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
Core Courses
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.
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3225-00LIntroduction to Lie GroupsW8 credits4GP. D. Nelson
AbstractTopological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups.
ObjectiveThe goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it.
LiteratureA. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser)
A.Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73)
F.Warner: "Foundations of differentiable manifolds and Lie groups" (Springer)
H. Samelson: "Notes on Lie algebras" (Springer, '90)
S.Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78)
A.Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press)
Prerequisites / NoticeTopology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester.

Course webpage: http://www.math.ethz.ch/education/bachelor/lectures/hs2014/math/introlg
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields
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NumberTitleTypeECTSHoursLecturers
401-3651-00LNumerical Methods for Elliptic and Parabolic Partial Differential Equations Information
Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.
W10 credits4V + 1UC. Schwab
AbstractThis course gives a comprehensive introduction into the numerical treatment of linear and non-linear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.
ObjectiveParticipants of the course should become familiar with
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
ContentA selection of the following topics will be covered:

* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
Lecture notesCourse slides will be made available to the audience.
Literaturen.a.
Prerequisites / NoticePractical exercises based on MATLAB
401-3621-00LFundamentals of Mathematical StatisticsW10 credits4V + 1UF. Balabdaoui
AbstractThe course covers the basics of inferential statistics.
Objective
401-4889-00LMathematical FinanceW11 credits4V + 2UM. Schweizer
AbstractAdvanced introduction to mathematical finance:
- absence of arbitrage and martingale measures
- option pricing and hedging
- optimal investment problems
- additional topics
ObjectiveAdvanced level introduction to mathematical finance, presupposing knowledge in probability theory and stochastic processes
ContentThis is an advanced level introduction to mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this in both discrete- and continuous-time models. Topics include absence of arbitrage and martingale measures, option pricing and hedging, optimal investment problems, and probably others.
Prerequisites are probability theory and stochastic processes (for which lecture notes are available).
Lecture notesNone available
LiteratureDetails will be announced in the course.
Prerequisites / NoticePrerequisites are probability theory and stochastic processes (for which lecture notes are available).
401-3901-00LMathematical Optimization Information W11 credits4V + 2UR. Weismantel
AbstractMathematical treatment of diverse optimization techniques.
ObjectiveAdvanced optimization theory and algorithms.
Content1. Linear optimization: The geometry of linear programming, the simplex method for solving linear programming problems, Farkas' Lemma and infeasibility certificates, duality theory of linear programming.

2. Nonlinear optimization: Lagrange relaxation techniques, Newton method and gradient schemes for convex optimization.

3. Integer optimization: Ties between linear and integer optimization, total unimodularity, complexity theory, cutting plane theory.

4. Combinatorial optimization: Network flow problems, structural results and algorithms for matroids, matchings and, more generally, independence systems.
(also Bachelor) Core Courses: Pure Mathematics
Further restrictions apply, but in particular:
401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme.
Analogously for:
401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II
401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II
401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry
401-3371-00L Dynamical Systems I - 401-3372-00L Dynamical Systems II
For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3461-00LFunctional Analysis I
This course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from neither 401-3461-00L Functional Analysis I nor 401-3462-00L Functional Analysis II for their Bachelor's degree still can have recognised this course for the Master's degree.
Furthermore, at most one of the three course units
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics.
W10 credits4V + 1UM. Struwe
AbstractBaire category; Banach and Hilbert spaces, bounded linear operators; three fundamental principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces.
Objective
Lecture notesLecture Notes on "Funktionalanalysis I" by Michael Struwe
401-3531-00LDifferential Geometry I
This course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from neither 401-3531-00L Differential Geometry I nor 401-3532-00L Differential Geometry II for their Bachelor's degree still can have recognised this course for the Master's degree.
Furthermore, at most one of the three course units
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics.
W10 credits4V + 1UU. Lang
AbstractCurves in R^n, inner geometry of hypersurfaces in R^n, curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, tangent bundle, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
ObjectiveIntroduction to elementary differential geometry and differential topology.
Content- Differential geometry in R^n: theory of curves, submanifolds and immersions, inner geometry of hypersurfaces, Gauss map and curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet, Poincaré Index Theorem.
- The hyperbolic space.
- Differential topology: differentiable manifolds, tangent bundle, immersions and embeddings in R^n, Sard's Theorem, transversality, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
LiteratureDifferential Geometry in R^n:
- Manfredo P. do Carmo: Differential geometry of curves and surfaces
- Wolfgang Kühnel: Differentialgeometrie. Curves-surfaces-manifolds
- Christian Bär: Elementary differential geometry
Differential Topology:
- Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds
- Victor Guillemin & Alan Pollack: Differential Topology
- Morris W. Hirsch: Differential Topology
401-3371-00LDynamical Systems IW10 credits4V + 1UW. Merry
AbstractThis course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics.
ObjectiveMastery of the basic methods and principal themes of some aspects of dynamical systems.
ContentTopics covered include:

1. Topological dynamics
(transitivity, attractors, chaos, structural stability)

2. Ergodic theory
(Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures)

3. Low-dimensional dynamics
(Poincare rotation number, dynamical systems on [0,1])
LiteratureThe most relevant textbook for this course is

Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.

I will also produce full lecture notes.
Prerequisites / NoticeThe material of the basic courses of the first two years of the program at ETH is assumed. In particular, you should be familiar with metric spaces and elementary measure theory.
401-3001-61LAlgebraic Topology IW8 credits4GP. S. Jossen
AbstractThis is an introductory course in algebraic topology. The course will cover the following main topics: introduction to homotopy theory, homology and cohomology of spaces.
Objective
Literature1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:
http://www.math.cornell.edu/%7ehatcher/AT/ATpage.html

See also:
http://www.math.cornell.edu/%7eehatcher/#anchor1772800

3) E. Spanier, "Algebraic topology", Springer-Verlag
Prerequisites / NoticeGeneral topology, linear algebra.

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
401-3132-00LCommutative Algebra Information W10 credits4V + 1UR. Pink
AbstractThis course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. The material in this course will be assumed in the lecture course "Algebraic Geometry" in the spring semester 2017.
ObjectiveWe shall cover approximately the material from
--- most of the textbook by Atiyah-MacDonald, or
--- the first half of the textbook by Bosch.
Topics include:
* Basics about rings, ideals and modules
* Localization
* Primary decomposition
* Integral dependence and valuations
* Noetherian rings
* Completions
* Basic dimension theory
LiteraturePrimary Reference:
1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)
Secondary Reference:
2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013)
Tertiary References:
3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989)
5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer)
Prerequisites / NoticePrerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
(also Bachelor) Core Courses: Applied Mathematics ...
Further restrictions apply, but in particular:
401-3601-00L Probability Theory can only be recognised for the Master Programme if neither 401-3642-00L Brownian Motion and Stochastic Calculus nor 401-3602-00L Applied Stochastic Processes has been recognised for the Bachelor Programme.
402-0205-00L Quantum Mechanics I is eligible as an applied core course, but only if 402-0224-00L Theoretical Physics (offered for the last time in FS 2016) isn't recognised for credits (neither in the Bachelor's nor in the Master's programme).
For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3601-00LProbability Theory
This course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from none of the three course units 401-3601-00L Probability Theory, 401-3642-00L Brownian Motion and Stochastic Calculus resp. 401-3602-00L Applied Stochastic Processes for their Bachelor's degree still can have recognised this course for the Master's degree.
Furthermore, at most one of the three course units
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics.
W10 credits4V + 1UA.‑S. Sznitman
AbstractBasics of probability theory and the theory of stochastic processes in discrete time
ObjectiveThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
ContentThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
Lecture notesavailable, will be sold in the course
LiteratureR. Durrett, Probability: Theory and examples, Duxbury Press 1996
H. Bauer, Probability Theory, de Gruyter 1996
J. Jacod and P. Protter, Probability essentials, Springer 2004
A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006
D. Williams, Probability with martingales, Cambridge University Press 1991
402-0205-00LQuantum Mechanics I Information W10 credits3V + 2UT. K. Gehrmann
AbstractIntroduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, wave functions and the description of observables as operators on a Hilbert space, and the formulation of symmetries will be discussed. Basic phenomena will be analysed and illustrated by generic examples.
ObjectiveIntroduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, perturbation theory) and generic examples and applications (bound states, tunneling, scattering states, in one- and three-dimensional settings). Ability to solve simple problems.
ContentKeywords: Schrödinger equation, basic formalism of quantum mechanics (states, operators, commutators, measuring process), symmetries (translations, rotations), quantum mechanics in one dimension, spherically symmetric problems in three dimensions, scattering theory, perturbation theory, variational techniques, spin, addition of angular momenta, relation between QM and classical physics.
LiteratureF. Schwabl: Quantum mechanics
J.J. Sakurai: Modern Quantum Mechanics
C. Cohen-Tannoudji: Quantum mechanics I
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.
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