# Search result: Catalogue data in Spring Semester 2020

Mathematics Master | ||||||

Course Units for Additional Admission Requirements The courses below are only available for MSc students with additional admission requirements. | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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406-2004-AAL | Algebra IIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 5 credits | 11R | R. Pink | |

Abstract | Galois theory and related topics. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

Learning objective | Introduction to fundamentals of field extensions, Galois theory, and related topics. | |||||

Content | The main topic is Galois Theory. Starting point is the problem of solvability of algebraic equations by radicals. Galois theory solves this problem by making a connection between field extensions and group theory. Galois theory will enable us to prove the theorem of Abel-Ruffini, that there are polynomials of degree 5 that are not solvable by radicals, as well as Galois' theorem characterizing those polynomials which are solvable by radicals. | |||||

Literature | Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society Galois Theory is the topic treated in Chapter A5. | |||||

Prerequisites / Notice | Algebra I, in Rotman's book this corresponds to the topics treated in the Chapters A3 and A4. | |||||

406-2005-AAL | Algebra I and IIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 12 credits | 26R | R. Pink | |

Abstract | Introduction and development of some basic algebraic structures - groups, rings, fields including Galois theory, representations of finite groups, algebras. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

Learning objective | ||||||

Content | Basic notions and examples of groups; Subgroups, Quotient groups and Homomorphisms, Group actions and applications Basic notions and examples of rings; Ring Homomorphisms, ideals, and quotient rings, rings of fractions Euclidean domains, Principal ideal domains, Unique factorization domains Basic notions and examples of fields; Field extensions, Algebraic extensions, Classical straight edge and compass constructions Fundamentals of Galois theory Representation theory of finite groups and algebras | |||||

Literature | Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society | |||||

406-2284-AAL | Measure and IntegrationEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | F. Da Lio | |

Abstract | Introduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces | |||||

Learning objective | Basic acquaintance with the abstract theory of measure and integration | |||||

Content | Introduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces | |||||

Lecture notes | no lecture notes | |||||

Literature | 1. P.R. Halmos, "Measure Theory", Springer 2. Extra material: Lecture Notes by Emmanuel Kowalski and Josef Teichmann from spring semester 2012, http://www.math.ethz.ch/~jteichma/measure-integral_120615.pdf 3. Extra material: P. Cannarsa & T. D'Aprile, "Lecture Notes on Measure Theory and Functional Analysis", http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf | |||||

Prerequisites / Notice | The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

406-2303-AAL | Complex AnalysisAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | P. Biran | |

Abstract | Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, conformal mappings, Riemann mapping theorem. | |||||

Learning objective | ||||||

Literature | L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co. B. Palka: "An introduction to complex function theory." Undergraduate Texts in Mathematics. Springer-Verlag, 1991. R.Remmert: Theory of Complex Functions.. Springer Verlag E.Hille: Analytic Function Theory. AMS Chelsea Publication | |||||

Prerequisites / Notice | The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

406-2554-AAL | TopologyAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | A. Carlotto | |

Abstract | Topics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces. | |||||

Learning objective | An introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures. | |||||

Lecture notes | See lecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-2554-00L/ | |||||

Literature | James Munkres: Topology | |||||

Prerequisites / Notice | The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

406-2604-AAL | Probability and StatisticsAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 7 credits | 15R | M. Schweizer | |

Abstract | - Discrete probability spaces - Continuous models - Limit theorems - Introduction to statistics | |||||

Learning objective | The goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. This includes a mathematically rigorous treatment as well as intuition and getting acquainted with the ideas behind the definitions. The course does not use measure theory systematically, but does point out where this is required and what the connections are. | |||||

Content | - Probability: Chapters 1-12 from the book by DasGupta - Statistics: Chapters 8-11 from the book by Rice | |||||

Lecture notes | There will be lecture notes (in German) that are continuously updated during the semester. | |||||

Literature | A. DasGupta, Fundamentals of Probability: A First Course, Springer (2010) J. A. Rice, Mathematical Statistics and Data Analysis, Duxbury Press, second edition (1995) | |||||

Prerequisites / Notice | Some of the exercise classes associated to the original course (which is in German) will be offered in English. |

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