Content

1. Additions to Analysis I. Non-authentic integrals
2. Uniform convergence of function sequences. Power series. Sentence of Taylor.
3. Elements of topology. Standardized and metric spaces. Open quantities. Convergence. Completed quantities. Consistency. Compactness
4. Differential calculus of several variables. Partial, total and continuous differentiability. Block via the inverse function. Block of implicit functions.
5. Iterated integrals.
6. Ordinary differential equations. Basic terms, elementary solvable differential equations, existential and unambiguous results for systems.

Content:
This is the first part of a three semester introduction into the basic mathematical field of Analysis. Differential and integral calculus in a real variable will be covered. Topics:

1. fundamentals, elementary logic, ordered pairs, relations, functions, domain and range of a function, inverse functions (injectivity, surjectivity)
2. numbers, induction, calculations in R, C
3. arrangement of R, maximum and minimum, supremum and infimum of real sets, supremum / infimum completeness of R, absolute value of a real number, Q is dense in R
4. sequences and series, limits, cauchy sequences, convergence criteria, series and basic principles of convergence
5. topological aspects of R, open, closed, and compact real sets
6. sequences of functions, series of functions, power series
7. properties of functions, boundedness, monotony, convexity
8. continuity, limits and continuity of functions, uniform continuity, intermediate value theorems, continuity and compactness
9. differentiability, concept of the derivative, differentiation rules, mean value theorem, local and global extrema, curvature, monotony, convexity
10. elementary functions, rational functions, root functions, exponential functions, angular functions, hyperbolic functions, real logarithm, inverse trigonometric functions, curve sketching
11. beginnings of integral calculus

Detailed Information can be found on the Homepage of 19202801 Analysis I.

Literature:

• Bröcker, Theodor: Analysis 1, Spektrum der Wissenschaft-Verlag.
• Forster, Otto: Analysis 1, Vieweg-Verlag.
• Spivak, Michael: Calculus, 4th Edition.

Viele Analysis Bücher sind auch über die Fachbibliothek der FU Berlin elektronisch verfügbar.

Bei Schwierigkeiten mit den Grundbegriffen Menge, Abbildung etc. ist die folgende Ausarbeitung empfehlenswert:

• Scheerer, Hans: Brückenkurs, Skript FU Berlin 2006.

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Content:
Functional analysis is the branch of mathematics dealing with the study of normalized (or general topological) vector spaces and continuous images between them. Analysis, topology and algebra are linked.
The lecture deals with Banach and Hilbert spaces, linear operators and functional as well as spectral theory of compact operators.

Target group: Students from the 3rd/4th semester on.

Requirements: Good command of the material of the lectures Analysis I/II and Linear Algebra I/II.

Literatur:

• Dirk Werner: Funktionalanalysis, 7. Auflage, Springer-Verlag 2011, ISBN 978-3-642-21016-7

Will be announced in class.

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

This lecture will continue the introduction to algebraic topology. The course will cover a selection from the following topics

• homotopy sets, homotopy groups and the Hurewicz theorem
• fibrations and cofibrations
• bundles and classifying spaces

The methods presented are illustrated by applications to various classical problems in algebraic topology.

Books that can be used for some of these topics include:

1. G. Bredon: Topology and geometry, Springer GTM 139,
2. T. tom Dieck: Algebraic topology, EMS 2008
3. P. May: A concise course in algebraic topology, Chicago Lecture Notes in Mathematics, UChicago Press 1999
4. Allen Hatcher: Vector Bundles & K-theory, online book
   http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

The goal of the seminar is for students to get introduced to beautiful and important proofs from all areas of mathematics for which deep mathematical knowledge is not necessary. In the seminar we will read and present different chapters of the celebrated book "Proofs from THE BOOK" by Martin Aigner and Günter Ziegler. It is to our advantage that the book is available in multiple languages and is written in a very accessible way.
The plan is to meet once a week. Each weak there will be a talk from one of the students.
Further details will be discussed on the first meeting.

-- Literature --

Deutsche Version:
https://link.springer.com/book/10.1007/978-3-642-02259-3

English Version:
https://link.springer.com/book/10.1007/978-3-662-57265-8

Aigner & Ziegler: Das Buch der Beweise.

Deutsche Version:
https://link.springer.com/book/10.1007/978-3-642-02259-3

English Version:
https://link.springer.com/book/10.1007/978-3-662-57265-8

In this course we will study topology of symmetric products and ordered and unordered configuration spaces.  

1. T. tom Dieck: Algebraic topology, EMS 2008
2. P. May: A concise course in algebraic topology, Chicago Lecture Notes in Mathematics, UChicago Press 1999
3. John McCleary: A User's Guide to Spectral Sequences
4. Edward R. Fadell , Sufian Y. Husseini: Geometry and Topology of Configuration Spaces

Content: Homology, cohomology and applications, CW-complexes, basic notions of homotopy theory

Literatur

• Hatcher, Allen: Algebraic Topology; Cambridge University Press.
• http://www.math.cornell.edu/~hatcher/AT/ATpage.html
• Lück, Wolfgang: Algebraische Topologie, Homologie und Mannigfaltigkeiten; Vieweg.

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Course Overview This is a beginning course from the series of three courses Topology I—III:

1. Basic notions: topological spaces, continuous maps, connectedness, compactness, products, coproducts, quotients.
2. Groups acting on topological spaces
3. Gluing constructions.
4. Homotopies between continuous maps, degree of a map, fundamental group.
5. Seifert-van Kampen Theorem.
6. Covering spaces.

Literature:

1. Tammo tom Dieck: Topologie, De Gruyter Lehrbuch
2. Allen Hatcher: Algebraic Topology, Chapter I. Also available online from the author's website
3. Klaus Jänich: Topologie, Springer-Verlag
4. Gerd Laures, Markus Szymik: Grundkurs Topologie, Spektrum Akademischer Verlag
5. James R. Munkres: Topology, Prentice Hall

Contents:

• Determinants
• Eigenvalues and eigenvectors: diagonalizability, trigonalizability, set of Cayley-Hamilton, Jordanian normal form
• Bilinear forms
• Vectorräume with scalar product: Euclidean, unitary vectorräume, orthogonal projection, isometries, self-adjusted images, Gram-Schmidt orthonormalization methods, major axis transformation

Prerequisites:
Linear Algebra I
Literature:

Will be mentioned in the lecture.

Siehe http://page.mi.fu-berlin.de/werner99/.

The course will cover a selection from the following topics

• homotopy sets, homotopy groups and the Hurewicz theorem
• fibrations and cofibrations
• generalized homology theories and spectra
• bundles and classifying spaces
• characteristic classes
• low-dimensional topology
• geometric group theory
• spectral sequences and cohomology operations

Books that can be used for some of these topics include:

1. G. Bredon: Topology and geometry, Springer GTM 139,
2. T. tom Dieck: Algebraic topology, EMS 2008
3. P. May: A concise course in algebraic topology, Chicago Lecture Notes in Mathematics, UChicago Press 1999
4. J. Milnor, J.D. Sasheff: Characteristic classes, (AM-76) Princeton University Press
5. Allen Hatcher: Vector Bundles & K-theory, online book
   http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

This class will be taught as a block course from May 23 to June 3 in the framework of a BMS Spring School. Students who would like to receive ungraded credit for this course will have to submit a short paper by the end of the semester.

The class consists of three lecture series:

1. Equivariant topology in discrete geometry (P. Blagojević)

2. Intersection results in convex geometry and topology and their combinatorial applications (F. Frick)

3. Fixed point theorems and their combinatorial applications (S. Zerbib)

Additionally there will be recitations and a few research-focussed talks by invited speakers. The goal of the three lecture series will be to explain recent and current work on geometric and topological methods in combinatorics. 

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Content: Homology, cohomology and applications, CW-complexes, basic notions of homotopy theory

Literatur

• Hatcher, Allen: Algebraic Topology; Cambridge University Press.
• http://www.math.cornell.edu/~hatcher/AT/ATpage.html
• Lück, Wolfgang: Algebraische Topologie, Homologie und Mannigfaltigkeiten; Vieweg.

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics. All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

In the Pro seminar Topology I  topics  complementary to the  course Topology I will be discussed. In particular, we will talk about:

— Nets: Moore-Smith convergence

— Locally compact spaces

— Paracompact spaces

— Countability axioms

— Baire’s category theorem

— Urysohn’s lemma

— Urysohn’s metrication theorem 

— Tietze’s extension theorem

— Tychonov’s theorem for infinite products

— Compact open topology

— Compactly generates spaces

— Weak Haussdorf spaces

— Noetherian spaces

— Topological manifolds

— Partition of unity and imbedding of manifold 

— Cantor set and Peano curves

— Topology of SO(3)

— Fiber bundles

— Simplicial sets

— Classification of compact surfaces

It order to successful participate in this course it is recommended for the students to have some basic knowledge in Topology. Appropriate and complete literature for each of the topics will be provided.

Course Overview This is a beginning course from the series of three courses Topology I—III:

1. Basic notions: topological spaces, continuous maps, connectedness, compactness, products, coproducts, quotients.
2. Groups acting on topological spaces
3. Gluing constructions.
4. Homotopies between continuous maps, degree of a map, fundamental group.
5. Seifert-van Kampen Theorem.
6. Covering spaces.

Literature:

1. Tammo tom Dieck: Topologie, De Gruyter Lehrbuch
2. Allen Hatcher: Algebraic Topology, Chapter I. Also available online from the author's website
3. Klaus Jänich: Topologie, Springer-Verlag
4. Gerd Laures, Markus Szymik: Grundkurs Topologie, Spektrum Akademischer Verlag
5. James R. Munkres: Topology, Prentice Hall

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Contents:

• Eigenvalues and eigenvectors: diagonalizability, trigonalizability, set of Cayley-Hamilton, Jordanian normal form
• Vectorräume with scalar product: Euclidean, unitary vectorräume, orthogonal projection, isometries, self-adjusted images, Gram-Schmidt orthonormalization methods, major axis transformation

Prerequisites: Linear Algebra I

Literature:

Will be mentioned in the lecture.

Siehe http://page.mi.fu-berlin.de/werner99/.

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Content:

• Basic terms/concepts: sets, maps, equivalence relations, groups, rings,
• fields
• Linear equation systems: solvability criteria, Gauss algorithm
• Vector spaces: linear independence, generating systems and bases, dimension,
• subspaces, quotient spaces, cross products in R3
• Linear maps: image and rank, relationship to matrices, behaviour under
• change of basis
• Dual vector spaces: multilinear forms, alternating and symmetric bilinear
• forms, relationship to matices, change of basis
• Determinants: Cramer's rule, Eigenvalues and Eigenvectors

Prerequisites:

Participation in the preparatory course (Brückenkurs) is highly recommended.

Literatur:
Siegfried Bosch, Lineare Algebra, 4. Auflage, Springer-Verlag, 2008;
Tammo tom Dieck, Lineare Algebra, Skript, Universität Göttingen Version vom 7.10.2014

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

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.

Gute Kenntnisse der linearen Algebra werden vorausgesetzt. Vorbildung in Kombinatorik und Geometrie sind hilfreich.

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

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. .

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

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:

• 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, 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, Erdős-Szekeres
• Szemerédi--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.

Gute Kenntnisse der linearen Algebra werden vorausgesetzt. Vorbildung in Kombinatorik und Geometrie sind hilfreich.

Content:

• Homology and cohomology of manifolds proving Poincare duality isomorphism,
• Cohomology operations introducing in particular Steenrod squares,
• Vector bundles, and
• Characteristic classes, in particular we will introduce Stiefely-Whiteny classes using Steenrod squares.

Introduced methods will be illustrated by applications on different classical problems of algebraic topology.

The books that can be used for these topics are:

1. G. Bredon: Topology and Geometry, Springer GTM 139,
2. J. Milnor, J.D. Sasheff: Characteristic classes, (AM-76) Princeton University press
3. Allen Hatcher: Vector Bundles & K-theiry, online book
   http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Content: Homology, cohomology and applications, CW-complexes, basic notions of homotopy theory

Literatur

• Hatcher, Allen: Algebraic Topology; Cambridge University Press.
• http://www.math.cornell.edu/~hatcher/AT/ATpage.html
• Lück, Wolfgang: Algebraische Topologie, Homologie und Mannigfaltigkeiten; Vieweg.

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

http://wikis.fu-berlin.de/display/discgeom/Topology+I

-- this seminar will mostly take place in English --

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

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. .

In this seminar we treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

This course (in English, as the first part of a three semester sequence) will be an introduction to the theory of Convex Polytopes, a core field of Discrete Geometry. The lectures will

• introduce basic facts about polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin eliminiation, Schlegel diagrams, shellability, Gale diagrams, and oriented matroids
• discuss important example and elegant constructions (cyclic and neighborly polytopes, zonotopes, Minkowski sums, permuathedra and associahedra, fiber polytopes, and the Lawrence construction)
• show the excitement of current work in the field.

The course will be based on Günter M. Ziegler's 1995 text book "Lectures on Polytopes". This book has quickly become a standard text book for this field - but it also has triggered a lot of research, many of the open problems from 1995 have been solved by now, so the book does not any more cover "the excitement of current work in the field" - however, the course will try to provide this. In parallel to the course we will revise and update the book, towards a 2016 second edition of "Lectures on Polytopes". You are invited to get involved in this: By criticizing the old text, suggesting improvements and additions, researching new references, contributing problems, etc.

Here is the list of chapters of the book, and thus (roughly) the course contents:

1. Introduction and Examples
2. Polytopes, Polyhedra, and Cones
3. Faces of Polytopes
4. Graphs of Polytopes
5. Steinitz' Theorem for 3-Polytopes
6. Schlegel Diagrams for 4-Polytopes
7. Duality, Gale Diagrams, and Applications
8. Fans, Arrangements, Zonotopes, and Tilings
9. Shellability and the Upper Bound Theorem
10. Fiber Polytopes, and Beyond

References:

• Branko Grünbaum: "Convex Polytopes" (Original edition: Interscience 1967; 2nd edition, Graduate Texts in Math. 221, Springer 2003)
• Günter M. Ziegler "Lectures in Polytopes" (Graduate Texts in Math. 152, 1995; 7th revised printing 2007)

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

This course (in English) is an introduction in Metric Geometry. The target audience for this are students with a basic knowledge in geometry, discrete geometry and topology.  In the course we will introduce and develop basic notions and methods of Metric Geometry following presentation from the book of Burago, Burago and Ivanov: “A course in Metric Geometry.” Here is the list of some chapters of the book, and thus (roughly) an outline of the course contents:

1. Metric Spaces
2. Length Spaces
3. Constructions
4. Spaces of Bounded Curvature
5. Smooth Length Structures
6. Curvature of Riemannian Metrics
7. Spaces of Metric Spaces

1. Burago, Burago and Ivanov: “A course in Metric Geometry” (Graduate Studies in Mathematics Vol. 33; AMS)

Zielgruppe: The target audience are students with a solid background in discrete geometry and/or convex geometry (en par with Discrete Geometry I & II). The topic of this course is a state-of-the-art of advanced topics in discrete geometry that find applications and incarnations in differential geometry, topology, combinatorics, and algebraic geometry.

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 treat current research problems in discrete and computational geometry that can be approached using methods from algebraic topology. The aim of the seminar is to make advances in research and introduce students to potential research topics.
All seminars take place in the seminar room of the "Villa", Arnimallee 2, 14195 Berlin.

Inhalt

Diese Vorlesung für das Bachelorstudium soll als natürliche Fortsetzung von Lineare Algebra I und II Fundamente legen für Vorlesungen/Zyklen wie Diskrete Geometrie, Algebraische Geometrie und Differenzialgeometrie.

Sie behandelt grundlegende Modelle der Geometrie, insbesondere

euklidische, affine, sphärische, projektive und hyperbolische Geometrie,Möbiusgeometrie, Polarität und Dualität Strukturgruppen, Messen (Längen, Winkel, Volumina), explizite Berechnungen und Anwendungen, Beispiele sowie Illustrationsthemen;

Dabei werden weitere Bezüge hergestellt, zum Beispiel zur Funktionentheorie und zur Numerik.

Literatur

1. Marcel Berger. Geometry I
2. David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. Geometry
3. Gerd Fischer. Analytische Geometrie
4. V.V. Prasolov und V.M. Tikhomirov. Geometry

Das Seminar findet statt in SR1 / Arnimallee 2

Inhalt: BMS Advanced Course, in English.
The Course will show how methods from (elementary and advanced) Algebraic Topology can be used to treat problems from Discrete Geometry.
The Tutorials (Übungen) will provide all the necessary tools from Algebraic Topology. Contents:
I. Configuration Space/Test Map schemes for Discrete Geometry Problems
II. Group Cohomology/Index Theory methods
III. Equivariant Obstruction Theory methods.

Zielgruppe: Studierende im Masterstudiengang Mathematik können diese Vorlesung im Rahmen des Moduls "Topologie III" oder "Diskrete Mathematik III" wählen.

Literatur: Suggested background/preparation reading:
J. Matousek: "Using the Borsuk-Ulam Theorem", Springer.

Content:

• Homology and cohomology of manifolds proving Poincare duality isomorphism,
• Cohomology operations introducing in particular Steenrod squares,
• Vector bundles, and
• Characteristic classes, in particular we will introduce Stiefely-Whiteny classes using Steenrod squares.

Introduced methods will be illustrated by applications on different classical problems of algebraic topology.

The books that can be used for these topics are:

1. G. Bredon: Topology and Geometry, Springer GTM 139,
2. J. Milnor, J.D. Sasheff: Characteristic classes, (AM-76) Princeton University press
3. Allen Hatcher: Vector Bundles & K-theiry, online book
   http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html