Search result: Catalogue data in Spring Semester 2020

Mathematics Master Information
Core Courses
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3146-12LAlgebraic Geometry Information W10 credits4V + 1UD. Johnson
AbstractThis course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
ObjectiveLearning Algebraic Geometry.
LiteraturePrimary 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, Link
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link

"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, Link

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.
Prerequisites / NoticeRequirement: Some knowledge of Commutative Algebra.
401-3002-12LAlgebraic Topology II Information W8 credits4GA. Sisto
AbstractThis 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.
Objective
Literature1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

The book can be downloaded for free at:
Link

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

3) E. Spanier, "Algebraic topology", Springer-Verlag
Prerequisites / NoticeGeneral 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.
401-3226-00LSymmetric Spaces Information W8 credits4GM. Burger
Abstract* Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples
* Symmetric spaces of non-compact type: flats and rank, roots and root spaces
* Iwasawa decomposition, Weyl group, Cartan decomposition
* Hints of the geometry at infinity of SL(n,R)/SO(n).
ObjectiveLearn the basics of symmetric spaces
401-3372-00LDynamical Systems IIW10 credits4V + 1UW. Merry
AbstractThis course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic and complex dynamics.
ObjectiveMastery of the basic methods and principal themes of some aspects of hyperbolic and complex dynamical systems.
ContentTopics 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.
Lecture notesI will provide full lecture notes, available here:

Link
LiteratureThe most useful textbook is

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

Link

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.
401-3532-08LDifferential Geometry II Information W10 credits4V + 1UU. Lang
AbstractIntroduction 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.
ObjectiveLearn the basics of Riemannian geometry and some elements of modern metric geometry.
Literature- 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
Prerequisites / NoticePrerequisite 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 credits4V + 1UM. Struwe
AbstractSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity
ObjectiveAcquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates
Lecture notesFunktionalanalysis II, Michael Struwe
LiteratureFunktionalanalysis II, Michael Struwe

Functional Analysis, Spectral Theory and Applications.
Manfred Einsiedler and Thomas Ward, GTM Springer 2017
Prerequisites / NoticeFunctional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces.
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