Autumn Semester 2020 takes place in a mixed form of online and classroom teaching.
Please read the published information on the individual courses carefully.

Search result: Catalogue data in Autumn Semester 2016

Mathematics Master Information
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
NumberTitleTypeECTSHoursLecturers
401-3117-66LIntroduction to the Circle MethodW6 credits2V + 1UE. Kowalski
AbstractThe circle method, invented by Hardy and Ramanujan and developped by Hardy and Littlewood and Kloosterman, is one of the most versatile methods currently available to determine the asymptotic behavior of the number of integral solutions to polynomial equations, when the number of solutions is sufficiently large.
Objective
ContentThe circle method, invented by Hardy and Ramanujan and developped by Hardy and
Littlewood and Kloosterman, is one of the most versatile methods currently available
to determine the asymptotic behavior of the number of integral solutions to
polynomial equations, when the number of solutions is sufficiently large.

The lecture will present an introduction to this method. In particular, it will
present the solution of Waring's Problem concerning the representability of integers
as sums of a bounded numbers of (fixed) powers of integers.
LiteratureH. Davenport, "Analytic methods for Diophantine equations and Diophatine
inequalities", Cambridge

H. Iwaniec and E. Kowalski, "Analytic number theory", chapter 20; AMS

R. Vaughan, "The Hardy-Littlewood method", Cambridge
401-4209-66LGroup and Representation Theory: Beyond an IntroductionW8 credits3V + 1UT. H. Willwacher
AbstractThe goal of the course is to study several classical and important (and beautiful!) topics in group and representation theory, that are otherwise often overlooked in a standard curriculum.
In particular, we plan to study reflection and Coxeter groups, classical invariant theory, and the theory of real semi simple Lie algebras and their representations.
ObjectiveDespite the title, the course will begin by a recollection of basic concepts of group and representation theory, in particular that of finite groups and Lie groups.
Hence the course should be accessible also for students who only had a brief exposure to representation theory, as for example in the MMP course.
401-3059-00LCombinatorics II
Does not take place this semester.
W4 credits2GN. Hungerbühler
AbstractThe course Combinatorics I and II is an introduction into the field of enumerative combinatorics.
ObjectiveUpon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.
ContentContents 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.
401-4145-66LReading Course: Abelian Varieties over Finite FieldsW2 credits4AJ. Fresán, P. S. Jossen
Abstract
Objective
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-4531-66LTopics in Rigidity TheoryW6 credits3GM. Burger
AbstractThe 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.
ObjectiveUnderstand the basic techniques of rigidity theory.
ContentThis 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 / NoticeFor 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.
401-3309-66LRiemann Surfaces (Part 2) Information W4 credits2VA. Buryak
AbstractThe program will be the following:

* Proof of the Serre duality;
* Riemann-Hurwitz formula;
* Functions and differential forms on a compact Riemann surface with prescribed principal parts;
* Weierstrass points on a compact Riemann surface;
* The Jacobian and the Picard group of a compact Riemann surface;
* Holomorphic vector bundles;
* Non-compact Riemann surfaces.
Objective
LiteratureO. Forster. Lectures on Riemann Surfaces.
Prerequisites / NoticeThis is a continuation of 401-3308-16L Riemann Surfaces that was taught in the spring semester (FS 2016), see Link for the lecture notes. The students are also assumed to be familiar with what would generally be covered in one semester courses on general topology and on algebra.
401-3057-00LFinite Geometries IIW4 credits2GN. Hungerbühler
AbstractFinite 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.
ObjectiveFinite 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.
ContentFinite 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
NumberTitleTypeECTSHoursLecturers
401-3536-11LGeometric Aspects of Hamiltonian DynamicsW6 credits3VP. Biran
AbstractThe course will concentrate on the geometry of the group of Hamiltonian diffeomorphisms introduced by Hofer in the early 1990's and its relations to various topics in symplectic geometry such as capacities, Lagrangian submanifolds, holomorphic curves, as well as recent algebraic structures on the group of Hamiltonian diffeomorphisms such as quasi-morphisms.
Objective
LiteratureBooks:
* L. Polterovich: "The geometry of the group of symplectic diffeomorphisms"
* H. Hofer & E. Zehnder: "Symplectic invariants and Hamiltonian dynamics"
Prerequisites / NoticePrerequisites. Good knowledge of undergraduate mathematics (analysis, complex functions, topology, and differential geometry). Some knowledge of elementary algebraic topology would be useful.
401-4767-66LPartial Differential Equations (Hyperbolic PDEs)W7 credits4VD. Christodoulou
AbstractThe course begins with characteristics, the definition of hyperbolicity,
causal structure and the domain of dependence theorem. The course then focuses
on nonlinear systems of equations in two independent variables, in particular the
Euler equations of compressible fluids with plane symmetry and the Einstein equations
of general relativity with spherical symmetry.
ObjectiveThe objective is to introduce students in mathematics and physics to an area of
mathematical analysis involving differential geometry which is of fundamental importance
for the development of classical macroscopic continuum physics.
ContentThe course shall begin with the basic structure associated to hyperbolic
partial differential equations, characteristic hypersurfaces and bicharacteristics,
causal structure, and the domain of dependence theorem. The course shall then focus
on nonlinear systems of equations in two independent variables. The first topic shall be
the Euler equations of compressible fluids under plane symmetry where we shall
study the formation of shocks, and second topic shall be the Einstein equations
of general relativity under spherical symmetry where we shall study the formation of
black holes and spacetime singularities.
Prerequisites / NoticeBasic real analysis and differential geometry.
401-4831-66LMathematical Themes in General Relativity IW4 credits2VA. Carlotto
AbstractFirst 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.
ObjectiveAcquisition 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.
ContentLorentzian geometry; geometric review of special relativity; the Einstein equations and their basic classes of special solutions; the Einstein equations as an initial-value problem; causality theory and hyperbolicity; singularities and trapped domains; Penrose diagrams; asymptotically flat spaces: ADM invariants, positive mass theorems, Penrose inequalities, geometric properties.
Lecture notesLecture notes written by the instructor will be provided to all enrolled students.
Prerequisites / NoticeThe 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.
401-4497-66LFree Boundary ProblemsW4 credits2VA. Figalli
Abstract
Objective
401-4463-62LFourier Analysis in Function Space TheoryW6 credits3VT. Rivière
AbstractIn the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
Objective
ContentDuring the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces.
In the second part of the course we will study fundamental pro­perties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occa­sion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings.
In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course.
If time permits we shall present the notion of Paraproduct, Para­compositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE.
Literature1) Elias M. Stein, "Singular Integrals and Differentiability Proper­ties of Functions" (PMS-30) Princeton University Press.
2) Javier Duoandikoetxea, "Fourier Analysis" AMS.
3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer.
4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer.
Prerequisites / NoticeNotions from ETH courses in Measure Theory, Functional Analysis I and II (Fun­damental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions)
401-4475-66LPartial Differential Equations and Semigroups of Bounded Linear Operators Information W4 credits2GA. Jentzen
AbstractIn this course we study the concept of a semigroup of bounded linear operators and we use this concept to investigate existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs) of the evolutionary type.
ObjectiveThe aim of this course is to teach the students a decent knowledge (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs.
ContentThe course includes content (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs. Key example PDEs that are treated in this course are heat and wave equations.
Lecture notesLecture Notes are available in the lecture homepage (please follow the link in the Learning materials section).
Literature1. Amnon Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983).

2. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations. Springer-Verlag, New York (2000).
Prerequisites / NoticeMandatory prerequisites: Functional analysis

Start of lectures: Friday, September 23, 2016
For more details, please follow the link in the Learning materials section.
401-3303-00LSpecial Topics in One Complex VariableW6 credits3VH. Knörrer
AbstractHypergeometric Functions, Boundary values of holomorphic functions,
Nevanlinna Theory and other special topics.
ObjectiveAdvanced methods of one complex variables
LiteratureR. Remmert: Funktionentheorie II. Springer Verlag
E.Titchmarsh: The Theory of Functions. Oxford University Press
C.Caratheodory: Funktionentheorie. Birkhaeuser
E.Hille: Analytic Function Theory. AMS Chelsea Publishing
A.Gogolin:Komplexe Integration. Springer
Prerequisites / NoticeFunktionentheorie
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-3502-66LReading Course Restricted registration - show details
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.
W2 credits4AProfessors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3503-66LReading Course Restricted registration - show details
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.
W3 credits6AProfessors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3504-66LReading Course Restricted registration - show details
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.
W4 credits9AProfessors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
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