Search result: Catalogue data in Autumn Semester 2016
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.
|Electives: Pure Mathematics|
|401-4531-66L||Topics in Rigidity Theory||W||6 credits||3G||M. Burger|
|Abstract||The aim of this course is to give detailed proofs of Margulis' normal subgroup theorem and his superrigidity theorem for lattices in higher rank Lie groups.|
|Objective||Understand the basic techniques of rigidity theory.|
|Content||This course gives an introduction to rigidity theory, which is a set of techniques initially invented to understand the structure of a certain class of discrete subgroups of Lie groups, called lattices, and currently used in more general contexts of groups arising as isometries of non-positively curved geometries. A prominent example of a lattice in the Lie group SL(n, R) is the group SL(n, Z) of integer n x n matrices with determinant 1. Prominent questions concerning this group are: |
- Describe all its proper quotients.
- Classify all its finite dimensional linear representations.
- More generally, can this group act by diffeomorphisms on "small" manifolds like the circle?
- Does its Cayley graph considered as a metric space at large scale contain enough information to recover the group structure?
In this course we will give detailed treatment for the answers to the first two questions; they are respectively Margulis' normal subgroup theorem and Margulis' superrigidity theorem. These results, valid for all lattices in simple Lie groups of rank at least 2 --like SL(n, R), with n at least 3-- lead to the arithmeticity theorem, which says that all lattices are obtained by an arithmetic construction.
|Literature||- R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984.|
- D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv
- Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage.
- M.Burger: "Rigidity and Arithmeticity", European School of Group Theory, 1996, handwritten notes, will be put online.
|Prerequisites / Notice||For this course some knowledge of elementary Lie theory would be good. We will however treat Lie groups by examples and avoid structure theory since this is not the point of the course nor of the techniques.|
|401-3309-66L||Riemann Surfaces (Part 2)||W||4 credits||2V||A. Buryak|
|Abstract||The program will be the following:|
* Proof of the Serre duality;
* Riemann-Hurwitz formula;
* Functions and differential forms on a compact Riemann surface with prescribed principal parts;
* Weierstrass points on a compact Riemann surface;
* The Jacobian and the Picard group of a compact Riemann surface;
* Holomorphic vector bundles;
* Non-compact Riemann surfaces.
|Literature||O. Forster. Lectures on Riemann Surfaces.|
|Prerequisites / Notice||This is a continuation of 401-3308-16L Riemann Surfaces that was taught in the spring semester (FS 2016), see Link for the lecture notes. The students are also assumed to be familiar with what would generally be covered in one semester courses on general topology and on algebra.|
|401-3057-00L||Finite Geometries II||W||4 credits||2G||N. Hungerbühler|
|Abstract||Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.|
|Objective||Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.|
|Content||Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design|
|Literature||- Max Jeger, Endliche Geometrien, ETH Skript 1988|
- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983
- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press
- Dembowski: Finite Geometries.
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