Marc Burger: Catalogue data in Autumn Semester 2016
|Name||Prof. Dr. Marc Burger|
ETH Zürich, HG G 37.1
|Telephone||+41 44 632 49 73|
|Fax||+41 44 632 10 85|
|401-3200-64L||Proofs from THE BOOK |
Number of participants limited to 26.
|4 credits||2S||M. Burger|
|Objective||Ziel des Seminares ist zu lernen wie man Mathematik vortraegt. Als|
Vorlage fuer dieses Seminar dient das Buch von Aigner und Ziegler "Proofs from the BOOK"
das aus allen Gebieten der Mathematik fundamentale Saetze und deren "schoensten" Beweise
praesentiert. Die Auswahl der Themen ist also gross und es gibt etwas fuer jeden Geschmack.
Vortraege koennen auf Deutsch, Franzoesisch oder Englisch gehalten werden.
|401-4531-66L||Topics in Rigidity Theory||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.|
|401-5000-00L||Zurich Colloquium in Mathematics||0 credits||W. Werner, P. L. Bühlmann, M. Burger, S. Mishra, R. Pandharipande, University lecturers|
|Abstract||The lectures try to give an overview of "what is going on" in important areas of contemporary mathematics, to a wider non-specialised audience of mathematicians.|
|401-5530-00L||Geometry Seminar||0 credits||1K||M. Burger, M. Einsiedler, U. Lang, University lecturers|