# Search result: Catalogue data in Autumn Semester 2019

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 | |
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401-3033-00L | Gödel's Theorems | W | 8 credits | 3V + 1U | L. Halbeisen | |

Abstract | Die Vorlesung besteht aus drei Teilen: Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe. Teil II behandelt den Gödel'schen Vollständigkeitssatz Teil III behandelt die Gödel'schen Unvollständigkeitssätze | |||||

Objective | Das Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln. | |||||

Content | Syntax und Semantik der Prädikatenlogik Gödel'scher Vollständigkeitssatz Gödel'sche Unvollständigkeitssätze | |||||

Literature | Ergänzende Literatur wird in der Vorlesung angegeben. | |||||

401-4037-69L | O-Minimality and Diophantine Applications | W | 4 credits | 2V | A. Forey | |

Abstract | O-minimal structures provide a framework for tame topology as envisioned by Grothendieck. Originally it was mainly a topic of interest for real algebraic geometers. However, since Pila and Wilkie proved their counting theorem for rational points of bounded height, many applications to diophantine and algebraic geometry have been found. | |||||

Objective | The overall goal of this course is to provide an introduction to o-minimality and to prove results needed for diophantine applications. | |||||

Content | The first part of the course will be devoted to the definition of o-minimal structures and to prove the cell decomposition theorem, which is crucial for describing the shape of subsets of an o-minimal structure. In the second part of the course, we will prove the Pila-Wilkie counting theorem. The last part will be devoted to diophantine applications, with the proof by Pila and Zanier of the Manin-Mumford conjecture and, if time permit, a sketch of the proof by Pila of the André-Oort conjecture for product of modular curves. | |||||

Literature | G. Jones and A. Wilkie: O-minimality and diophantine geometry, Cambridge University Press L. van den Dries: Tame topology and o-minimal structures, Cambridge University Press | |||||

Prerequisites / Notice | This course is appropriate for people with basic knowledge of commutative algebra and algebraic geometry. Knowledge of mathematical logic is welcomed but not required. | |||||

401-4117-69L | p-Adic Galois Representations | W | 4 credits | 2V | M. Mornev | |

Abstract | This course covers the structure theory of Galois groups of local fields, the rings of Witt vectors, the classification of p-adic representations via phi-modules, the tilting construction from the theory of perfectoid spaces, the ring of de Rham periods and the notion of a de Rham representation. | |||||

Objective | Understanding the construction of the ring of de Rham periods. | |||||

Content | In addition to the subjects mentioned in the abstract the course included the basic theory of local fields, l-adic local Galois representations, an oveview of perfectoid fields, the statements of the theorems of Fontaine-Winterberger and Faltings-Tsuji. | |||||

Literature | J.-M. Fontaine, Y. Ouyang. Theory of p-adic Galois representations. O. Brinon, B. Conrad. CMI summer school notes on p-adic Hodge theory. | |||||

Prerequisites / Notice | General topology, linear algebra, Galois theory. | |||||

401-3059-00L | Combinatorics II | 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 | |

401-4531-69L | Four-Manifolds | W | 4 credits | 2V | G. Smirnov | |

Abstract | Making use of theoretical physics methods, Witten came up with a novel approach to four-dimensional smooth structures, which made the constructing of exotic 4-manifolds somewhat routine. Today, Seiberg-Witten theory has become a classical topic in mathematics, which has a variety of applications to complex and symplectic geometry. We will go through some of these applications. | |||||

Objective | This introductory course has but one goal, namely to familiarize the students with the basics in the Seiberg-Witten theory. | |||||

Content | The course will begin with an introduction to Freedman’s classification theorem for simply-connected topological 4-manifolds. We then will move to the Seiberg-Witten equations and prove the Donaldson theorem of positive-definite intersection forms. Time permitting we may discuss some applications of SW-theory to real symplectic 4-manifolds. | |||||

Prerequisites / Notice | Some knowledge of homology, homotopy, vector bundles, moduli spaces of something, elliptic operators would be an advantage. | |||||

401-3057-00L | Finite Geometries IIDoes not take place this semester. | 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 | |

401-4351-69L | Optimal Transport | W | 4 credits | 2V | A. Figalli | |

Abstract | In this course I plan to give an introduction to optimal transport: I'll first introduce the optimal transport problem and explain how to solve it in some important cases of interest. Then I'll show a series of applications to geometry and to gradient flows. | |||||

Objective | The aim of the course is to provide a self contained introduction to optimal transport. The students are expected to know the basic concepts of measure theory. Although not strictly required, some basic knowledge of Riemannian geometry may be useful. | |||||

Literature | Topics in Optimal Transportation (Graduate Studies in Mathematics, Vol. 58), by Cédric Villani Optimal Transport for Applied Mathematicians (Calculus of Variations, PDEs, and Modeling), by Filippo Santambrogio Optimal transport and curvature, available at Link | |||||

401-4461-69L | Reading Course: Functional Analysis III, Unitary RepresentationsLimited number of participants. Please contact andreas.wieser@math.ethz.ch | W | 3 credits | 6A | M. Einsiedler, further speakers | |

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Selection: Further Realms | ||||||

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

401-3502-69L | 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 | ||||||

401-3503-69L | 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 | ||||||

401-3504-69L | 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: https://www.merry.io/teaching/ | |||||

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

401-0000-99L | Communication in Mathematics (Upgrade 2018 → 2019)This course unit is only for students who got 1 ECTS credit from last year's course unit 401-0000-00L CiM. (Registration now closed.) | W | 1 credit | 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: https://www.merry.io/teaching/ | |||||

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

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