Suchergebnis: Katalogdaten im Frühjahrssemester 2020

Mathematik Bachelor Information
Kernfächer aus Bereichen der reinen Mathematik
401-3532-08LDifferential Geometry II Information W10 KP4V + 1UU. Lang
KurzbeschreibungIntroduction to Riemannian geometry in combination with some elements of modern metric geometry. Contents: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of Riemannian manifolds.
LernzielLearn the basics of Riemannian geometry and some elements of modern metric geometry.
Literatur- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- B. O'Neill, Semi-Riemannian Geometry, With Applications to Relativity, Academic Press 1983
Voraussetzungen / BesonderesPrerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, and differential forms.
401-3462-00LFunctional Analysis II Information W10 KP4V + 1UM. Struwe
KurzbeschreibungSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity
LernzielAcquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates
SkriptFunktionalanalysis II, Michael Struwe
LiteraturFunktionalanalysis II, Michael Struwe

Functional Analysis, Spectral Theory and Applications.
Manfred Einsiedler and Thomas Ward, GTM Springer 2017
Voraussetzungen / BesonderesFunctional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces.
401-3146-12LAlgebraic Geometry Information W10 KP4V + 1UD. Johnson
KurzbeschreibungThis course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
LernzielLearning Algebraic Geometry.
LiteraturPrimary reference:
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.

Secondary reference:
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.
* Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013).

Other good textbooks and online texts are:
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer.
* Ravi Vakil, Foundations of Algebraic Geometry,
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry

"Classical" Algebraic Geometry over an algebraically closed field:
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.
* J.S. Milne, Algebraic Geometry,

Further readings:
* Günter Harder: Algebraic Geometry 1 & 2
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
Voraussetzungen / BesonderesRequirement: Some knowledge of Commutative Algebra.
401-3002-12LAlgebraic Topology II Information W8 KP4GA. Sisto
KurzbeschreibungThis is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including:
cohomology of spaces, operations in homology and cohomology, duality.
Literatur1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

The book can be downloaded for free at:

2) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / BesonderesGeneral topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I").

Some knowledge of differential geometry and differential topology
is useful but not absolutely necessary.
401-3372-00LDynamical Systems IIW10 KP4V + 1UW. Merry
KurzbeschreibungThis course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic and complex dynamics.
LernzielMastery of the basic methods and principal themes of some aspects of hyperbolic and complex dynamical systems.
InhaltTopics covered include:

- Hyperbolic linear dynamical systems, hyperbolic fixed points, the Hartman-Grobman Theorem.
- Hyperbolic sets, Anosov diffeomorphisms.
- The (Un)stable Manifold Theorem.
- Shadowing Lemmas and stability.
- The Lambda Lemma.
- Transverse homoclinic points, horseshoes, and chaos.
- Complex dynamics of rational maps on the Riemann sphere
- Julia sets and Fatou sets.
- Fractals and the Mandelbrot set.
SkriptI will provide full lecture notes, available here:
LiteraturThe most useful textbook is

- Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.
Voraussetzungen / BesonderesIt will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here:

However we will only really use material covered in the first 10 lectures of Dynamical Systems I, so if you did not attend Dynamical Systems I, it is sufficient to read through the notes from the first 10 lectures.

In addition, it would be useful to have some familiarity with basic differential geometry and complex analysis.
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