Suchergebnis: Katalogdaten im Frühjahrssemester 2016

Mathematik Master Information
Kernfächer
Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Kernfächer aus Bereichen der reinen Mathematik
NummerTitelTypECTSUmfangDozierende
401-3146-12LAlgebraic GeometryW10 KP4V + 1UR. Pandharipande
KurzbeschreibungThis course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
Lernziel
LiteraturThe main reference for the course is
* Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer.

For the exercises we will also use
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.

There are also some very good texts that are freely available online. I recommend two of them:
* J.S. Milne, Algebraic Geometry, Link (mainly about abstract algebraic varieties - schemes only appear in the very end)
* Ravi Vakil, Foundations of Algebraic Geometry, Link (quite abstract)

Further readings:
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Ulrich Görtz and Torsten Wedhorn, Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link
Voraussetzungen / BesonderesRequirement: Commutative Algebra course.
401-3226-01LUnitary Representations of Lie Groups and Discrete Subgroups of Lie GroupsW8 KP4GM. Einsiedler
KurzbeschreibungThis course will contain three parts:
* Classification of simple Lie algebras
* Introduction to unitary representations of Lie groups
* Introduction to the study of discrete subgroups of Lie groups, the quotient space, and some applications.
LernzielThe goal is to acquire familiarity with the basic formalism and results concerning Lie groups and their unitary representations, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups.
Inhalt* Classification using Dynkin diagrams
* 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
LiteraturBekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press.
Voraussetzungen / BesonderesFunctional analysis I and
Lie Groups I or Differential geometry I.
401-3002-12LAlgebraic Topology IIW8 KP4GP. Biran
KurzbeschreibungThis 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.
Lernziel
Literatur1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:
Link

See also:
Link

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.
Voraussetzungen / BesonderesGeneral 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.
401-3372-00LDynamical Systems IIW10 KP4V + 1UW. Merry
KurzbeschreibungThis course is intended as an introduction to hyperbolic dynamical systems.
LernzielThis course is intended as an introduction to hyperbolic dynamical systems.
InhaltThe course begins with the basic notions and definitions of dynamical systems. We then move on hyperbolic dynamical systems. Roughly speaking, hyperbolic systems are dynamical systems that can exhibit the so-called deterministic chaotic behavior - the appearance of chaotic motions in purely deterministic dynamical systems.

We will cover some of the fundamental results on hyperbolic systems, including the Anosov closing lemma and
the stable manifold theorem. The last part of the course moves onto structural stability theory.

Familiarity with basic differential geometry is essential. It is not necessary to have taken Dynamical Systems I last semester.
LiteraturUseful textbooks include:

- Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.
- Introduction to the Modern Theory of Dynamical Systems, Katok and Hasselblatt, CUP (1994)
- Global Stability of Dynamical Systems, Shub, Springer-Verlag (1987)

I will also provide partial lecture notes.
401-3532-08LDifferential Geometry IIW10 KP4V + 1UM. Burger
KurzbeschreibungThe aim of this course is to give an introduction to Riemannian Geometry and modern metric geometry.
LernzielRiemannian Geometry, metric geometry.
InhaltThe aim of this course is to give an introduction to Riemannian Geometry and modern metric geometry. We will present the basics on affine and riemannian connections, discuss existence and properties of geodesics; then we proceed to the central concept of riemannian curvature tensor and its various avatars, like sectional curvature and scalar curvature. We will then move to Topogonov's comparison theorems. This constitutes the bridge with metric geometry and the modern notion of negative curvature, which applies to singular spaces, and constitutes the topic of the second part of this course.
SkriptWill be made available.
LiteraturM.P. do Carmo, "Riemannian Geometry", Birkhauser, 1992

M. Bridson, A. Haefliger, "Metric Spaces of Non-Positive Curvature",
Springer 1999.
Voraussetzungen / BesonderesPrerequisite are the sections concerning manifolds and tangent bundles of the Differential Geometry I course, Fall Semester 2015.
401-3462-00LFunctional Analysis IIW10 KP4V + 1UD. A. Salamon
KurzbeschreibungSobolev spaces, Calderon-Zygmund inequality,
elliptic regularity, strongly continuous semigroups,
parabolic pde's.
LernzielThe lecture course will begin with an introduction to Sobolev spaces
and Sobolev embedding theorems, a proof of the Calderon-Zygmund
inequality, and regularity theorems for second order elliptic operators,
followed by an introduction to the theory of strongly continuous
operator semigroups and some basic results about parabolic regularity.
Applications to geometry will be included if time allows.
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