# Search result: Catalogue data in Autumn Semester 2020

Mathematics Master | ||||||

Electives 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 | ||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|---|

401-3119-70L | p-Adic Numbers | W | 4 credits | 2V | P. Bengoechea Duro | |

Abstract | This course is an introduction to the p-adic numbers. We will see how the field of p-adic numbers Q_p is build. We will explore the (strange) topology and the arithmetic of Q_p, as well as some elementary analytic concepts such as functions, continuity, integrals, etc. We will explain an algebraic and an analytic reasons of interest for the existence of p-adic numbers. | |||||

Objective | ||||||

Content | - Absolute values on Q and Completions - Topology and Arithmetic of Q_p, p-adic Integers - Equations over p-adic numbers and Hensel's Lemma - Local-global principle - Hasse-Minkowski's Theorem on binary quadratic forms - Elementary Analysis in Q_p - the p-adic Riemann zeta function | |||||

Literature | "p-adic Numbers. An Introduction", Fernando Q. Gouvea (Springer) "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Neal Koblitz (Springer) "p-adic numbers and Diophantine equations", Yuri Bilu (online notes 2013) | |||||

Prerequisites / Notice | The courses Topology, Measure and Integration, Algebra I/II are required prerequisites. | |||||

401-3059-00L | Combinatorics IIDoes not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | |||||

Objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | |||||

Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | |||||

Selection: Geometry | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-3533-70L | Differential Geometry III | W | 4 credits | 2V | U. Lang | |

Abstract | Topics in Riemannian geometry in the large: the structure of complete, non-compact Riemannian manifolds of non-negative sectional curvature, including Perelman's (1994) proof of the Cheeger-Gromoll soul conjecture; the Besson-Courtois-Gallot barycenter method (1996) and the proofs of the minimal entropy theorem and the Mostow rigidity theorem for rank one locally symmetric spaces. | |||||

Objective | ||||||

401-4531-66L | Topics in Rigidity Theory | W | 6 credits | 3V | 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. | |||||

401-4141-70L | Curves, Jacobians, and Modern Abel-Jacobi Theory | W | 6 credits | 3V | R. Pandharipande | |

Abstract | ||||||

Objective | ||||||

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. | |||||

Selection: Analysis | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-4355-70L | Elliptic Regularity Theory | W | 8 credits | 4V | M. Struwe | |

Abstract | We extend the theory developed in Functional Analysis II in various directions, including variants of the maximum principle, Harnack's inequality, L^p-theory, and systems. Certain limit cases will be discussed. Examples, including the harmonic map system, will illustrate the use of these methods. | |||||

Objective | ||||||

Literature | Giaquinta, Mariano: Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993. Gilbarg, David; Trudinger, Neil S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 2001. Further references will be given in the lectures. | |||||

Selection: Further Realms | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-3502-70L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 2 credits | 4A | Supervisors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective | ||||||

401-3503-70L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 3 credits | 6A | Supervisors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective | ||||||

401-3504-70L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective | ||||||

401-0000-00L | Communication in Mathematics | W | 2 credits | 1V | W. Merry | |

Abstract | Don't hide your Next Great Theorem behind bad writing. This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. In addition, the course will help you establish a working knowledge of LaTeX. | |||||

Objective | Knowing how to present written mathematics in a structured and clear manner. | |||||

Content | Topics covered include: - Language conventions and common errors. - How to write a thesis (more generally, a mathematics paper). - How to use LaTeX. - How to write a personal statement for Masters and PhD applications. | |||||

Lecture notes | Full lecture notes will be made available on my website: Link | |||||

Prerequisites / Notice | There are no formal mathematical prerequisites. |

- Page 1 of 1