This seminar will be about polytopes.

Themenvergabe und speziellere Literaturangaben in der Vorbesprechung zum Seminar.

Research Seminar and Colloquium of the "Discrete Geometry" group at FU Berlin: Guests as well as members of the group report about their own research, new developments, problems and insights.

Topics include: Discrete Geometry (polytopes, point configurations and arrangements, linear and integer programs, integer points, etc.), Topological and Algebraic Combinatorics, Real Algebraic Geometry, Discrete, Linear and Integer Optimization, Computational Geometry, and more. 

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

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.

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. .

Will be announced in class.

Research Seminar and Colloquium of the "Discrete Geometry" group at FU Berlin: Guests as well as members of the group report about their own research, new developments, problems and insights.

Topics include: Discrete Geometry (polytopes, point configurations and arrangements, linear and integer programs, integer points, etc.), Topological and Algebraic Combinatorics, Real Algebraic Geometry, Discrete, Linear and Integer Optimization, Computational Geometry, and more. 

Es werden insbesondere folgende Inhalte vermittelt.
–  Diskrete Wahrscheinlichkeitsräume und -maße
–  Diskrete und stetige Zufallsvariablen und ihre Verteilungen, wichtige Beispiele
–  Erwartungswert, (Ko-)Varianz, Korrelation
–  Bedingte Wahrscheinlichkeit, Unabhängigkeit
–  Schwaches Gesetz der großen Zahl
–  Zentraler Grenzwertsatz
–  Datenanalyse und deskriptive Statistik: Histogramme; empirische Verteilung; Kenngrößen von Stichprobenver-teilungen; Beispiele irreführender deskriptiver Statistiken; lineare Regression
–  Elementare Begriffe und Techniken des Testens und Schätzens: Maximum-Likelihood-Prinzip; Konfidenzinter-valle; Hypothesentests; Fehler erster und zweiter Art 

 

E. Behrends: Elementary Stochastics, Springer, 2013
    H.-O. Georgii: Stochastics: Introduction to Probability Theory and Statistics, De Gruyter, 2007
    U. Krengel: Introduction to probability theory and statistics, Vieweg, 2005
    D. Meintrup, S. Schäffler, Stochastics: Theory and Applications, Springer, 2005.
    Most of the books listed below are available online at the UB. For this purpose, there is an extensive hand apparatus for stochastics in the mathematic library.

Research Seminar and Colloquium of the "Discrete Geometry" group at FU Berlin: guests as well as members of the group report about their own research, new developments, problems and insights.
Topics include: Point configurations and arrangements, convex polytopes, linear and integer programs, topological methods, etc.

This is the third in a series of three courses on discrete geometry. This advanced course discusses the interaction of convex bodies with lattices (known as the "Geometry of Numbers"), lattice polytopes and lattice point counting (known as "Ehrhart Theory"), and other selected topics of current interest.

will be announced in class.

Voraussetzungen: Preferably Discrete Geometry I and II. Background in discrete geometry (polytopes, subdivisions, h-vectors) and convex geometry (mixed volumes, Brunn-Minkowski, Aleksandrov-Fenchel) should suffice.

In this seminar we will study some remarkable examples of polytopes, discuss the construction techniques and derive the most important properties. Some of these were used to solve problems, disprove conjectures, or are examples to support such. Some also have unexplored or unexplained properties that we will look at.

The seminar will probably take place mostly in English.

MR1608265 (2001f:52009) Daniel Klain; Gian-Carlo Rota Introduction to geometric probability. Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1997. ISBN: 0-521-59362-X; 0-521-59654-8

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

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.

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. .

Inhalt:

Extensions of polytopes The extension complexity of a polytope P is the minimal number of facets of a polytope Q that linearly projects onto P. This rather simple definition has interesting consequences and relations to areas such as discrete geometry, combinatorial optimization, information theory, and linear algebra. Determining the extension complexity of a polytope is extremely hard (even for polygons!) and obtaining exact values or even just bounds for special polytopes is an active area of research. The goal of the seminar is to develop a good understanding of extension complexity and the notions related to it. Topics might include

• geometry of extensions: sections, projections, and duality
• relations to the nonnegative rank of matrices
• lower bounds via coverings and chromatic numbers
• bounds via communication protocols
• special instances: permutahedra, matching polytopes, etc.
• other notions of extensions: the positive semidefinite and cone ranks

The seminar is aimed at students with an interest in discrete and convex geometry, discrete mathematics / combinatorial optimization, and linear algebra. The prerequisites for most topics is a basic understanding of polytopes (such as Discrete Geometry I). The first meeting of the seminar will take place during the first week of the semester. Extensions of polytopes

This is the first in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures including analysis and proof techniques. The material will be a selection of the following topics:
Basic structures in discrete geometry

polyhedra and polyhedral complexes

configurations of points, hyperplanes, subspaces

Subdivisions and triangulations (including Delaunay and Voronoi)

Polytope theory

Representations and the theorem of Minkowski-Weyl

polarity, simple/simplicial polytopes, shellability

shellability, face lattices, f-vectors, Euler- and Dehn-Sommerville

graphs, diameters, Hirsch (ex-)conjecture

Geometry of linear programming

linear programs, simplex algorithm, LP-duality

Combinatorial geometry / Geometric combinatorics

Arrangements of points and lines, Sylvester-Gallai, Erdos-Szekeres,

Szemeredi--Trotter

Arrangements, zonotopes, zonotopal tilings, oriented matroids

Examples, examples, examples

regular polytopes, centrally symmetric polytopes

extremal polytopes, cyclic/neighborly polytopes, stacked polytopes

combinatorial optimization and 0/1-polytopes

For students with an interest in discrete mathematics and geometry, this is the starting point to specialize 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.

G.M. Ziegler "Lectures in Polytopes"

J. Matousek "Lectures on Discrete Geometry"

Further literature will be announced in class.

Solid background in linear algebra. Knowledge in combinatorics and geometry is advantageous.

Stipendiaten, Dozenten und Gäste des Graduiertenkollegs halten wissenschaftliche Vorträge über ihre eigene Arbeit zu speziellen Themen des Kollegs. Dazu gehören insbesondere algorithmische und diskrete Geometrie, algorithmische Kombinatorik, Codierungstheorie, Graphentheorie und Graphenalgorithmen, kombinatorische Optimierung, konstruktive Approximation, Mustererkennung und zufällige diskrete Strukturen. Die Themen des Kolloquiums werden auf der Webseite des Kollegs angekündigt.