-dimensions
- Minkowski's representation theorem for polytopes 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
- Theorem of Minkowski-Hlawka
- analytic methods
Literaturhinweise
* G.M. Ziegler "Lectures on Polytopes" * R. Schneider "Convex Bodies: The Brunn-Minkowski Theory" * P. Gruber "Convex and Discrete Geometry" * J. Matousek "Lectures on Discrete Geometry" More literature will be announced in class.
Schließen26 Termine
Regelmäßige Termine der Lehrveranstaltung
Di, 09.04.2013 10:00 - 12:00
Di, 16.04.2013 10:00 - 12:00
Di, 23.04.2013 10:00 - 12:00
Di, 30.04.2013 10:00 - 12:00
Di, 07.05.2013 10:00 - 12:00
Di, 14.05.2013 10:00 - 12:00
Di, 21.05.2013 10:00 - 12:00
Di, 28.05.2013 10:00 - 12:00
Di, 04.06.2013 10:00 - 12:00
Di, 11.06.2013 10:00 - 12:00
Di, 18.06.2013 10:00 - 12:00
Di, 25.06.2013 10:00 - 12:00
Di, 02.07.2013 10:00 - 12:00
Di, 09.07.2013 10:00 - 12:00
Do, 18.04.2013 12:00 - 14:00
Do, 25.04.2013 12:00 - 14:00
Do, 02.05.2013 12:00 - 14:00
Do, 16.05.2013 12:00 - 14:00
Do, 23.05.2013 12:00 - 14:00
Do, 30.05.2013 12:00 - 14:00
Do, 06.06.2013 12:00 - 14:00
Do, 13.06.2013 12:00 - 14:00
Do, 20.06.2013 12:00 - 14:00
Do, 27.06.2013 12:00 - 14:00
Do, 04.07.2013 12:00 - 14:00
Do, 11.07.2013 12:00 - 14:00
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 convex ...
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