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 ... Lesen Sie weiter
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 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. There will be brief lecture notes available for course ... Lesen Sie weiter
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.
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 ... Lesen Sie weiter