# Search result: Catalogue data in Spring Semester 2017

Mathematics Master | ||||||

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 | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3146-12L | Algebraic Geometry | W | 10 credits | 4V + 1U | R. Pink | |

Abstract | This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes). | |||||

Objective | Learning Algebraic Geometry. | |||||

Literature | Primary 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, http://math.stanford.edu/~vakil/216blog/ * Jean Gallier and Stephen S. Shatz, Algebraic Geometry http://www.cis.upenn.edu/~jean/algeom/steve01.html "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, http://www.jmilne.org/math/CourseNotes/AG.pdf 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 / Notice | Requirement: Some knowledge of Commutative Algebra. | |||||

401-3002-12L | Algebraic Topology II | W | 8 credits | 4G | P. S. Jossen | |

Abstract | This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as: products, duality, cohomology operations, characteristic classes, spectral sequences etc. | |||||

Objective | ||||||

Literature | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: http://www.math.cornell.edu/~hatcher/AT/ATpage.html See also: http://www.math.cornell.edu/~hatcher/#anchor1772800 2) E. Spanier, "Algebraic topology", Springer-Verlag 3) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 4) R. Bott & L. Tu, "Differential forms in algebraic topology", Graduate Texts in Mathematics, 82. Springer-Verlag, 1982. 5) J. Milnor & J. Stasheff, "Characteristic classes", Annals of Mathematics Studies, No. 76. Princeton University Press, 1974. | |||||

Prerequisites / Notice | General topology, linear algebra. Basic knowledge of singular homolgoy and cohomology 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. | |||||

401-3226-01L | Representation Theory of Lie Groups | W | 8 credits | 4G | E. Kowalski | |

Abstract | This course will contain two parts: * Introduction to unitary representations of Lie groups * Introduction to the study of discrete subgroups of Lie groups and some applications. | |||||

Objective | The goal is to acquire familiarity with the basic formalism and results concerning unitary representations of Lie groups, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups. | |||||

Content | * Unitary representations of compact Lie groups: Peter-Weyl theory, weights, Weyl character formula * Introduction to unitary representations of non-compact Lie groups: the examples of SL(2,R), SL(2,C) * Example: Property (T) for SL(n,R) * Discrete subgroups of Lie groups: examples and some applications | |||||

Literature | Bekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press. | |||||

Prerequisites / Notice | Differential geometry, Functional analysis, Introduction to Lie Groups (or equivalent). Notice that this course has a large overlap with 401-3226-01L Unitary Representations of Lie Groups and Discrete Subgroups of Lie Groups taught in FS 2016. Therefore it is not possible to acquire credits for both courses. | |||||

401-3372-00L | Dynamical Systems II | W | 10 credits | 4V + 1U | W. Merry | |

Abstract | This course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic dynamics. | |||||

Objective | Mastery of the basic methods and principal themes of some aspects of hyperbolic dynamical systems. | |||||

Content | Topics covered include: - Circle homeomorphisms and rotation numbers. - 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. | |||||

Lecture notes | I will provide full lecture notes, available here: http://www.merry.io/dynamical-systems/ | |||||

Literature | The most useful textbook is - Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002. Another (more advanced) useful book is - Introduction to the Modern Theory of Dynamical Systems, Katok and Hasselblatt, CUP, 1995. | |||||

Prerequisites / Notice | It will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here: http://www.merry.io/dynamical-systems/ However we will only really use material covered in the first 12 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 12 lectures. In addition, it would be useful to have some familiarity with basic differential geometry. | |||||

401-3532-08L | Differential Geometry II | W | 10 credits | 4V + 1U | U. Lang | |

Abstract | Introduction 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, curvature and topology, spaces of riemannian manifolds. | |||||

Objective | The aim of this course is to give an introduction to Riemannian Geometry in combination with some elements of modern metric geometry. | |||||

Content | Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form of submanifolds, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of riemannian manifolds. | |||||

Literature | Riemannian Geometry: - 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 Metric Geometry: - M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999 - D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Amer. Math. Soc. 2001 | |||||

Prerequisites / Notice | Prerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, tangent and tensor bundles, and differential forms. | |||||

401-3462-00L | Functional Analysis II | W | 10 credits | 4V + 1U | M. Struwe | |

Abstract | Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity theory, Schauder estimates | |||||

Objective | The lecture course will focus on weak solutions of elliptic boundary value problems in Sobolev spaces and discuss their regularity properties, possibly followed by a proof of the Calderon-Zygmund inequality and some basic results on parabolic regularity, with applications to geometry, if time allows. |

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