# Search result: Catalogue data in Spring Semester 2017

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, Topology, Discrete Mathematics, Logic | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-4142-17L | Algebraic Curves | W | 6 credits | 3G | R. Pandharipande | |

Abstract | I will discuss the classical theory of algebraic curves. The topics will include: divisors, Riemann-Roch, linear systems, differentials, Clifford's theorem, curves on surfaces, singularities, curves in projective space, elliptic curves, hyperelliptic curves, families of curves, moduli, and enumerative geometry. There will be many examples and calculations. | |||||

Objective | ||||||

Content | Lecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-4142-17L/ | |||||

Literature | Forster, "Lectures on Riemann Surfaces" Arbarello, Cornalba, Griffiths, Harris, "Geometry of Algebraic Curves" Mumford, "Curves and their Jacobians" | |||||

Prerequisites / Notice | For background, a semester course in algebraic geometry should be sufficient (perhaps even if taken concurrently). You should know the definitions of algebraic varieties and algebraic morphisms and their basic properties. | |||||

401-3106-17L | Class Field Theory | W | 6 credits | 2V + 1U | J. Fresán | |

Abstract | Class Field Theory aims at describing the Galois group of the maximal abelian extension of global and local fields. | |||||

Objective | ||||||

Literature | [1] D. Harari, Cohomologie galoisienne et théorie du corps de classes, EDP Sciences, CNRS Éditions, Paris, 2017. [2] K. Kato, N. Kurokawa, T. Saito, Number theory 2. Introduction to class field theory, Translations of Mathematical Monographs 240, AMS, 2011. [3] J. S. Milne, Class Field Theory (available at http://www.jmilne.org/math/CourseNotes/cft.html) [4] J-P. Serre, Local fields, Grad. Texts Math. 67. Springer-Verlag, 1979. | |||||

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. | |||||

401-3058-00L | Combinatorics I | 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. | |||||

Prerequisites / Notice | Recognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008). | |||||

401-3112-17L | Introduction to Number Theory | W | 4 credits | 2V | C. Busch | |

Abstract | This course gives an introduction to number theory. The focus will be on algebraic number theory. | |||||

Objective | ||||||

Content | The following subjects will be covered: - Euclidean algorithm, greatest common divisor, ... - Congruences, Chinese Remainder Theorem - Quadratic residues, Legendre symbol, law of quadratic reciprocity - Quadratic number fields, integers and primes - Units of quadratic number fields, Pell's equation, Dirichlet unit theorem - Continued fractions and quadratic irrationalities, Theorem of Euler Lagrange, relation to units. | |||||

Literature | - A. Fröhlich, M.J. Taylor, Algebraic number theory, Cambridge studies in advanced mathematics 27, Cambridge University Press, 1991 - S. Lang, Algebraic Number Theory, Second Edition, Graduate Texts in Mathematics, 110, Springer, 1994 - J. Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften 322, Springer 1999 - R. Remmert, P. Ullrich, Elementare Zahlentheorie, Grundstudium Mathematik, Basel Birkhäuser, 2008 - P. Samuel, Algebraic Theory of Numbers, Kershaw Publishing Company LTD, 1972 (Original edition in French at Hermann) - J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer 1973 (Original edition in French at Presses Universitaires de France) | |||||

Prerequisites / Notice | Basic knowledge of Algebra as taught in a course Algebra I + II. | |||||

Selection: Geometry | ||||||

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

401-4206-17L | Group Actions on Trees | W | 4 credits | 2V | N. Lazarovich | |

Abstract | As a main theme, we will explain how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure. After introducing the general theory, we will cover various topics in this general theme. | |||||

Objective | Introduction to the general theory of group actions on trees, also known as Bass-Serre theory, and various important results on decompositions of groups. | |||||

Content | Depending on time we will cover some of the following topics. - Free groups and their subgroups. - The general theory of actions on trees, i.e, Bass-Serre theory. - Trees as 1-dimensional buildings. - Stallings' theorem. - Grushko's and Dunwoody's accessibility results. - Actions on R-trees and the Rips machine. | |||||

Literature | J.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9 C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101 | |||||

Prerequisites / Notice | Familiarity with the basics of fundamental group (and covering theory). | |||||

401-4148-17L | Moduli of Maps and Gromov-Witten invariants | W | 2 credits | 4A | G. Bérczi | |

Abstract | Enumerative questions motivated the development of algebraic geometry for centuries. This course is a short tour to some ideas which have revolutionised enumerative geometry in the last 30 years: stable maps, Gromov-Witten invariants and quantum cohomology. | |||||

Objective | The aim of the course is to understand the concept of stable maps, their moduli and quantum cohomology. We prove Kontsevich's celebrated formula on the number of plane rational curves of degree d passing through 3d-1 given points in general position. | |||||

Content | Topics covered: 1) Brief survey on moduli spaces: fine and coarse moduli. 2) Stable n-pointed curves 3) Stable maps 4) Enumerative geometry via stable maps 5) Gromov-Witten invariants 6) Quantum cohomology and quantum product 7) Kontsevich's formula | |||||

Literature | The main reference for the course is: J. Kock and I.Vainsencher: Kontsevich's Formula for Rational Plane Curves www.math.utah.edu/%7eyplee/teaching/gw/Koch.pdf Background material: -Algebraic varieties: I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag. -Moduli of curves: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag -Moduli spaces (fine and coarse): Peter. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag | |||||

Prerequisites / Notice | Some minimal background in algebraic geometry (varieties, line bundles, Grassmannians, curves). Basic concepts of moduli spaces (fine and coarse) and group actions will be explained mainly through examples. | |||||

401-3056-00L | Finite Geometries IDoes 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. | |||||

401-3574-61L | Introduction to Knot Theory Does not take place this semester. | W | 6 credits | 3G | ||

Abstract | Introduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school. | |||||

Objective | The aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school. | |||||

Content | Definition of a knot and of equivalent knots. Definition of a knot invariant and some elementary examples. Various operations on knots. Knot polynomials (Jones, ev. Alexander.....) | |||||

Literature | An extensive bibliography will be handed out in the course. | |||||

Prerequisites / Notice | Prerequisites are some elementary knowledge of algebra and topology. | |||||

Selection: Analysis | ||||||

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

401-4832-17L | Mathematical Themes in General Relativity II | W | 4 credits | 2V | A. Carlotto | |

Abstract | Second part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems. | |||||

Objective | Acquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis. | |||||

Content | Analysis of Jang's equation and application to the proof of the spacetime positive energy theorem; the conformal method for the Einstein constraint equations and links with the Yamabe problem; gluing methods for the Einstein constraint equations: canonical asymptotics, N-body solutions, gravitational shielding. | |||||

Lecture notes | Lecture notes written by the instructor will be provided to all enrolled students. | |||||

Prerequisites / Notice | The content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable. **This course is the sequel of the one offered during the first semester.** | |||||

401-3352-09L | An Introduction to Partial Differential Equations | W | 6 credits | 3V | F. Da Lio | |

Abstract | This course aims at being an introduction to first and second order partial differential equations (in short PDEs). We will present the so called method of characteristics to solve quasilinear PDEs and some basic properties of classical solutions to second order linear PDEs. | |||||

Objective | ||||||

Content | A preliminary plan is the following - Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem. - Weak and strong maximum principle for elliptic operators. - Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity. - Wave equation, existence of the solution, D'Alembert formula, solutions by spherical means, main properties, uniqueness by energy methods. - The Method of characteristics for first order equations, linear and nonlinear, transport equation, Hamilton-Jacobi equation, scalar conservation laws. - A brief introduction to viscosity solutions. | |||||

Lecture notes | The teacher provides the students with personal notes. | |||||

Literature | Bibliography - L.Evans Partial Differential Equations, AMS 2010 (2nd edition) - D. Gilbarg, N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer, 1998. - E. Di Benedetto Partial Differential Equations, Birkauser, 2010 (2nd edition). - W. A. Strauss Partial Differential Equations. An Introduction, Wiley, 1992. | |||||

Prerequisites / Notice | Differential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory. | |||||

401-3496-17L | Topics in the Calculus of Variations | W | 4 credits | 2V | A. Figalli | |

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Selection: Further Realms | ||||||

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

401-3502-17L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 2 credits | 4A | Professors | |

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

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401-3503-17L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 3 credits | 6A | Professors | |

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

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401-3504-17L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 4 credits | 9A | Professors | |

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

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