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 Spring Semester 2017

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
401-3146-12LAlgebraic Geometry Information W10 credits4V + 1UR. Pink
AbstractThis course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
ObjectiveLearning Algebraic Geometry.
LiteraturePrimary reference:
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.

Secondary reference:
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.
* Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013).

Other good textbooks and online texts are:
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer.
* Ravi Vakil, Foundations of Algebraic Geometry, http://math.stanford.edu/~vakil/216blog/
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry http://www.cis.upenn.edu/~jean/algeom/steve01.html

"Classical" Algebraic Geometry over an algebraically closed field:
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.
* J.S. Milne, Algebraic Geometry, http://www.jmilne.org/math/CourseNotes/AG.pdf

Further readings:
* Günter Harder: Algebraic Geometry 1 & 2
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
Prerequisites / NoticeRequirement: Some knowledge of Commutative Algebra.
401-3002-12LAlgebraic Topology IIW8 credits4GP. S. Jossen
AbstractThis is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as: products, duality, cohomology operations, characteristic classes, spectral sequences etc.
Objective
Literature1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:
http://www.math.cornell.edu/~hatcher/AT/ATpage.html

See also:
http://www.math.cornell.edu/~hatcher/#anchor1772800

2) E. Spanier, "Algebraic topology", Springer-Verlag

3) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

4) R. Bott & L. Tu, "Differential forms in algebraic topology",
Graduate Texts in Mathematics, 82. Springer-Verlag, 1982.

5) J. Milnor & J. Stasheff, "Characteristic classes",
Annals of Mathematics Studies, No. 76.
Princeton University Press, 1974.
Prerequisites / NoticeGeneral topology, linear algebra.
Basic knowledge of singular homolgoy and cohomology of topological spaces (e.g. as taught in "Algebraic topology I").

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
401-3226-01LRepresentation Theory of Lie GroupsW8 credits4GE. Kowalski
AbstractThis course will contain two parts:
* Introduction to unitary representations of Lie groups
* Introduction to the study of discrete subgroups of Lie groups and some applications.
ObjectiveThe goal is to acquire familiarity with the basic formalism and results concerning unitary representations of Lie groups, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups.
Content* Unitary representations of compact Lie groups: Peter-Weyl theory, weights, Weyl character formula
* Introduction to unitary representations of non-compact Lie groups: the examples of SL(2,R), SL(2,C)
* Example: Property (T) for SL(n,R)
* Discrete subgroups of Lie groups: examples and some applications
LiteratureBekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press.
Prerequisites / NoticeDifferential geometry, Functional analysis, Introduction to Lie Groups (or equivalent).
Notice that this course has a large overlap with 401-3226-01L Unitary Representations of Lie Groups and Discrete Subgroups of Lie Groups taught in FS 2016. Therefore it is not possible to acquire credits for both courses.
401-3372-00LDynamical Systems IIW10 credits4V + 1UW. Merry
AbstractThis course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic dynamics.
ObjectiveMastery of the basic methods and principal themes of some aspects of hyperbolic dynamical systems.
ContentTopics covered include:

- Circle homeomorphisms and rotation numbers.
- Hyperbolic linear dynamical systems, hyperbolic fixed points, the Hartman-Grobman Theorem.
- Hyperbolic sets, Anosov diffeomorphisms.
- The (Un)stable Manifold Theorem.
- Shadowing Lemmas and stability.
- The Lambda Lemma.
- Transverse homoclinic points, horseshoes, and chaos.
Lecture notesI will provide full lecture notes, available here:

http://www.merry.io/dynamical-systems/
LiteratureThe most useful textbook is

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

Another (more advanced) useful book is

- Introduction to the Modern Theory of Dynamical Systems, Katok and Hasselblatt, CUP, 1995.
Prerequisites / NoticeIt will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here:

http://www.merry.io/dynamical-systems/

However we will only really use material covered in the first 12 lectures of Dynamical Systems I, so if you did not attend Dynamical Systems I, it is sufficient to read through the notes from the first 12 lectures.

In addition, it would be useful to have some familiarity with basic differential geometry.
401-3532-08LDifferential Geometry IIW10 credits4V + 1UU. Lang
AbstractIntroduction to Riemannian Geometry in combination with some elements of modern metric geometry. Contents: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, curvature and topology, spaces of riemannian manifolds.
ObjectiveThe aim of this course is to give an introduction to Riemannian Geometry in combination with some elements of modern metric geometry.
ContentRiemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form of submanifolds, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of riemannian manifolds.
LiteratureRiemannian Geometry:
- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- B. O'Neill, Semi-Riemannian Geometry, With Applications to Relativity, Academic Press 1983
Metric Geometry:
- M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999
- D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Amer. Math. Soc. 2001
Prerequisites / NoticePrerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, tangent and tensor bundles, and differential forms.
401-3462-00LFunctional Analysis IIW10 credits4V + 1UM. Struwe
AbstractSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity theory, Schauder estimates
ObjectiveThe lecture course will focus on weak solutions of elliptic boundary value problems in Sobolev spaces and discuss their regularity properties, possibly followed by a proof of the Calderon-Zygmund
inequality and some basic results on parabolic regularity, with
applications to geometry, if time allows.
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields
¬
NumberTitleTypeECTSHoursLecturers
401-3052-10LGraph Theory Information W10 credits4V + 1UB. Sudakov
AbstractBasics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem
ObjectiveThe students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.
Lecture notesLecture will be only at the blackboard.
LiteratureWest, D.: "Introduction to Graph Theory"
Diestel, R.: "Graph Theory"

Further literature links will be provided in the lecture.
401-3652-00LNumerical Methods for Hyperbolic Partial Differential Equations Information W10 credits4V + 1UU. S. Fjordholm
AbstractThis course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB.
ObjectiveThe goal of this course is familiarity with the fundamental ideas and mathematical
consideration underlying modern numerical methods for conservation laws and wave equations.
Content* Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering.

* Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes.

* Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes.

* Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods.

* Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes.

* Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory.
Lecture notesLecture slides will be made available to participants. However, additional material might be covered in the course.
LiteratureH. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online.

R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online.

E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.
Prerequisites / NoticeHaving attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite.

Programming exercises in MATLAB

Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations"
401-3642-00LBrownian Motion and Stochastic CalculusW10 credits4V + 1UM. Larsson
AbstractThis course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
ObjectiveThis course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
Lecture notesLecture notes will be distributed in class.
Literature- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).
- D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006).
Prerequisites / NoticeFamiliarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in
- J. Jacod, P. Protter, Probability Essentials, Springer (2004).
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).
401-3632-00LComputational Statistics Information W10 credits3V + 2UM. Mächler, P. L. Bühlmann
Abstract"Computational Statistics" deals with modern methods of data analysis (aka "data science") for prediction and inference. An overview of existing methodology is provided and also by the exercises, the student is taught to choose among possible models and about their algorithms and to validate them using graphical methods and simulation based approaches.
ObjectiveGetting to know modern methods of data analysis for prediction and inference.
Learn to choose among possible models and about their algorithms.
Validate them using graphical methods and simulation based approaches.
ContentCourse Synopsis:
multiple regression, nonparametric methods for regression and classification (kernel estimates, smoothing splines, regression and classification trees, additive models, projection pursuit, neural nets, ridging and the lasso, boosting). Problems of interpretation, reliable prediction and the curse of dimensionality are dealt with using resampling, bootstrap and cross validation.
Details are available via https://stat.ethz.ch/lectures/ .

Exercises will be based on the open-source statistics software R (http://www.R-project.org/). Emphasis will be put on applied problems. Active participation in the exercises is strongly recommended.
More details are available via the webpage https://stat.ethz.ch/lectures/ (-> "Computational Statistics").
Lecture noteslecture notes are available online; see
http://stat.ethz.ch/lectures/ (-> "Computational Statistics").
Literature(see the link above, and the lecture notes)
Prerequisites / NoticeBasic "applied" mathematical calculus (incl. simple two-dimensional) and linear algebra (including Eigenvalue decomposition) similar to two semester "Analysis" in an ETH (math or) engineer's bachelor.

At least one semester of (basic) probability and statistics, as e.g., taught in an ETH engineer's or math bachelor.

Programming experience in either a compiler-based computer language (such as C++) or a high-level language such as python, R, julia, or matlab. The language used in the exercises and the final exam will be R (https://www.r-project.org) exclusively. If you don't know it already, some extra effort will be required for the exercises.
401-3602-00LApplied Stochastic ProcessesW8 credits3V + 1UA.‑S. Sznitman
AbstractPoisson processes; renewal processes; Markov chains in discrete and in continuous time; some applications.
ObjectiveStochastic processes are a way to describe and study the behaviour of systems that evolve in some random way. In this course, the evolution will be with respect to a scalar parameter interpreted as time, so that we discuss the temporal evolution of the system. We present several classes of stochastic processes, analyse their properties and behaviour and show by some examples how they can be used. The main emphasis is on theory; in that sense, "applied" should be understood to mean "applicable".
LiteratureR. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: http://epubs.siam.org/doi/book/10.1137/1.9780898718997
R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: http://link.springer.com/book/10.1007/978-1-4614-3615-7/page/1
M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: http://link.springer.com/book/10.1007/978-0-387-48976-6/page/1
S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005)
Prerequisites / NoticePrerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).
401-3622-00LRegression
Does not take place this semester.
W8 credits4Gnot available
AbstractIn regression, the dependency of a random response variable on other variables is examined. We consider the theory of linear regression with one or more covariates, nonlinear models and generalized linear models, robust methods, model choice and nonparametric models. Several numerical examples will illustrate the theory.
ObjectiveEinführung in Theorie und Praxis eines umfassenden und vielbenutzten Teilgebiets der angewandten Statistik, unter Berücksichtigung neuerer Entwicklungen.
ContentIn der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression, Numerik, Ridge Regression. Durchrechnung und Diskussion von Anwendungsbeispielen.
Lecture notesLecture notes
Prerequisites / NoticeCredits cannot be recognised for both courses 401-3622-00L Regression and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden).
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
401-4142-17LAlgebraic CurvesW6 credits3GR. Pandharipande
AbstractI 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
ContentLecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-4142-17L/
LiteratureForster, "Lectures on Riemann Surfaces"

Arbarello, Cornalba, Griffiths, Harris, "Geometry of Algebraic Curves"

Mumford, "Curves and their Jacobians"
Prerequisites / NoticeFor 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.
401-3106-17LClass Field TheoryW6 credits2V + 1UJ. Fresán
AbstractClass 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.
401-3033-00LGödel's TheoremsW8 credits3V + 1UL. Halbeisen
AbstractDie 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
ObjectiveDas Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln.
ContentSyntax und Semantik der Prädikatenlogik
Gödel'scher Vollständigkeitssatz
Gödel'sche Unvollständigkeitssätze
LiteratureErgänzende Literatur wird in der Vorlesung angegeben.
401-3058-00LCombinatorics IW4 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.
Prerequisites / NoticeRecognition 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).
401-3112-17LIntroduction to Number TheoryW4 credits2VC. Busch
AbstractThis course gives an introduction to number theory. The focus will be on algebraic number theory.
Objective
ContentThe 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 / NoticeBasic knowledge of Algebra as taught in a course Algebra I + II.
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-4206-17LGroup Actions on TreesW4 credits2VN. Lazarovich
AbstractAs 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.
ObjectiveIntroduction to the general theory of group actions on trees, also known as Bass-Serre theory, and various important results on decompositions of groups.
ContentDepending 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.
LiteratureJ.-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 / NoticeFamiliarity with the basics of fundamental group (and covering theory).
401-4148-17LModuli of Maps and Gromov-Witten invariantsW2 credits4AG. Bérczi
AbstractEnumerative 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.
ObjectiveThe 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.
ContentTopics 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
LiteratureThe 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 / NoticeSome 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.
401-3056-00LFinite Geometries I
Does not take place this semester.
W4 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.
401-3574-61LIntroduction to Knot Theory Information
Does not take place this semester.
W6 credits3G
AbstractIntroduction 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.
ObjectiveThe 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.
ContentDefinition 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.....)
LiteratureAn extensive bibliography will be handed out in the course.
Prerequisites / NoticePrerequisites are some elementary knowledge of algebra and topology.
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-4832-17LMathematical Themes in General Relativity IIW4 credits2VA. Carlotto
AbstractSecond 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.
ContentAnalysis 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 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.
**This course is the sequel of the one offered during the first semester.**
401-3352-09LAn Introduction to Partial Differential EquationsW6 credits3VF. Da Lio
AbstractThis 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
ContentA 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 notesThe teacher provides the students with personal notes.
LiteratureBibliography
- 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 / NoticeDifferential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory.
401-3496-17LTopics in the Calculus of VariationsW4 credits2VA. Figalli
Abstract
Objective
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-3502-17LReading 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
401-3503-17LReading 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
401-3504-17LReading 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
Electives: Applied Mathematics and Further Application-Oriented Fields
¬
Selection: Numerical Analysis
NumberTitleTypeECTSHoursLecturers
401-4606-00LNumerical Analysis of Stochastic Partial Differential Equations
Does not take place this semester.
W8 credits4Gnot available
AbstractIn this course solutions of semilinear stochastic partial differential equations (SPDEs) of the evolutionary type and some of their numerical approximation methods are investigated. Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences.
ObjectiveThe aim of this course is to teach the students a decent knowledge on solutions of semilinear stochastic partial differential equations (SPDEs), on some numerical approximation methods for such equations and on the functional analytic and probabilistic concepts used to formulate and study such equations.
ContentThe course includes content (i) on the (functional) analytic concepts used to study semilinear stochastic partial differential equations (SPDEs) (e.g., nuclear operators, Hilbert-Schmidt operators, diagonal linear operators on Hilbert spaces, interpolation spaces associated to a diagonal linear operator, semigroups of bounded linear operators, Gronwall-type inequalities), (ii) on the probabilistic concepts used to study SPDEs (e.g., Hilbert space valued random variables, Hilbert space valued stochastic processes, infinite dimensional Wiener processes, stochastic integration with respect to infinite dimensional Wiener processes, infinite dimensional jump processes), (iii) on solutions of SPDEs (e.g., existence, uniqueness and regularity properties of mild solutions of SPDEs, applications involving SPDEs), and (iv) on numerical approximations of SPDEs (e.g., spatial and temporal discretizations, strong convergence, weak convergence). Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. They appear, for example, in models from neurobiology for the approximative description of the propagation of electrical impulses along nerve cells, in models from financial engineering for the approximative pricing of financial derivatives, in models from fluid mechanics for the approximative description of velocity fields in fully developed turbulent flows, in models from quantum field theory for describing the temporal dynamics associated to Euclidean quantum field theories, and in models from chemistry for the approximative description of the temporal evolution of the concentration of an undesired chemical contaminant in the groundwater system.
Lecture notesLecture notes will be available as a PDF file.
Literature1. Stochastic Equations in Infinite Dimensions
G. Da Prato and J. Zabczyk
Cambridge Univ. Press (1992)

2. Taylor Approximations for Stochastic Partial Differential Equations
A. Jentzen and P.E. Kloeden
Siam (2011)

3. Numerical Solution of Stochastic Differential Equations
P.E. Kloeden and E. Platen
Springer Verlag (1992)

4. A Concise Course on Stochastic Partial Differential Equations
C. Prévôt and M. Röckner
Springer Verlag (2007)

5. Galerkin Finite Element Methods for Parabolic Problems
V. Thomée
Springer Verlag (2006)
Prerequisites / NoticeMandatory prerequisites: Functional analysis, probability theory;
Recommended prerequisites: stochastic processes;
401-4658-00LComputational Methods for Quantitative Finance: PDE Methods Information Restricted registration - show details W6 credits3V + 1UC. Schwab
AbstractIntroduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming
and knowledge of numerical mathematics at ETH BSc level.
ObjectiveIntroduce the main methods for efficient numerical valuation of derivative contracts in a
Black Scholes as well as in incomplete markets due Levy processes or due to stochastic volatility
models. Develop implementation of pricing methods in MATLAB.
Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.
Content1. Review of option pricing. Wiener and Levy price process models. Deterministic, local and stochastic
volatility models.
2. Finite Difference Methods for option pricing. Relation to bi- and multinomial trees.
European contracts.
3. Finite Difference methods for Asian, American and Barrier type contracts.
4. Finite element methods for European and American style contracts.
5. Pricing under local and stochastic volatility in Black-Scholes Markets.
6. Finite Element Methods for option pricing under Levy processes. Treatment of
integrodifferential operators.
7. Stochastic volatility models for Levy processes.
8. Techniques for multidimensional problems. Baskets in a Black-Scholes setting and
stochastic volatility models in Black Scholes and Levy markets.
9. Introduction to sparse grid option pricing techniques.
Lecture notesThere will be english, typed lecture notes as well as MATLAB software for registered participants in the course.
LiteratureR. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004.

Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005.

D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008.

J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000.

N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013.
Prerequisites / NoticeStart of the lecture: Wednesday, March 1, 2017 (second week of the semester).
401-4788-16LMathematics of (Super-Resolution) Biomedical ImagingW8 credits4GH. Ammari
AbstractThe aim of this course is to review different methods used to address challenging problems in biomedical imaging. The emphasis will be on scale separation techniques, hybrid imaging, spectroscopic techniques, and nanoparticle imaging. These approaches allow one to overcome the ill-posedness character of imaging reconstruction in biomedical applications and to achieve super-resolution imaging.
ObjectiveSuper-resolution imaging is a collective name for a number of emerging techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other.

In this course we describe recent advances in scale separation techniques, spectroscopic approaches, multi-wave imaging, and nanoparticle imaging. The objective is fivefold:
(i) To provide asymptotic expansions for both internal and boundary perturbations that are due to the presence
of small anomalies;
(ii) To apply those asymptotic formulas for the purpose of identifying the material parameters and certain geometric features of the anomalies;
(iii) To design efficient inversion algorithms in multi-wave modalities;
(iv) to develop inversion techniques using multi-frequency measurements;
(v) to develop a mathematical and numerical framework for nanoparticle imaging.

In this course we shall consider both analytical and computational
matters in biomedical imaging. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, inverse problems, mathematical imaging, optimal control, stochastic modelling, and analysis of physical phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in biomedical imaging, requires a deep understanding of the different scales in the physical models, an accurate mathematical modelling of the imaging techniques, and fine analysis of complex physical phenomena.

An emphasis is put on mathematically analyzing acoustic-electric imaging, thermo-elastic imaging, Lorentz force based imaging, elastography, multifrequency electrical impedance tomography, and plasmonic resonant nanoparticles.
252-0504-00LNumerical Methods for Solving Large Scale Eigenvalue Problems Information
Does not take place this semester.
W4 credits3GP. Arbenz
AbstractIn this lecture algorithms are investigated for solving eigenvalue problems
with large sparse matrices. Some of these eigensolvers have been developed
only in the last few years. They will be analyzed in theory and practice (by means
of MATLAB exercises).
ObjectiveKnowing the modern algorithms for solving large scale eigenvalue problems, their numerical behavior, their strengths and weaknesses.
ContentThe lecture starts with providing examples for applications in which
eigenvalue problems play an important role. After an introduction
into the linear algebra of eigenvalue problems, an overview of
methods (such as the classical QR algorithm) for solving small to
medium-sized eigenvalue problems is given.

Afterwards, the most important algorithms for solving large scale,
typically sparse matrix eigenvalue problems are introduced and
analyzed. The lecture will cover a choice of the following topics:

* vector and subspace iteration
* trace minimization algorithm
* Arnoldi and Lanczos algorithms (including restarting variants)
* Davidson and Jacobi-Davidson Algorithm
* preconditioned inverse iteration and LOBPCG
* methods for nonlinear eigenvalue problems

In the exercises, these algorithm will be implemented (in simplified forms)
and analysed in MATLAB.
Lecture notesLecture notes,
Copies of slides
LiteratureZ. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000.

Y. Saad: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, 1994.

G. H. Golub and Ch. van Loan: Matrix Computations, 3rd ed. Johns Hopkins University Press, Baltimore 1996.
Prerequisites / NoticePrerequisite: linear agebra
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-3919-60LAn Introduction to the Modelling of ExtremesW4 credits2VP. Embrechts
AbstractThis course yields an introduction into the MATHEMATICAL THEORY of one-dimensional extremes, and this mainly from a more probabilistic point of view.
ObjectiveIn this course, students learn to distinguish between so-called normal models, i.e. models based on the normal or Gaussian distribution, and so-called heavy-tailed or power-tail models.
They learn to do probabilistic modelling of extremes in one-dimensional data. The probabilistic key theorems are the Fisher-Tippett Theorem and the Balkema-de Haan-Pickands Theorem. These lead to the statistical techniques for the analysis of extremes or rare events known as the Block Method, and Peaks Over Threshold method, respectively.
Content- Introduction to rare or extreme events
- Regular Variation
- The Convergence to Types Theorem
- The Fisher-Tippett Theorem
- The Method of Block Maxima
- The Maximal Domain of Attraction
- The Fre'chet, Gumbel and Weibull distributions
- The POT method
- The Point Process Method: a first introduction
- The Pickands-Balkema-de Haan Theorem and its applications
- Some extensions and outlook
Lecture notesThere will be no script available, students are required to take notes from the blackboard lectures. The course follows closely Extreme Value Theory as developed in:
P. Embrechts, C. Klueppelberg and T. Mikosch (1997)
Modelling Extremal Events for Insurance and Finance.
Springer.
LiteratureThe main text on which the course is based is:
P. Embrechts, C. Klueppelberg and T. Mikosch (1997)
Modelling Extremal Events for Insurance and Finance.
Springer.
Further relevant literature is:
S. I. Resnick (2007) Heavy-Tail Phenomena. Probabilistic and
Statistical Modeling. Springer.
S. I. Resnick (1987) Extreme Values, Regular Variation,
and Point Processes. Springer.
401-4632-15LCausalityW4 credits2GM. H. Maathuis
AbstractIn statistics, we are used to search for the best predictors of some random variable. In many situations, however, we are interested in predicting a system's behavior under manipulations. For such an analysis, we require knowledge about the underlying causal structure of the system. In this course, we study concepts and theory behind causal inference.
ObjectiveAfter this course, you should be able to
- understand the language and concepts of causal inference
- know the assumptions under which one can infer causal relations from observational and/or interventional data
- describe and apply different methods for causal structure learning
- given data and a causal structure, derive causal effects and predictions of interventional experiments
Prerequisites / NoticePrerequisites: basic knowledge of probability theory and regression
401-6102-00LMultivariate StatisticsW4 credits2GN. Meinshausen
AbstractMultivariate Statistics deals with joint distributions of several random variables. This course introduces the basic concepts and provides an overview over classical and modern methods of multivariate statistics. We will consider the theory behind the methods as well as their applications.
ObjectiveAfter the course, you should be able to:
- describe the various methods and the concepts and theory behind them
- identify adequate methods for a given statistical problem
- use the statistical software "R" to efficiently apply these methods
- interpret the output of these methods
ContentVisualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis
Lecture notesNone
LiteratureThe course will be based on class notes and books that are available electronically via the ETH library.
Prerequisites / NoticeTarget audience: This course is the more theoretical version of "Applied Multivariate Statistics" (401-0102-00L) and is targeted at students with a math background.

Prerequisite: A basic course in probability and statistics.

Note: The courses 401-0102-00L and 401-6102-00L are mutually exclusive. You may register for at most one of these two course units.
401-3822-17LPercolation and Ising ModelW4 credits2VV. Tassion
AbstractIn this course we will provide a general introduction to the Ising model on the hypercubic lattice Z^d (main properties, standard mathematical tools).
In order to answer important questions regarding the Ising model, we will exploit a deep connection of the model with two (dependent) percolation processes, the Fortuin-Kastelen percolation and the random-current model.
Objective- Discover important models of statistical mechanics: the Ising model (and its random current representation) and FK percolation.
- Learn some important techniques in statistical mechanics (e.g. coupling methods, monotonicity properties, the use of differential inequalities, to name few).
Prerequisites / Notice- Probability Theory.
- The course "Recent Development in Percolation Theory" (Autumn 2017) of Pierre Nolin is advised but not necessary (the overlap with Nolin's course will be minimal).
401-3597-64LConcentration of MeasureW4 credits2VJ. Aru, T. Lupu
Abstract
Objective
ContentThe basic examples of the concentration of measure phenomena are the following:

1) The visual distance of a N-dimensional unit sphere is only of order N^{-0.5}. In other words, more than 99% of the measure on the sphere lies at distance of at most O(N^{-0.5}) of a fixed hyperplane through the origin.

2) The suprema of a centred Gaussian process G(t) even with a possibility infinite index set T is always concentrated around its expected value with a Gaussian tail that only depends on the highest variance among the Gaussians G(t).

In this course we will try to understand these two slightly puzzling examples and related phenomena. We try to approach and understand the concentration of measure phenomena from different directions: through elementary martingale inequalities like Azuma-Hoeffding or McDiarmid inequality; through exact isoperimetry; through Poincaré and log-Sobolev inequalities.

On the way we aim to discuss several applications and connections to different topics, including Dvoretzky's theorem, convergence of Markov chains to their stationary measure, entropy, threshold phenomena, empirical processes etc...
401-3616-17LAn Introduction to Stochastic Partial Differential EquationsW8 credits4GA. Jentzen
AbstractIn this course solutions of semilinear stochastic partial differential equations (SPDEs) of the evolutionary type are investigated. Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences.
ObjectiveThe aim of this course is to teach the students a decent knowledge on solutions of semilinear stochastic partial differential equations (SPDEs) and on the functional analytic and probabilistic concepts used to formulate and study such equations.
ContentThe course includes content (i) on the (functional) analytic concepts used to study semilinear stochastic partial differential equations (SPDEs), (ii) on the probabilistic concepts used to study SPDEs, and (iii) on solutions of SPDEs (e.g., existence, uniqueness and regularity properties of mild solutions of SPDEs, applications involving SPDEs). Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. They appear, for example, in models from neurobiology for the approximative description of the propagation of electrical impulses along nerve cells, in models from financial engineering for the approximative pricing of financial derivatives, in models from fluid mechanics for the approximative description of velocity fields in fully developed turbulent flows, in models from quantum field theory for describing the temporal dynamics associated to Euclidean quantum field theories, and in models from chemistry for the approximative description of the temporal evolution of the concentration of an undesired chemical contaminant in the groundwater system.
Lecture notesThe current version of the lecture notes is available as a PDF file here:
https://polybox.ethz.ch/index.php/s/bI884u6tz9mO9Vz/download
Literature1. Stochastic Equations in Infinite Dimensions
G. Da Prato and J. Zabczyk
Cambridge Univ. Press (1992)

2. Taylor Approximations for Stochastic Partial Differential Equations
A. Jentzen and P.E. Kloeden
Siam (2011)

3. Numerical Solution of Stochastic Differential Equations
P.E. Kloeden and E. Platen
Springer Verlag (1992)

4. A Concise Course on Stochastic Partial Differential Equations
C. Prévôt and M. Röckner
Springer Verlag (2007)

5. Galerkin Finite Element Methods for Parabolic Problems
V. Thomée
Springer Verlag (2006)
Prerequisites / NoticeMandatory prerequisites: Functional analysis, probability theory;
Recommended prerequisites: stochastic processes;
Selection: Financial and Insurance Mathematics
NumberTitleTypeECTSHoursLecturers
401-3629-00LQuantitative Risk Management Information W4 credits2VP. Cheridito
AbstractThis course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, multivariate models, copulas and dependence structures, extreme value theory, risk measures, aggregation of risk, and risk allocation.
ObjectiveThe goal is to learn the most important methods from probability theory and statistics used to model financial risks.
Content1. Risk in Perspective
2. Basic Concepts
3. Multivariate Models
4. Copulas and Dependence
5. Aggregate Risk
6. Extreme Value Theory
7. Operational Risk and Insurance Analytics
Lecture notesCourse material is available on https://people.math.ethz.ch/~patrickc/qrm
LiteratureQuantitative Risk Management: Concepts, Techniques and Tools
AJ McNeil, R Frey and P Embrechts
Princeton University Press, Princeton, 2015 (Revised Edition)
http://press.princeton.edu/titles/10496.html
Prerequisites / NoticeThe course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.
401-4938-14LStochastic Optimal Control Information W4 credits2VM. Soner
AbstractDynamic programming approach to stochastic optimal control problems will be developed. In addition to the general theory, detailed analysis of several important control problems will be given.
ObjectiveGoals are to achieve a deep understanding of

1. Dynamic programming approach to optimal control;
2. Several classes of important optimal control problems and their solutions.
3. To be able to use this models in engineering and economic modeling.
ContentIn this course, we develop the dynamic programming approach for the stochastic optimal control problems. The general approach will be described and several subclasses of problems will also be discussed in including:
1. Standard exit time problems;
2. Finite and infinite horizon problems;
3. Optimal stoping problems;
4. Singular problems;
5. Impulse control problems.

After the general theory is developed, it will be applied to several classical problems including:
1. Linear quadratic regulator;
2. Merton problem for optimal investment and consumption;
3. Optimal dividend problem of (Jeanblanc and Shiryayev);
4. Finite fuel problem;
5. Utility maximization with transaction costs;
6. A deterministic differential game related to geometric flows.

Textbook will be

Controlled Markov Processes and Viscosity Solutions, 2nd edition, (W.H. Fleming and H.M. Soner) Springer-Verlag, (2005).

And lecture notes will be provided.
LiteratureControlled Markov Processes and Viscosity Solutions, 2nd edition, (W.H. Fleming and H.M. Soner) Springer-Verlag, (2005).

And lecture notes will be provided.
Prerequisites / NoticeBasic knowledge of Brownian motion, stochastic differential equations and probability theory is needed.
401-3923-00LSelected Topics in Life Insurance MathematicsW4 credits2VM. Koller
AbstractStochastic Models for Life insurance
1) Markov chains
2) Stochastic Processes for demography and interest rates
3) Cash flow streams and reserves
4) Mathematical Reserves and Thiele's differential equation
5) Theorem of Hattendorff
6) Unit linked policies
Objective
401-3917-00LStochastic Loss Reserving MethodsW4 credits2VR. Dahms
AbstractLoss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to calculate those loss reserves.
ObjectiveOur goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.
ContentWe will present the following stochastic claims reserving methods/models:
- Stochastic Chain-Ladder Method
- Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods
- Distributional Models
- Linear Stochastic Reserving Models, with and without inflation
- Bootstrap Methods
- Claims Development Result (solvency view)
- Coupling of portfolios
LiteratureM. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination periods.

This course will be held in English and counts towards the diploma "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Basic knowledge in probability theory is assumed, in particular conditional expectations.
401-3956-00LEconomic Theory of Financial MarketsW4 credits2VM. V. Wüthrich
AbstractThis lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
ObjectiveThis lecture aims at providing the fundamental financial and economic concepts to insurance mathematicians and actuaries. It focuses on portfolio theory, cash flow valuation and deflator techniques.
ContentWe treat the following topics:
- Fundamental concepts in economics
- Portfolio theory
- Mean variance analysis, capital asset pricing model
- Arbitrage pricing theory
- Cash flow theory
- Valuation principles
- Stochastic discounting, deflator techniques
- Interest rate modeling
- Utility theory
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-4920-00LMarket-Consistent Actuarial Valuation
Does not take place this semester.
W4 credits2VM. V. Wüthrich
AbstractIntroduction to market-consistent actuarial valuation.
Topics: Stochastic discounting, full balance sheet approach, valuation portfolio in life and non-life insurance, technical and financial risks, risk management for insurance companies.
ObjectiveGoal is to give the basic mathematical tools for describing insurance products within a financial market and economic environment and provide the basics of solvency considerations.
ContentIn this lecture we give a full balance sheet approach to the task of actuarial valuation of an insurance company. Therefore we introduce a multidimensional valuation portfolio (VaPo) on the liability side of the balance sheet. The basis of this multidimensional VaPo is a set of financial instruments. This approach makes the liability side of the balance sheet directly comparable to its asset side.

The lecture is based on four sections:
1) Stochastic discounting
2) Construction of a multidimensional Valuation Portfolio for life insurance products (with guarantees)
3) Construction of a multidimensional Valuation Portfolio for a run-off portfolio of a non-life insurance company
4) Measuring financial risks in a full balance sheet approach (ALM risks)
LiteratureMarket-Consistent Actuarial Valuation, 2nd edition.
Wüthrich, M.V., Bühlmann, H., Furrer, H.
EAA Series Textbook, Springer, 2010.
ISBN: 978-3-642-14851-4

Wüthrich, M.V., Merz, M.
Claims Run-Off Uncertainty: The Full Picture
SSRN Manuscript ID 2524352 (2015).

Wüthrich, M.V., Embrechts, P., Tsanakas, A.
Risk margin for a non-life insurance run-off.
Statistics & Risk Modeling 28 (2011), no. 4, 299--317.

Financial Modeling, Actuarial Valuation and Solvency in Insurance.
Wüthrich, M.V., Merz, M.
Springer Finance 2013.
ISBN: 978-3-642-31391-2
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

This course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3888-00LIntroduction to Mathematical Finance Information
A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.
W10 credits4V + 1UJ. Teichmann
AbstractThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.
ObjectiveThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation, and maybe other topics. We prove the fundamental theorem of asset pricing and the hedging duality theorems in discrete time, and also study convex duality in utility maximization.
ContentThis course focuses on discrete-time financial markets and presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course will be offered every year in the Spring semester. The textbook by Föllmer and Schied or lecture notes similar to that will be used.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MFII), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MFI, is taken prior to MFII.
Lecture notesThe textbook by Föllmer and Schied or lecture notes similar to that will be used. However, actual lecture notes will not be available.
LiteratureRecommended textbook:

Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter
Prerequisites / NoticeA related course is "Mathematical Foundations for Finance" (MFF), 401-3913-01. Although both courses can be taken independently of each other, only one will be given credit points for the Bachelor and the Master degree. In other words, it is also not possible to earn credit points with one for the Bachelor and with the other for the Master degree.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MFII), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MFI, is taken prior to MFII.
401-3928-00LReinsurance Analytics
Does not take place this semester.
W4 credits2V
AbstractHistory and motivation. Basic Risk Theory applied to reinsurance. The reinsurance market and lines of business. Pricing reinsurance contracts. Solvency and capital considerations. Alternative Risk Transfer.
ObjectiveUnderstanding the economic value creation through reinsurance. Knowing the most common types of reinsurance and being able to represent the reinsured losses in terms of random variables. Understanding the economic and mathematic principles underlying the premium calculations for reinsurance contracts.
ContentHistory of reinsurance. Historic examples of large events. Fundamentals of reinsurance & contract types. Overview of large reinsurance companies & market places. Lines of business explained: Property, Casualty, Life & Health, Credit & Surety. Risk theoretical principles including frequency/severity models, stop-loss transforms and exposure curves. Exposure and experience rating. Capital impact of reinsurance: Swiss Solvency Test and Solvency 2, rating agency view, insurance vs. investment risks. Insurance Linked Securities: cat bonds, industry loss warranties.
Lecture notesA script will be made available in electronic form.
Prerequisites / NoticeFormer course title: "Insurance Analytics"
Selection: Mathematical Physics, Theoretical Physics
NumberTitleTypeECTSHoursLecturers
401-3814-00LQuantum Physics for MathematiciansW6 credits3GG. Felder
AbstractIntroduction to quantum mechanics aimed at mathematics students
ObjectiveBasic notions of quantum mechanics: states, observables, semiclassical limit, Schrödinger and Heisenberg picture, examples, angular momentum and spin, Pauli equation, perturbation theory, introduction to path integrals.
Prerequisites / NoticePrerequisite: Allgemeine Mechanik / Classical Mechanics
402-0206-00LQuantum Mechanics II Information W10 credits3V + 2UT. K. Gehrmann
AbstractIntroduction to many-particle quantum mechanics and quantum statistics. Basic concepts: symmetrised many-body wave functions for fermions and bosons, the Pauli principle, Bose- and Fermi-statistic and second quantisation. Applications include the description of atoms, and the interaction between radiation and matter.
ObjectiveIntroduction to many-particle quantum mechanics and quantum statistics. In particular basic concepts such as symmetrised many-body wave functions for fermions and bosons, the Pauli principle, Bose- and Fermi-statistics and second quantisation will be discussed. Applications include
the description of atoms, and the interaction between radiation and matter.
ContentThe description of identical particles leads us to the
introduction of symmetrised wave functions for fermions
and bosons. We discuss simple few-body problems and
proceed with a systematic description of fermionic many
body problems in terms of second quantisation.
We also discuss basic concepts of quantum statistics.
Applications include the description of
atoms, and the interaction between radiation and matter.
LiteratureF. Schwabl, Quantenmechanik (Springer)
F. Schwabl, Quantenmechanik fuer Fortgeschrittene (Springer)
J.J. Sakurai, Advanced Quantum mechanics (Addison Wesley)
402-0844-00LQuantum Field Theory IIW10 credits3V + 2UN. Beisert
AbstractThe subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.
Objective
ContentThe following topics will be covered:
- path integral quantization
- non-abelian gauge theories and their quantization
- systematics of renormalization, including BRST symmetries,
Slavnov-Taylor Identities and the Callan Symanzik equation
- gauge theories with spontaneous symmetry breaking and
their quantization
- renormalization of spontaneously broken gauge theories and
quantum effective actions
LiteratureM.E. Peskin and D.V. Schroeder,
An introduction to Quantum Field Theory, Perseus (1995).
L.H. Ryder,
Quantum Field Theory, CUP (1996).
S. Weinberg,
The Quantum Theory of Fields (Volume 2), CUP (1996).
M. Srednicki,
Quantum Field Theory, CUP (2006).
Selection: Mathematical Optimization
NumberTitleTypeECTSHoursLecturers
401-3903-11LGeometric Integer Programming
Does not take place this semester.
W6 credits2V + 1Unot available
AbstractInteger programming is the task of minimizing a linear function over all the integer points in a polyhedron. This lecture introduces the key concepts of an algorithmic theory for solving such problems.
ObjectiveThe purpose of the lecture is to provide a geometric treatment of the theory of integer optimization.
ContentKey topics are:
- lattice theory and the polynomial time solvability of integer optimization problems in fixed dimension,
- the theory of integral generating sets and its connection to totally dual integral systems,
- finite cutting plane algorithms based on lattices and integral generating sets.
Lecture notesnot available, blackboard presentation
LiteratureBertsimas, Weismantel: Optimization over Integers, Dynamic Ideas 2005.
Schrijver: Theory of linear and integer programming, Wiley, 1986.
Prerequisites / Notice"Mathematical Optimization" (401-3901-00L)
401-4904-00LCombinatorial Optimization Information W6 credits2V + 1UR. Zenklusen
AbstractCombinatorial Optimization deals with efficiently finding a provably strong solution among a finite set of options. This course discusses key combinatorial structures and techniques to design efficient algorithms for combinatorial optimization problems. We put a strong emphasis on polyhedral methods, which proved to be a powerful and unifying tool throughout combinatorial optimization.
ObjectiveThe goal of this lecture is to get a thorough understanding of various modern combinatorial optimization techniques with an emphasis on polyhedral approaches. Students will learn a general toolbox to tackle a wide range of combinatorial optimization problems.
ContentKey topics include:
- Polyhedral descriptions;
- Combinatorial uncrossing;
- Ellipsoid method;
- Equivalence between separation and optimization;
- Design of efficient approximation algorithms for hard problems.
Lecture notesNot available.
Literature- Bernhard Korte, Jens Vygen: Combinatorial Optimization. 5th edition, Springer, 2012.
- Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency, Springer, 2003. This work has 3 volumes.
Prerequisites / NoticeWe recommend that students interested in Combinatorial Optimization first attend the course "Mathematical Optimization" (401-3901-00L).
Auswahl: Theoretical Computer Science, Discrete Mathematics
In the Master's programme in Mathematics 401-3052-05L Graph Theory is eligible as an elective course, but only if 401-3052-10L Graph Theory 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
252-0407-00LCryptography Foundations Information
Does not take place this semester.
W7 credits3V + 2U + 1AU. Maurer
AbstractFundamentals and applications of cryptography. Cryptography as a mathematical discipline: reductions, constructive cryptography paradigm, security proofs. The discussed primitives include cryptographic functions, pseudo-randomness, symmetric encryption and authentication, public-key encryption, key agreement, and digital signature schemes. Selected cryptanalytic techniques.
ObjectiveThe goals are:
(1) understand the basic theoretical concepts and scientific thinking in cryptography;
(2) understand and apply some core cryptographic techniques and security proof methods;
(3) be prepared and motivated to access the scientific literature and attend specialized courses in cryptography.
ContentSee course description.
Lecture notesyes.
Prerequisites / NoticeFamiliarity with the basic cryptographic concepts as treated for
example in the course "Information Security" is required but can
in principle also be acquired in parallel to attending the course.
252-0408-00LCryptographic Protocols Information W5 credits2V + 2UM. Hirt
AbstractThe course presents a selection of hot research topics in cryptography. The choice of topics varies and may include provable security, interactive proofs, zero-knowledge protocols, secret sharing, secure multi-party computation, e-voting, etc.
ObjectiveIndroduction to a very active research area with many gems and paradoxical
results. Spark interest in fundamental problems.
ContentThe course presents a selection of hot research topics in cryptography. The choice of topics varies and may include provable security, interactive proofs, zero-knowledge protocols, secret sharing, secure multi-party computation, e-voting, etc.
Lecture notesthe lecture notes are in German, but they are not required as the entire
course material is documented also in other course material (in english).
Prerequisites / NoticeA basic understanding of fundamental cryptographic concepts
(as taught for example in the course Information Security or
in the course Cryptography) is useful, but not required.
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
227-0434-00LHarmonic Analysis: Theory and Applications in Advanced Signal Processing Information W6 credits2V + 2UH. Bölcskei, E. Riegler
AbstractThis course is an introduction to the field of applied harmonic analysis with emphasis on applications in signal processing such as transform coding, inverse problems, imaging, signal recovery, and inpainting. We will consider theoretical, applied, and algorithmic aspects.
ObjectiveThis course is an introduction to the field of applied harmonic analysis with emphasis on applications in signal processing such as transform coding, inverse problems, imaging, signal recovery, and inpainting. We will consider theoretical, applied, and algorithmic aspects.
ContentFrame theory: Frames in finite-dimensional spaces, frames for Hilbert spaces, sampling theorems as frame expansions

Spectrum-blind sampling: Sampling of multi-band signals with known support set, density results by Beurling and Landau, unknown support sets, multi-coset sampling, the modulated wideband converter, reconstruction algorithms

Sparse signals and compressed sensing: Uncertainty principles, recovery of sparse signals with unknown support set, recovery of sparsely corrupted signals, orthogonal matching pursuit, basis pursuit, the multiple measurement vector problem

High-dimensional data and dimension reduction: Random projections, the Johnson-Lindenstrauss Lemma, the Restricted Isometry Property, concentration inequalities, covering numbers, Kashin widths
Lecture notesLecture notes, problem sets with documented solutions.
LiteratureS. Mallat, ''A wavelet tour of signal processing: The sparse way'', 3rd ed., Elsevier, 2009

I. Daubechies, ''Ten lectures on wavelets'', SIAM, 1992

O. Christensen, ''An introduction to frames and Riesz bases'', Birkhäuser, 2003

K. Gröchenig, ''Foundations of time-frequency analysis'', Springer, 2001

M. Elad, ''Sparse and redundant representations -- From theory to applications in signal and image processing'', Springer, 2010
Prerequisites / NoticeThe course is heavy on linear algebra, operator theory, and functional analysis. A solid background in these areas is beneficial. We will, however, try to bring everybody on the same page in terms of the mathematical background required, mostly through reviews of the mathematical basics in the discussion sessions. Moreover, the lecture notes contain detailed material on the advanced mathematical concepts used in the course. If you are unsure about the prerequisites, please contact C. Aubel or H. Bölcskei.
401-3502-17LReading 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
401-3503-17LReading 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
401-3504-17LReading 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
Application Area
Only necessary and eligible for the Master degree in Applied Mathematics.
One of the application areas specified must be selected for the category Application Area for the Master degree in Applied Mathematics. At least 8 credits are required in the chosen application area.
Atmospherical Physics
NumberTitleTypeECTSHoursLecturers
701-1216-00LNumerical Modelling of Weather and Climate Information W4 credits3GU. Lohmann, L. Schlemmer
AbstractThe guiding principle of this lecture is that students can understand how weather and climate models are formulated from the governing physical principles and how they are used for climate and weather prediction purposes.
ObjectiveThe guiding principle of this lecture is that students can understand how weather and climate models are formulated from the governing physical principles and how they are used for climate and weather prediction purposes.
ContentThe course provides an introduction into the following themes: numerical methods (finite differences and spectral methods); adiabatic formulation of atmospheric models (vertical coordinates, hydrostatic approximation); parameterization of physical processes (e.g. clouds, convection, boundary layer, radiation); atmospheric data assimilation and weather prediction; predictability (chaos-theory, ensemble methods); climate models (coupled atmospheric, oceanic and biogeochemical models); climate prediction.

Hands-on experience with simple models will be acquired in the tutorials.
Lecture notesSlides and lecture notes will be made available at
Link
LiteratureList of literature will be provided.
Prerequisites / NoticePrerequisites: to follow this course, you need some basic background in atmospheric science, numerical methods (e.g., "Numerische Methoden in der Umweltphysik", 701-0461-00L) as well as experience in programming
Biology
NumberTitleTypeECTSHoursLecturers
551-0016-00LBiology II Information W2 credits2VM. Stoffel, E. Hafen, K. Köhler
AbstractThe lecture course Biology II, together with the course Biology I of the previous winter semester, is a basic introductory course into biology for students of materials sciences, of chemistry and of chemical engineering.
ObjectiveThe objective of the lecture course Biology II is the understanding of form, function, and development of animals and of the basic underlying mechanisms.
ContentThe following numbers of chapters refer to the text-book "Biology" (Campbell & Rees, 10th edition, 2015) on which the course is based.

Chapters 1-4 are a basic prerequisite. The sections "Structure of the Cell" (Chapters 5-10, 12, 17) and "General Genetics" (Chapters 13-16, 18, 46) are covered by the lecture Biology I.

1. Genomes, DNA Technology, Genetic Basis of Development

Chapter 19: Eukaryotic Genomes: Organization, Regulation, and Evolution
Chapter 20: DNA Technology and Genomics
Chapter 21: The Genetic Basis of Development

2. Form, Function, and Development of Animals I

Chapter 40: Basic Principles of Animal Form and Function
Chapter 41: Animal Nutrition
Chapter 44: Osmoregulation and Excretion
Chapter 47: Animal Development

3. Form, Function, and Develeopment of Animals II

Chapter 42: Circulation and Gas Exchange
Chapter 43: The Immune System
Chapter 45: Hormones and the Endocrine System
Chapter 48: Nervous Systems
Chapter 49: Sensory and Motor Mechanisms
Lecture notesThe course follows closely the recommended text-book. Additional handouts may be provided by the lecturers.
LiteratureThe following text-book is the basis for the courses Biology I and II:

Biology, Campbell and Rees, 10th Edition, 2015, Pearson/Benjamin Cummings, ISBN 978-3-8632-6725-4
Prerequisites / NoticePrerequisite: Lecture course Biology I of autumn semester
Computational Electromagnetics
"Computational Electromagnetics" is no longer offered as an application area.
Participants of this former application area who need more credits can register for the course unit 227-0707-00L Optimization Methods for Engineers (Link ).
Control and Automation
NumberTitleTypeECTSHoursLecturers
151-0660-00LModel Predictive Control Information W4 credits2V + 1UM. Zeilinger
AbstractModel predictive control is a flexible paradigm that defines the control law as an optimization problem, enabling the specification of time-domain objectives, high performance control of complex multivariable systems and the ability to explicitly enforce constraints on system behavior. This course provides an introduction to the theory and practice of MPC and covers advanced topics.
ObjectiveDesign and implement Model Predictive Controllers (MPC) for various system classes to provide high performance controllers with desired properties (stability, tracking, robustness,..) for constrained systems.
Content- Review of required optimal control theory
- Basics on optimization
- Receding-horizon control (MPC) for constrained linear systems
- Theoretical properties of MPC: Constraint satisfaction and stability
- Computation: Explicit and online MPC
- Practical issues: Tracking and offset-free control of constrained systems, soft constraints
- Robust MPC: Robust constraint satisfaction
- Nonlinear MPC: Theory and computation
- Hybrid MPC: Modeling hybrid systems and logic, mixed-integer optimization
- Simulation-based project providing practical experience with MPC
Lecture notesScript / lecture notes will be provided.
Prerequisites / NoticeOne semester course on automatic control, Matlab, linear algebra.
Courses on signals and systems and system modeling are recommended. Important concepts to start the course: State-space modeling, basic concepts of stability, linear quadratic regulation / unconstrained optimal control.

Expected student activities: Participation in lectures, exercises and course project; homework (~2hrs/week).
227-0207-00LNonlinear Systems and Control Information
Prerequisite: Control Systems (227-0103-00L)
W6 credits4GE. Gallestey Alvarez, P. F. Al Hokayem
AbstractIntroduce students to the area of nonlinear systems and their control. Familiarize them with tools for modelling and analysis of nonlinear systems. Provide an overview of the various nonlinear controller design methods.
ObjectiveOn completion of the course, students understand the difference between linear and nonlinear systems, know the the mathematical techniques for modeling and analysing these systems, and have learnt various methods for designing controllers for these systems.
Course puts the student in the position to deploy nonlinear control techniques in real applications. Theory and exercises are combined for better understanding of virtues and drawbacks in the different methods.
ContentVirtually all practical control problems are of nonlinear nature. In some cases the application of linear control methods will lead to satisfying controller performance. In many other cases however, only application of nonlinear analysis and synthesis methods will guarantee achievement of the desired objectives. During the past decades a number of mature nonlinear controller design methods have been developed and have proven themselves in applications. After an introduction of the basic methods for modelling and analysing nonlinear systems, these methods will be introduced together with a critical discussion of their pros and cons, and the students will be familiarized with the basic concepts of nonlinear control theory.

This course is designed as an introduction to the nonlinear control field and thus no prior knowledge of this area is required. The course builds, however, on a good knowledge of the basic concepts of linear control.
Lecture notesAn english manuscript will be made available on the course homepage during the course.
LiteratureH.K. Khalil: Nonlinear Systems, Prentice Hall, 2001.
Prerequisites / NoticePrerequisites: Linear Control Systems, or equivalent.
227-0224-00LStochastic Systems Information W4 credits2V + 1UF. Herzog
AbstractProbability. Stochastic processes. Stochastic differential equations. Ito. Kalman filters. St Stochastic optimal control. Applications in financial engineering.
ObjectiveStochastic dynamic systems. Optimal control and filtering of stochastic systems. Examples in technology and finance.
Content- Stochastic processes
- Stochastic calculus (Ito)
- Stochastic differential equations
- Discrete time stochastic difference equations
- Stochastic processes AR, MA, ARMA, ARMAX, GARCH
- Kalman filter
- Stochastic optimal control
- Applications in finance and engineering
Lecture notesH. P. Geering et al., Stochastic Systems, Measurement and Control Laboratory, 2007 and handouts
151-0530-00LNonlinear Dynamics and Chaos II Information W4 credits4GG. Haller
AbstractThe internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems
ObjectiveThe course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.
ContentI. The internal structure of chaos: symbolic dynamics, Bernoulli shift map, sub-shifts of finite type; chaos is numerical iterations.

II.Hamiltonian dynamical systems: conservation and recurrence, stability of fixed points, integrable systems, invariant tori, Liouville-Arnold-Jost Theorem, KAM theory.

III. Normally hyperbolic invariant manifolds: Crash course on differentiable manifolds, existence, persistence, and smoothness, applications.
IV. Geometric singular perturbation theory: slow manifolds and their stability, physical examples. V. Finite-time dynamical system; detecting Invariant manifolds and coherent structures in finite-time flows
Lecture notesStudents have to prepare their own lecture notes
LiteratureBooks will be recommended in class
Prerequisites / NoticeNonlinear Dynamics I (151-0532-00) or equivalent
151-0566-00LRecursive Estimation Information W4 credits2V + 1UR. D'Andrea
AbstractEstimation of the state of a dynamic system based on a model and observations in a computationally efficient way.
ObjectiveLearn the basic recursive estimation methods and their underlying principles.
ContentIntroduction to state estimation; probability review; Bayes' theorem; Bayesian tracking; extracting estimates from probability distributions; Kalman filter; extended Kalman filter; particle filter; observer-based control and the separation principle.
Lecture notesLecture notes available on course website: http://www.idsc.ethz.ch/education/lectures/recursive-estimation.html
Prerequisites / NoticeRequirements: Introductory probability theory and matrix-vector algebra.
Economics
NumberTitleTypeECTSHoursLecturers
363-0552-00LEconomic Growth and Resource UseW3 credits2GA. Schäfer
AbstractThe lecture focuses on the economics of non-renewable resources and deals with the main economic issues regarding such commodities.
ObjectiveThe objective of the lecture is to make students familiar with the main topics in the economics of non-renewable natural resources so that they become able to autonomously read much of the academic literature on the issue. The economics of natural resources adds an intertemporal dimension to the classical static theory. The analyses provided in the lecture will use basic dynamic optimization tools; students are also expected to develop or consolidate their related technical skills.
ContentThe lecture focuses on the economics of non-renewable resources and deals with the main economic issues regarding such commodities. Two peculiarities of natural resources make them interesting economic objects. The intertemporal dimension of resource exploitation is absent in standard static treatments of classical economic theory. The non-renewability of natural resources further implies long-term supply limitations, unlike conventional goods that are indefinitely reproducible. Because of those peculiarities, many well-known economic results do not apply to the case of resources.

As it is appropriate in most chapters, priority will be given to a synthetic partial equilibrium setting. Elementary knowledge of microeconomics (like what is provided by H. Varian, Intermediate Microeconomics) is considered as a prerequisite. Moreover, an introduction to standard partial equilibrium analysis will be provided at the beginning of the lecture. General equilibrium effects should be introduced as they become crucial, as will be the case in the chapters on the interplay between economic growth and resource depletion.

The questions addressed in the lecture will be the following ones:
The intertemporal theory of non-renewable resource supply; the dynamic market equilibrium allocation; the exploration and development of exploitable reserves; the heterogenous quality of resource deposits; pollution and other externalities arising from the use of fossil fuels; the exercise of market power by resource suppliers and market structures; socially optimum extraction patterns and sustainability; the taxation of non-renewable resources; the international strategic dimension of resource taxation; the uncertainty about future reserves and market conditions; economic growth, resource limitations, and the innovation process...
Lecture notesLecture Notes of the course will be sent by email to officially subscribed students.
LiteratureThe main reference of the course is the set of lecture notes; students will also be encouraged to read some influential academic articles dealing with the issues under study.
Prerequisites / NoticeElementary knowledge of microeconomics (like what is provided by H. Varian, Intermediate Microeconomics) is considered as a prerequisite.
363-0514-00LEnergy Economics and Policy
It is recommended for students to have taken a course in introductory microeconomics. If not, they should be familiar with microeconomics as in, for example,"Microeconomics" by Mankiw & Taylor and the appendices 4 and 7 of the book "Microeconomics" by Pindyck & Rubinfeld.
W3 credits2GM. Filippini
AbstractAn introduction to principles of energy economics and applications using energy policies: demand analysis, economic analysis of energy investments and cost analysis, economics of fossil fuels, economics of electricity, economics of renewable energy, market failures and energy policy, market-based and non-market based instruments, demand side management and regulation of energy industries.
ObjectiveThe students will develop the understanding of economic principles and tools necessary to analyze energy issues and to formulate energy policy instruments. Emphasis will be put on empirical analysis of energy demand and supply, market failures, energy policy instruments, investments in power plants and in energy efficiency technologies and the reform of the electric power sector.
ContentThe course provides an introduction to energy economics principles and policy applications. The core topics are
-Demand analysis
-Economic analysis of energy investments and cost analysis
-Economics of fossil fuels
-Economics of electricity
-Economics of renewable energies
-Market failures and energy policy
-Market oriented and non-market oriented instruments
-Demand side management
-Regulation of energy industries
Literature- Joanne Evans (Editor) and Lester C. Hunt (Editor), 2009, International Handbook on the Economics of Energy, Edward Elgar Publishing.

- Bhattacharyya, Subhes C., Energy Economics, 2011, Energy Economics Concepts, Issues, Markets and Governance, 1st Edition, Springer.
Prerequisites / NoticeIt is recommended for students to have taken a course in introductory microeconomics. If not, they should be familiar with microeconomics as in, for example, "Microeconomics" by Mankiw & Taylor and the appendices 4 and 7 of the book "Microeconomics" by Pindyck & Rubinfeld.
364-0576-00LAdvanced Sustainability Economics Information
PhD course, open for MSc students
W3 credits2GL. Bretschger, A. Brausmann
AbstractThe course covers current resource and sustainability economics, including ethical foundations of sustainability, intertemporal optimisation in capital-resource economies, sustainable use of non-renewable and renewable resources, pollution dynamics, population growth, and sectoral heterogeneity. A final part is on empirical contributions, e.g. the resource curse, energy prices, and the EKC.
ObjectiveUnderstanding of the current issues and economic methods in sustainability research; ability to solve typical problems like the calculation of the growth rate under environmental restriction with the help of appropriate model equations.
363-0575-00LEconomic Growth, Cycles and PolicyW3 credits2GH. Gersbach
AbstractThis intermediate macroeconomics course focuses on topics in macroeconomics and monetary economics, like economic growth, financial markets and expectations, the goods market in an open economy, monetary policy, and fiscal policy.
ObjectiveStudents obtain a deeper understanding of some important macroeconomic issues.
ContentThis intermediate macroeconomics course focuses on topics in macroeconomics and monetary economics, like economic growth, financial markets and expectations, the goods market in an open economy, monetary policy, and fiscal policy.
Lecture notesCopies of the slides will be made available.
LiteratureChapters in
Manfred Gärtner (2009), Macroeconomics, Third Edition, Prentice Hall.
and selected chapters in other books and/or papers
Prerequisites / NoticeIt is required that participants have attended the lecture "Principles of Macroeconomics" (351-0565-00L).
363-0515-00LDecisions and MarketsW3 credits2VD. Harenberg
AbstractThis course provides an introduction to microeconomics. The course is open to students who have completed an undergraduate course in economics principles and an undergraduate course in multivariate calculus. The course emphasizes the conceptual foundations of microeconomics and contains concrete examples of their application.
ObjectiveMicroeconomics is an element of nearly every subfield in economic analysis today. Model building in economics relies on a number of fundamental frameworks, many of which are introduced for the first time in intermediate microeconomics.

The purpose of this course is to provide MTEC master students with an introduction to graduate-level microeconomics, particularly for students considering further graduate work in economics, business administration or management science. The course provides the fundamental concepts and tools for graduate courses in economics offered at ETH and UZH.
ContentThe lectures will cover consumer choice, producer theory, markets and market failure. The course will include concrete examples of the use of choice theory in applied economics.
Lecture notesThe course is mostly based on the textbook by R. Serrano and A. Feldman: "A short Course in Intermediate Economics with Calculus" (Cambridge University Press, 2013)
Another textbook of interest is "Intermediate Microeconomics: A Modern Approach" by H. Varian (Norton, 2009).
LiteratureExercises are available in the textbook by R. Serrano and A. Feldman on which the lecture is based ("A short Course in Intermediate Economics with Calculus", Cambridge University Press, 2013).
More exercises can be found in the book "Workouts in Intermediate Microeconomics" by T. Bergstrom and H. Varian (Norton, 2010).
363-1017-00LRisk and Insurance EconomicsW4 credits3VW. Mimra
AbstractThe course covers economics of risk and insurance. Topics covered are fundamentals of insurance, risk measures and risk management, demand and supply of insurance and asymmetric information in insurance markets.
ObjectiveThe goal is to introduce students to basic concepts of risk, risk management and economics of insurance.
Content- fundamentals of insurance
- what is the rationale for corporate risk management?
- measures of risk and methods of risk management
- demand for insurance
- supply of insurance
- information problems in insurance markets: moral hazard, adverse selection, fraud
Literature- Peter Zweifel and Roland Eisen (2012), Insurance Economics, Springer.
- S. Hun Seog (2010), The Economics of Risk and Insurance, Wiley-Blackwell.
- Ray Rees and Achim Wambach (2008), The Microeconomics of Insurance, Foundations and Trends in Microeconomics: Vol. 4: No 1-2.
- Eeckhoudt/Gollier/Schlesinger (2007), Economic and Financial Decisions under Risk, Princeton University Press.
- introductory background reading: Harrington/Niehaus (2003), Risk Management and Insurance, McGraw Hill.
Environmental Science
NumberTitleTypeECTSHoursLecturers
701-1334-00LModelling of Processes in Soils and Aquifers Information Restricted registration - show details
Number of participants limited to 18.
First come, first serve.
W3 credits2GG. Furrer, W. Pfingsten
AbstractComputational modelling of biogeochemical processes and transport of water and solutes in soils and aquifers.
ObjectiveRationale:
The content of the course builds on the students' basic knowledge in soil and aquatic chemistry as well as in soil physics (see below: Prerequisites).
This course addresses the modelling of impacts by pollutants on terrestrial and aquatic environments. It helps to acquire to model hydrological, geochemical and microbial processes in soils and aquifers in order to predict the mobility of contaminants in heterogeneous environmental systems. Computer models used will be provided by the internet platform PolyQL (http://www.polyql.ethz.ch).

Aims:
- Conveying the fact that there are different modelling approaches
- Learning how to parameterize physically-based models
- Developing skills for critical judgement of modelling results
- Applying theoretical models to real systems
- Gaining competence with web-aided learning
Content- Applying computer models for biogeochemical and transport processes
- Chemical equilibria, speciation in aqueous systems
- Chemical kinetics, biogeochemical processes, redox processes
- Steady-state approach, serial-box models, sensitivity analysis
- Basic concepts in modelling water flow and solute transport
- Hydraulic processes in variably saturated soils
- Using models for pollutant transport in soils and aquifers
Lecture notesAvailable as hardcopy and on-line material.
(http://www.polyql.ethz.ch)
Literature- CAJ Appelo and D Postma, 2005. Geochemistry, Groundwater and Pollution. Taylor & Francis
- D Hillel, 2004. Introduction to environmental soil physics. Elsevier
Prerequisites / NoticePrerequisites: Courses (or equivalent knowledge)
- Soil Chemistry (701-0533-00, autumn semester, German)
- Environmental Soil Physics/Vadose Zone Hydrology (701-0535-00, autumn semester, English)
Finance
NumberTitleTypeECTSHoursLecturers
401-8916-00LAdvanced Corporate Finance II (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MFOEC144

Mind the enrolment deadlines at UZH:
http://www.uzh.ch/studies/application/mobilitaet_en.html
W3 credits2VUniversity lecturers
AbstractTo provide the students with good understanding of the problems and issues in corporate finance.
ObjectiveTo provide the students with good understanding of the problems and issues in corporate finance.
ContentThe following topics are covered in this course: the role of information and incentives in determining the forms of financing a firm chooses; hedging; venture capital; initial public offerings; investment in very large projects; the setting up of a "bad" bank; the securitisation of commercial and industrial loans; the transfer of catastrophe risk to financial markets; agency in insurance; and dealing with a run on an insurance company.
Lecture notesSee: http://www.isb.uzh.ch/institut/staff/habib.michel/teaching/
LiteratureSee: http://www.isb.uzh.ch/institut/staff/habib.michel/teaching/
401-8915-00LAdvanced Financial Economics (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MFOEC105

Mind the enrolment deadlines at UZH:
http://www.uzh.ch/studies/application/mobilitaet_en.html
W3 credits2VUniversity lecturers
AbstractPortfolio Theory, CAPM, Financial Derivatives, Incomplete Markets, Corporate Finance, Behavioural Finance, Evolutionary Finance, Asymmetric Information.
ObjectiveStudents should get familiar with the cornerstones of modern finance.
LiteratureLecture Notes.
Image Processing and Computer Vision
No course offerings in this semester
Information and Communication Technology
NumberTitleTypeECTSHoursLecturers
227-0438-00LFundamentals of Wireless Communication Information W6 credits2V + 2UH. Bölcskei
AbstractThe class focuses on fundamental communication-theoretic aspects of modern wireless communication systems. The main topics covered are the system-theoretic characterization of wireless channels, the principle of diversity, information theoretic aspects of communication over fading channels, and the basics of multi-user communication theory and cellular systems.
ObjectiveAfter attending this lecture, participating in the discussion sessions, and working on the homework problem sets, students should be able to
- understand the nature of the fading mobile radio channel and its implications for the design of communication systems
- analyze existing communication systems
- apply the fundamental principles to new wireless communication systems, especially in the design of diversity techniques and coding schemes
ContentThe goal of this course is to study the fundamental principles of wireless communication, enabling students to analyze and design current and future wireless systems. The outline of the course is as follows:

Wireless Channels
What differentiates wireless communication from wired communication is the nature of the communication channel. Motion of the transmitter and the receiver, the environment, multipath propagation, and interference render the channel model more complex. This part of the course deals with modeling issues, i.e., the process of finding an accurate and mathematically tractable formulation of real-world wireless channels. The model will turn out to be that of a randomly time-varying linear system. The statistical characterization of such systems is given by the scattering function of the channel, which in turn leads us to the definition of key propagation parameters such as delay spread and coherence time.

Diversity
In a wireless channel, the time varying destructive and constructive addition of multipath components leads to signal fading. The result is a significant performance degradation if the same signaling and coding schemes as for the (static) additive white Gaussian noise (AWGN) channel are used. This problem can be mitigated by diversity techniques. If several independently faded copies of the transmitted signal can be combined at the receiver, the probability of all copies being lost--because the channel is bad--decreases. Hence, the performance of the system will be improved. We will look at different means to achieve diversity, namely through time, frequency, and space. Code design for fading channels differs fundamentally from the AWGN case. We develop criteria for designing codes tailored to wireless channels. Finally, we ask the question of how much diversity can be obtained by any means over a given wireless channel.

Information Theory of Wireless Channels
Limited spectral resources make it necessary to utilize the available bandwidth to its maximum extent. Information theory answers the fundamental question about the maximum rate that can reliably be transmitted over a wireless channel. We introduce the basic information theoretic concepts needed to analyze and compare different systems. No prior experience with information theory is necessary.

Multiple-Input Multiple-Output (MIMO) Wireless Systems
The major challenges in future wireless communication system design are increased spectral efficiency and improved link reliability. In recent years the use of spatial (or antenna) diversity has become very popular, which is mostly due to the fact that it can be provided without loss in spectral efficiency. Receive diversity, that is, the use of multiple antennas on the receive side of a wireless link, is a well-studied subject. Driven by mobile wireless applications, where it is difficult to deploy multiple antennas in the handset, the use of multiple antennas on the transmit side combined with signal processing and coding has become known under the name of space-time coding. The use of multiple antennas at both ends of a wireless link (MIMO technology) has been demonstrated to have the potential of achieving extraordinary data rates. This chapter is devoted to the basics of MIMO wireless systems.

Cellular Systems: Multiple Access and Interference Management
This chapter deals with the basics of multi-user communication. We start by exploring the basic principles of cellular systems and then take a look at the fundamentals of multi-user channels. We compare code-division multiple-access (CDMA) and frequency-division multiple access (FDMA) schemes from an information-theoretic point of view. In the course of this comparison an important new concept, namely that of multiuser diversity, will emerge. We conclude with a discussion of the idea of opportunistic communication and by assessing this concept from an information-theoretic point of view.
Lecture notesLecture notes will be handed out during the lectures.
LiteratureA set of handouts covering digital communication basics and mathematical preliminaries is available on the website. For further reading, we recommend
- J. M. Wozencraft and I. M. Jacobs, "Principles of Communication Engineering," Wiley, 1965
- A. Papoulis and S. U. Pillai, "Probability, Random Variables, and Stochastic Processes," McGraw Hill, 4th edition, 2002
- G. Strang, "Linear Algebra and its Applications," Harcourt, 3rd edition, 1988
- T.M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley, 1991
Prerequisites / NoticeThis class will be taught in English. The oral exam will be in German (unless you wish to take it in English, of course).

A prerequisite for this course is a working knowledge in digital communications, random processes, and detection theory.
227-0420-00LInformation Theory II Information W6 credits2V + 2UA. Lapidoth
AbstractThis course builds on Information Theory I. It introduces additional topics in single-user communication, connections between Information Theory and Statistics, and Network Information Theory.
ObjectiveThe course has two objectives: to introduce the students to the key information theoretic results that underlay the design of communication systems and to equip the students with the tools that are needed to conduct research in Information Theory.
ContentDifferential entropy, maximum entropy, the Gaussian channel and water filling, the entropy-power inequality, Sanov's Theorem, Fisher information, the broadcast channel, the multiple-access channel, Slepian-Wolf coding, and the Gelfand-Pinsker problem.
Lecture notesn/a
LiteratureT.M. Cover and J.A. Thomas, Elements of Information Theory, second edition, Wiley 2006
Material Modelling and Simulation
NumberTitleTypeECTSHoursLecturers
327-2201-00LTransport Phenomena II Information W4 credits4GH. C. Öttinger
AbstractNumerical methods for real-world "Transport Phenomena"; atomistic understanding of transport properties based on kinetic theory and mesoscopic models; fundamentals, applications, and simulations
ObjectiveThe teaching goals of this course are on five different levels:
(1) Deep understanding of fundamentals: kinetic theory, mesoscopic models, ...
(2) Ability to use the fundamental concepts in applications
(3) Insight into the role of boundary conditions
(4) Knowledge of a number of applications
(5) Flavor of numerical techniques: finite elements, lattice Boltzmann, ...
ContentThermodynamics of Interfaces
Interfacial Balance Equations
Interfacial Force-Flux Relations
Polymer Processing
Transport Around a Sphere
Semi-Conductor Processing
Refreshing Topics in Equilibrium Statistical Mechanics
Transport in Biological Systems
Kinetic Theory of Polymeric Liquids
Dynamic Light Scattering
Lecture notesA detailed manuscript is available; this manuscript will be developed into a book entitled "A Modern Course in Transport Phenomena" by David C. Venerus and Hans Christian Öttinger
Literature1. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd Ed. (Wiley, 2001)
2. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, 2nd Ed. (Dover, 1984)
3. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287
4. R. Phillips, J. Kondev, and J. Theriot, Physical Biology of the Cell (Garland, 2008)
5. G. A. Truskey, F. Yuan, and D. F. Katz, Transport Phenomena in Biological Systems (Prentice Hall, 2004)
Prerequisites / NoticeComplex numbers. Vector analysis (integrability; Gauss' divergence theorem). Laplace and Fourier transforms. Ordinary differential equations (basic ideas). Linear algebra (matrices; functions of matrices; eigenvectors and eigenvalues; eigenfunctions). Probability theory (Gaussian distributions; Poisson distributions; averages; moments; variances; random variables). Numerical mathematics (integration). Statistical thermodynamics (Gibbs' fundamental equation; thermodynamic potentials; Legendre transforms; Gibbs' phase rule; ergodicity; partition functions; Einstein's fluctuation theory). Linear irreversible thermodynamics (forces and fluxes; Fourier's, Newton's and Fick's laws for fluxes). Hydrodynamics (local equilibrium; balance equations for mass, momentum, energy and entropy). Programming and simulation techniques (Matlab, Monte Carlo simulations).
151-0515-00LContinuum Mechanics 2
Prerequisites: A course in Linear Continuum Mechanics
W4 credits2V + 1UE. Mazza, B. Röhrnbauer
AbstractAn introduction to finite deformation continuum mechanics and nonlinear material behavior. Coverage of basic tensor- manipulations and calculus, descriptions of kinematics, and balance laws . Discussion of invariance principles and mechanical response functions for elastic materials.
ObjectiveTo provide a modern introduction to the foundations of continuum mechanics and prepare students for further studies in solid
mechanics and related disciplines.
Content1. Tensors: algebra, linear operators
2. Tensors: calculus
3. Kinematics: motion, gradient, polar decomposition
4. Kinematics: strain
5. Kinematics: rates
6. Global Balance: mass, momentum
7. Stress: Cauchy's theorem
8. Stress: alternative measures
9. Invariance: observer
10. Material Response: elasticity
Lecture notesnone
LiteratureRecommended texts:
(1) Nonlinear solid mechanics, G.A. Holzapfel (2000).
(2) An introduction to continuum mechanics, M.B. Rubin (2003).
Quantum Chemistry
NumberTitleTypeECTSHoursLecturers
529-0474-00LQuantum ChemistryW6 credits3GM. Reiher
AbstractIntroduction into the basic concepts of electronic structure theory and into numerical methods of quantum chemistry. Exercise classes are designed to deepen the theory; practical case studies using quantum chemical software to provide a 'hands-on' expertise in applying these methods.
ObjectiveNowadays, chemical research can be carried out in silico, an intellectual achievement for which Pople and Kohn have been awarded the Nobel prize of the year 1998. This lecture shows how that has been accomplished. It works out the many-particle theory of many-electron systems (atoms and molecules) and discusses its implementation into computer programs. A complete picture of quantum chemistry shall be provided that will allow students to carry out such calculations on molecules (for accompanying experimental work in the wet lab or as a basis for further study of the theory).
ContentBasic concepts of many-particle quantum mechanics. Derivation of the many-electron theory for atoms and molecules; starting with the harmonic approximation for the nuclear problem and with Hartree-Fock theory for the electronic problem to Moeller-Plesset perturbation theory and configuration interaction and to coupled cluster and multi-configurational approaches. Density functional theory. Case studies using quantum mechanical software.
Lecture notesHand outs will be provided for each lecture (this script has been completely revised in spring 2014 anf has been supplemented by (computer) examples that continuously illustrate how the theory works).
LiteratureTextbooks on Quantum Chemistry:
F.L. Pilar, Elementary Quantum Chemistry, Dover Publications
I.N. Levine, Quantum Chemistry, Prentice Hall

Hartree-Fock in basis set representation:
A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill

Textbooks on Computational Chemistry:
F. Jensen, Introduction to Computational Chemistry, John Wiley & Sons
C.J. Cramer, Essentials of Computational Chemistry, John Wiley & Sons
Prerequisites / Noticebasic knowledge in quantum mechanics (e.g. through course physical chemistry III - quantum mechanics) required
Simulation of Semiconductor Devices
Simulation of Semiconductor Devices
NumberTitleTypeECTSHoursLecturers
227-0158-00LSemiconductor Devices: Transport Theory and Monte Carlo Simulation Information
Does not take place this semester.
W4 credits2V + 1U
AbstractThe first part deals with semiconductor transport theory including the necessary quantum mechanics.
In the second part, the Boltzmann equation is solved with the stochastic methods of Monte Carlo simulation.
The exercises address also TCAD simulations of MOSFETs. Thus the topics include theoretical physics,
numerics and practical applications.
ObjectiveOn the one hand, the link between microscopic physics and its concrete application in device simulation is established; on the other hand, emphasis is also laid on the presentation of the numerical techniques involved.
ContentQuantum theoretical foundations I (state vectors, Schroedinger and Heisenberg picture). Band structure (Bloch theorem, one dimensional periodic potential, density of states). Pseudopotential theory (crystal symmetries, reciprocal lattice, Brillouin zone).
Semiclassical transport theory (Boltzmann transport equation (BTE), scattering processes, linear transport).<br>
Monte Carlo method (Monte Carlo simulation as solution method of the BTE, algorithm, expectation values).<br>
Implementational aspects of the Monte Carlo algorithm (discretization of the Brillouin zone, self-scattering according to Rees, acceptance- rejection method etc.). Bulk Monte Carlo simulation (velocity-field characteristics, particle generation, energy distributions, transport parameters). Monte Carlo device simulation (ohmic boundary conditions, MOSFET simulation).
Quantum theoretical foundations II (limits of semiclassical transport theory, quantum mechanical derivation of the BTE, Markov-Limes).
Lecture notesLecture notes (in German)
Simulation of Semiconductor Devices (not eligible for credits)
NumberTitleTypeECTSHoursLecturers
227-0056-00LSemiconductor Devices Information E-4 credits2V + 2UC. Bolognesi
AbstractThe course covers the basic principles of semiconductor devices in micro-, opto-, and power electronics. It imparts knowledge both of the basic physics and on the operation principles of pn-junctions, diodes, contacts, bipolar transistors, MOS devices, solar cells, photodetectors, LEDs and laser diodes.
ObjectiveUnderstanding of the basic principles of semiconductor devices in micro-, opto-, and power electronics.
ContentBrief survey of the history of microelectronics. Basic physics: Crystal structure of solids, properties of silicon and other semiconductors, principles of quantum mechanics, band model, conductivity, dispersion relation, equilibrium statistics, transport equations, generation-recombination (G-R), Quasi-Fermi levels. Physical and electrical properties of the pn-junction. pn-diode: Characteristics, small-signal behaviour, G-R currents, ideality factor, junction breakdown. Contacts: Schottky contact, rectifying barrier, Ohmic contact, Heterojunctions. Bipolar transistor: Operation principles, modes of operation, characteristics, models, simulation. MOS devices: Band diagram, MOSFET operation, CV- and IV characteristics, frequency limitations and non-ideal behaviour. Optoelectronic devices: Optical absorption, solar cells, photodetector, LED, laser diode.
Lecture notesScript of the slides.
LiteratureThe lecture course follows the book Neamen, Semiconductor Physics and Devices, ISBN 978-007-108902-9, Fr. 89.00
Prerequisites / NoticeQualifications: Physics I+II
Systems Design
NumberTitleTypeECTSHoursLecturers
151-0530-00LNonlinear Dynamics and Chaos II Information W4 credits4GG. Haller
AbstractThe internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems
ObjectiveThe course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.
ContentI. The internal structure of chaos: symbolic dynamics, Bernoulli shift map, sub-shifts of finite type; chaos is numerical iterations.

II.Hamiltonian dynamical systems: conservation and recurrence, stability of fixed points, integrable systems, invariant tori, Liouville-Arnold-Jost Theorem, KAM theory.

III. Normally hyperbolic invariant manifolds: Crash course on differentiable manifolds, existence, persistence, and smoothness, applications.
IV. Geometric singular perturbation theory: slow manifolds and their stability, physical examples. V. Finite-time dynamical system; detecting Invariant manifolds and coherent structures in finite-time flows
Lecture notesStudents have to prepare their own lecture notes
LiteratureBooks will be recommended in class
Prerequisites / NoticeNonlinear Dynamics I (151-0532-00) or equivalent
363-0588-00LComplex Networks Information W4 credits2V + 1UI. Scholtes
AbstractThe course provides an overview of the methods and abstractions used in (i) the quantitative study of complex networks, (ii) empirical network analysis, (iii) the study of dynamical processes in networked systems, (iv) the analysis of robustness of networked systems, (v) the study of network evolution, and (vi) data mining techniques for networked data sets.
Objective* the network approach to complex systems, where actors are represented as nodes and interactions are represented as links
* learn about structural properties of classes of networks
* learn about feedback mechanism in the formation of networks
* learn about statistical inference and data mining techniques for data on networked systems
* learn methods and abstractions used in the growing literature on complex networks
ContentNetworks matter! This holds for social and economic systems, for technical infrastructures as well as for information systems. Increasingly, these networked systems are outside the control of a centralized authority but rather evolve in a distributed and self-organized way. How can we understand their evolution and what are the local processes that shape their global features? How does their topology influence dynamical processes like diffusion? And how can we characterize the importance of specific nodes?

This course provides a systematic answer to such questions, by developing methods and tools which can be applied to networks in diverse areas like infrastructure, communication, information systems, biology or (online) social networks. In a network approach, agents in such systems (like e.g. humans, computers, documents, power plants, biological or financial entities) are represented as nodes, whereas their interactions are represented as links.

The first part of the course, "Introduction to networks: basic and advanced metrics", describes how networks can be represented mathematically and how the properties of their link structures can be quantified empirically.

In a second part "Stochastic Models of Complex Networks" we address how analytical statements about crucial properties like connectedness or robustness can be made based on simple macroscopic stochastic models without knowing the details of a topology.

In the third part we address "Dynamical processes on complex networks". We show how a simple model for a random walk in networks can give insights into the authority of nodes, the efficiency of diffusion processes as well as the existence of community structures.

A fourth part "Network Optimisation and Inference" introduces models for the emergence of complex topological features which are due to stochastic optimization processes, as well as statistical methods to detect patterns in large data sets on networks.

In a fifth part, we address "Network Dynamics", introducing models for the emergence of complex features that are due to (i) feedback phenomena in simple network growth processes or (iii) order correlations in systems with highly dynamic links.

A final part "Research Trends" introduces recent research on the application of data mining and machine learning techniques to relational data.
Lecture notesThe lecture slides are provided as handouts - including notes and literature sources - to registered students only.
All material is to be found on Moodle at the following URL: https://moodle-app2.let.ethz.ch/course/view.php?id=2678
LiteratureSee handouts. Specific literature is provided for download - for registered students, only.
Prerequisites / NoticeThere are no pre-requisites for this course. Self-study tasks (to be solved analytically and by means of computer simulations) are provided as home work. Weekly exercises (45 min) are used to discuss selected solutions. Active participation in the exercises is strongly suggested for a successful completion of the final exam.
363-0543-00LAgent-Based Modelling of Social Systems Information
Does not take place this semester.
W3 credits2V + 1UF. Schweitzer
AbstractAgent-based modelling is introduced as a bottom-up approach to understand the dynamics of complex social systems. The course focuses on agents as the fundamental constituents of a system and their theoretical formalisation and on quantitative analysis of a wide range of social phenomena-cooperation and competition, opinion dynamics, spatial interactions and behaviour in online social networks.
ObjectiveA successful participant of this course is able to
- understand the rationale of agent-centered models of social systems
- understand the relation between rules implemented at the individual level and the emerging behaviour at the global level
- learn to choose appropriate model classes to characterise different social systems
- grasp the influence of agent heterogeneity on the model output
- efficiently implement agent-based models using Python and visualise the output
ContentAgent-based modelling (ABM) provides a bottom-up approach to understand the complex dynamics of social systems. In ABM, agents are the basic constituents of any social system. Depending on the granularity of the analysis, an agent could represent a single individual, a household, a firm, a country, etc. Agents have internal states or degrees of freedom opinions, strategies, etc.), the ability to perceive and change their environment, and the ability to interact with other agents. Their individual (microscopic) actions and interactions with other agents, result in macroscopic (collective, system) dynamics with emergent properties. As more and more accurate individual-level data about online and offline social systems become available, our formal, quantitative understanding of the collective dynamics of these systems needs to progress in the same manner.

We focus on a minimalistic description of the agents' behaviour which relates individual interaction rules to the dynamics on the collective level and complements engineering and machine learning approaches.

The course is structured in three main parts. The first two parts introduce two main agent concepts - Boolean agents and Brownian agents, which differ in how the internal dynamics of agents is represented. Boolean agents are characterized by binary internal states, e.g. yes/no opinion, while Brownian agents can have a continuous spectrum of internal states, e.g. preferences and attitudes. The last part introduces models in which agents interact in physical space, e.g. migrate or move collectively.

Throughout the course, we will discuss a wide variety of application areas, such as:
- opinion dynamics and social influence,
- cooperation and competition,
- online social networks,
- systemic risk
- emotional influence and communication
- swarming behavior
- spatial competition

While the lectures focus on the theoretical foundations of agent-based modelling, weekly exercise classes provide practical skills. Using the Python programming language, the participants implement agent-based models in guided and autonomous projects, which they present and jointly discuss.
Lecture notesThe lecture slides will be available on the Moodle platform, for registered students only.
LiteratureSee handouts. Specific literature is provided for download, for registered students only.
Prerequisites / NoticeParticipants of the course should have some background in mathematics and an interest in formal modelling and computer simulations, and should be motivated to learn about social systems from a quantitative perspective.

Prior knowledge of Python is not necessary.

Self-study tasks are provided as home work for small teams (2-4 members).
Weekly exercises (45 min) are used to discuss the solutions and guide the student.
During the second half of the semester, teams need to complete a course project in which they will implement and discuss an agent-based model to characterise a system chosen jointly with the course organisers. This project will be evaluated, and its grade will count as 25% of the final grade.
Theoretical Physics
In the Master's programme in Applied Mathematics 402-0204-00L Electrodynamics is eligible as a course unit in the application area Theoretical Physics, but only if 402-0224-00L Theoretical Physics wasn't or 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
402-0812-00LComputational Statistical Physics Information W8 credits2V + 2UM. Henkel, M. Lukovic, M. Mendoza Jimenez
AbstractComputer simulation methods in statistical physics. Classical Monte-Carlo-simulations: finite-size scaling, cluster algorithms, histogram-methods. Molecular dynamics simulations: long range interactions, Ewald summation, discrete elements, parallelization.
ObjectiveThe lecture will give a deeper insight into computer simulation methods in statistical physics. Thus, it is an ideal continuation of the lecture
"Introduction to Computational Physics" of the autumn semester focusing on the following topics. Classical Monte-Carlo-simulations: finite-size scaling, cluster algorithms, histogram-methods. Molecular dynamics simulations: long range interactions, Ewald summation, discrete elements, parallelization.
ContentComputer simulation methods in statistical physics.
Classical Monte-Carlo-simulations: finite-size scaling, cluster algorithms, histogram-methods. Molecular dynamics simulations: long range interactions, Ewald summation, discrete elements, parallelization.
402-0810-00LComputational Quantum PhysicsW8 credits2V + 2UG. Carleo
AbstractThis course provides an introduction to simulation methods for quantum systems, starting with the one-body problem and finishing with quantum field theory, with special emphasis on quantum many-body systems. Both approximate methods (Hartree-Fock, density functional theory) and exact methods (exact diagonalization, quantum Monte Carlo) are covered.
ObjectiveThe goal is to become familiar with computer simulation techniques for quantum physics, through lectures and practical programming exercises.
402-0206-00LQuantum Mechanics II Information W10 credits3V + 2UT. K. Gehrmann
AbstractIntroduction to many-particle quantum mechanics and quantum statistics. Basic concepts: symmetrised many-body wave functions for fermions and bosons, the Pauli principle, Bose- and Fermi-statistic and second quantisation. Applications include the description of atoms, and the interaction between radiation and matter.
ObjectiveIntroduction to many-particle quantum mechanics and quantum statistics. In particular basic concepts such as symmetrised many-body wave functions for fermions and bosons, the Pauli principle, Bose- and Fermi-statistics and second quantisation will be discussed. Applications include
the description of atoms, and the interaction between radiation and matter.
ContentThe description of identical particles leads us to the
introduction of symmetrised wave functions for fermions
and bosons. We discuss simple few-body problems and
proceed with a systematic description of fermionic many
body problems in terms of second quantisation.
We also discuss basic concepts of quantum statistics.
Applications include the description of
atoms, and the interaction between radiation and matter.
LiteratureF. Schwabl, Quantenmechanik (Springer)
F. Schwabl, Quantenmechanik fuer Fortgeschrittene (Springer)
J.J. Sakurai, Advanced Quantum mechanics (Addison Wesley)
402-0871-00LSolid State TheoryW10 credits4V + 1UV. Geshkenbein
AbstractThe course is addressed to students in experimental and theoretical condensed matter physics and provides a theoretical introduction to a variety of important concepts used in this field.
ObjectiveThe course provides a theoretical frame for the understanding of basic pinciples in solid state physics. Such a frame includes the topics of symmetries, band structures, many body interactions, Landau Fermi-liquid theory, and specific topics such as transport, superconductivity, or magnetism. The exercises illustrate the various themes in the lecture and help to develop problem-solving skills. The student understands basic concepts in solid state physics and is able to solve simple problems. No diagrammatic tools will be developed.
ContentThe course is addressed to students in experimental and theoretical condensed matter physics and provides a theoretical introduction to a variety of important concepts used in this field. A selection is made from topics such as: Symmetries and their handling via group theoretical concepts, electronic structure in crystals, insulators-semiconductors-metals, phonons, interaction effects, (un-)screened Fermi-liquids, linear response theory, collective modes, screening, transport in semiconductors and metals, magnetism, Mott-insulators, quantum-Hall effect, superconductivity.
Lecture notesin English
402-0844-00LQuantum Field Theory IIW10 credits3V + 2UN. Beisert
AbstractThe subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.
Objective
ContentThe following topics will be covered:
- path integral quantization
- non-abelian gauge theories and their quantization
- systematics of renormalization, including BRST symmetries,
Slavnov-Taylor Identities and the Callan Symanzik equation
- gauge theories with spontaneous symmetry breaking and
their quantization
- renormalization of spontaneously broken gauge theories and
quantum effective actions
LiteratureM.E. Peskin and D.V. Schroeder,
An introduction to Quantum Field Theory, Perseus (1995).
L.H. Ryder,
Quantum Field Theory, CUP (1996).
S. Weinberg,
The Quantum Theory of Fields (Volume 2), CUP (1996).
M. Srednicki,
Quantum Field Theory, CUP (2006).
402-0394-00LTheoretical Astrophysics and CosmologyW10 credits4V + 2UL. M. Mayer, A. Refregier
AbstractThis is the second of a two course series which starts with "General Relativity" and continues in the spring with "Theoretical Astrophysics and Cosmology", where the focus will be on applying general relativity to cosmology as well as developing the modern theory of structure formation in a cold dark matter Universe.
Objective
ContentThe course will cover the following topics:
- Homogeneous cosmology
- Thermal history of the universe, recombination, baryogenesis and nucleosynthesis
- Dark matter and Dark Energy
- Inflation
- Perturbation theory: Relativistic and Newtonian
- Model of structure formation and initial conditions from Inflation
- Cosmic microwave background anisotropies
- Spherical collapse and galaxy formation
- Large scale structure and cosmological probes
LiteratureSuggested textbooks:
H.Mo, F. Van den Bosch, S. White: Galaxy Formation and Evolution
S. Carroll: Space-Time and Geometry: An Introduction to General Relativity
S. Dodelson: Modern Cosmology
Secondary textbooks:
S. Weinberg: Gravitation and Cosmology
V. Mukhanov: Physical Foundations of Cosmology
E. W. Kolb and M. S. Turner: The Early Universe
N. Straumann: General relativity with applications to astrophysics
A. Liddle and D. Lyth: Cosmological Inflation and Large Scale Structure
Prerequisites / NoticeKnowledge of General Relativity is recommended.
» Electives Theoretical Physics
Transportation Science
NumberTitleTypeECTSHoursLecturers
101-0478-00LMeasurement and Modelling of Travel BehaviourW6 credits4GK. W. Axhausen
AbstractComprehensive introduction to survey methods in transport planning and modeling of travel behavior, using advanced discrete choice models.
ObjectiveEnabling the student to understand and apply the various measurement approaches and models of modelling travel behaviour.
ContentBehavioral model and measurement; travel diary, design process, hypothetical markets, discrete choice model, parameter estimation, pattern of travel behaviour, market segments, simulation, advanced discrete choice models
Lecture notesVarious papers and notes are distributed during the course.
Prerequisites / NoticeRequirement: Transport I
Seminars and Semester Papers
Seminars
Early enrolments for seminars in myStudies are encouraged, so that we will recognize need for additional seminars in a timely manner. Some seminars have waiting lists. Nevertheless, register for at most two mathematics seminars. In this case, you express a stronger preference for the seminar for which you register earlier.
NumberTitleTypeECTSHoursLecturers
401-3140-17LHyperbolic Surfaces Restricted registration - show details
Number of participants limited to 13.
W4 credits2SA. Sisto, P. D. Nelson
AbstractAny surface of genus at least 2 admits a metric locally modeled on the hyperbolic plane, and in fact it admits many such metrics. These metric structures play a fundamental roles in the study of surfaces and more in general in low-dimensional topology.
ObjectiveIn the first few meetings, we will briefly review the hyperbolic plane and then construct hyperbolic metrics.
I will then propose a few topics related to hyperbolic metrics and we will decide which one to pursue.
401-3370-17LArithmetic of Quadratic Forms Restricted registration - show details
Number of participants limited to 12. Registration to the seminar will only be effective once confirmed by the organisers. Please contact andreas.wieser@math.ethz.ch .
This seminar is fully occupied. Unconfirmed registrations have been deleted.
W4 credits2SM. Akka Ginosar
AbstractIntroductory seminar about rational quadratic forms. P-adic numbers, Hasse's local to global principle and the finiteness of the genus will be discussed.
ObjectiveQuadratic forms and the numbers they represent have been of interest to mathematicians for a long time. For example, which integers can be expressed as a sum of two squares of integers? Or as a sum of three squares? Lagrange's four-squares theorem for instance states that any positive integer can be expressed as a sum of four squares. Such questions motivated the development of many aspects of algebraic number theory.

In this seminar we follow the beautiful monograph of Cassels "Rational quadratic forms" and will treat the fundamental results concerning quadratic forms over the integers and the rationals such as Hasse's local to global principle and finiteness of the genus.
ContentThe seminar will mostly follow the book "Rational quadratic forms" by J.W.S. Cassels, particularly Chapters 1-9. Exercises in this book are an integral part of the seminar. Towards the end of the semester additional topics may be treated.
Lecture notesCassels, John William Scott. Rational quadratic forms. Vol. 13. Academic Pr, 1978.
LiteratureMain reference:
Cassels, John William Scott. Rational quadratic forms. Vol. 13. Academic Pr, 1978.
Additional references:

Kitaoka, Yoshiyuki. Arithmetic of quadratic forms. Vol. 106. Cambridge University Press, 1999.
Schulze-Pillot, Rainer. "Representation by integral quadratic forms - a survey." Contemporary Mathematics 344 (2004): 303-322.
Prerequisites / NoticeThe student is assumed to have attended courses on linear algebra and algebra (as taught at ETH for instance). Previous knowledge on p-adic numbers is not assumed.
401-3350-17LProducts and Nonlinearities in Function Space Theory Restricted registration - show details
Number of participants limited to 20.
W4 credits1SL. Keller, T. Rivière
AbstractThe seminar is continuing on from the course ''Fourier analysis in Function Space theory'' given in HS 2016.

We will first briefly recall the general Littlewood-Paley theory introduced in that course. After this review, we shall study applications of this theory to important problems relevant to the analysis of Partial Differential Equations which constitutes the main topics of the seminar.
ObjectiveThe first hours of the seminar will be given by the organizer himself. The participants will then be asked to take over the rest of the seminar by choosing thematics out of the topics listed above.
401-3600-17LStudent Seminar in Probability Theory: Information Restricted registration - show details
Limited number of participants.
Registration to the seminar will only be effective once confirmed by email from the organizers.
W4 credits2SA.‑S. Sznitman, J. Bertoin, P. Nolin, V. Tassion
AbstractSelected topics from Probability Theory will be discussed.
ObjectiveThe seminar is a natural complement to the material discussed in the lecture on probability theory in the 5th semester.
ContentThe seminar is centered around a topic in probability theory which changes each semester. Example of topics are random walks and electric networks, Markov chains, stochastic integrals, coupling methods, etc.
Prerequisites / NoticeThere is only a limited number of slots available for this seminar. Participation will only be effective once confirmed by the organizers.
401-3620-17LStudent Seminar in Statistics: Statistical Inference under Shape Restrictions Restricted registration - show details
Number of participants limited to 22.

Mainly for students from the Mathematics Bachelor and Master Programmes who, in addition to the introductory course unit 401-2604-00L Probability and Statistics, have heard at least one core or elective course in statistics.
W4 credits2SF. Balabdaoui, P. L. Bühlmann, M. H. Maathuis, N. Meinshausen, S. van de Geer
AbstractStatistical inference based on a random sample can be performed under additional shape restrictions on the unknown entity to be estimated (regression curve, probability density,...). Under shape restrictions, we mean a variety of constraints. Examples thereof include monotonicity, bounded variation, convexity, k-monotonicity or log-concavit.
ObjectiveThe main goal of this Student Seminar is to get acquainted with the existing approaches in shape constrained estimation. The students will get to learn that specific estimation techniques can be used under shape restrictions to obtain better estimators, especially for small/moderate sample sizes. Students will also have the opportunity to learn that one of the main merits of shape constrained inference is to avoid choosing some arbitrary tuning parameter as it is the case with bandwidth selection in kernel estimation methods.

Furthemore, students will get to read about some efficient algorithms that can be used to fastly compute the obtained estimators. One of the famous algoritms is the so-called PAVA (Pool Adjacent Violators Algorithm) used under monotonicity to compute a regression curve or a probability density.

During the Seminar, the students will have to study some selected chapters from the book "Statistical Inference under Order Restrictions" by Barlow, Bartholomew, Bremner and Brunk as well as some "famous" articles on the subject.
Prerequisites / NoticeWe require at least one course in statistics in addition to the 4th semester course Introduction to Probability and Statistics and basic knowledge in computer programming.

Topics will be assigned during the first meeting.
401-3920-17LNumerical Analysis Seminar: Mathematics for Biomimetics Restricted registration - show details
Number of participants limited to 8.
W4 credits2SH. Ammari
AbstractThe aim of this seminar is to explore how we can learn from Nature to provide new approaches to solving some of the most challenging problems in sensing systems and materials science. An emphasis will be put on the mathematical foundation of bio-inspired pressure and temperature sensing membranes and shape perception algorithms in electrolocation and echolocation.
Objective
401-3900-16LAdvanced Topics in Discrete Optimization Information Restricted registration - show details
Number of participants limited to 26.
W4 credits2SD. Adjiashvili, S. Chestnut
AbstractIn this seminar we will discuss selected topics in discrete optimization. The main focus is on modern approaches to combinatorial optimization, including linear programming and polyhedral methods. Additionally, the topics of linear and integer programming theory will be discussed.
ObjectiveThe goal of the seminar is twofold. On the one hand, the students will learn and practice presenting scientific papers to an audience. On the other hand, the students will be exposed to cutting-edge research in the field of combinatorial optimization. An active participation in the seminar should allow the student to later read and understand a paper in the topic of discrete optimization independently. Students intending to do a project in optimization are strongly encouraged to participate.
ContentThe selected topics will cover various classical and modern results in combinatorial optimization, focusing on papers that present important modern polyhederal tools.
Lecture notesThis seminar has no script.
LiteratureThe learning material will be in the form of scientific papers.
Prerequisites / NoticeRequirements: Mathematical Optimization or Introduction to Mathematical Optimization (or equivalent course) strongly suggested.
252-4102-00LSeminar on Randomized Algorithms and Probabilistic MethodsW2 credits2SA. Steger
AbstractThe aim of the seminar is to study papers which bring the students to the forefront of today's research topics. This semester we will study selected papers of the conference Symposium on Discrete Algorithms (SODA17).
ObjectiveRead papers from the forefront of today's research; learn how to give a scientific talk.
Prerequisites / NoticeThe seminar is open for both students from mathematics and students from computer science. As prerequisite we require that you passed the course Randomized Algorithms and Probabilistic Methods (or equivalent, if you come from abroad).
263-4203-00LGeometry: Combinatorics and Algorithms Information W2 credits2SB. Gärtner, M. Hoffmann, E. Welzl
AbstractThis seminar complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent.
ObjectiveEach student is expected to read, understand, and elaborate on a selected research paper. To this end, (s)he should give a 45-min. presentation about the paper. The process includes

* getting an overview of the related literature;
* understanding and working out the background/motivation:
why and where are the questions addressed relevant?
* understanding the contents of the paper in all details;
* selecting parts suitable for the presentation;
* presenting the selected parts in such a way that an audience
with some basic background in geometry and graph theory can easily understand and appreciate it.
ContentThis seminar is held once a year and complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent. The seminar is a good preparation for a master, diploma, or semester thesis in the area.
Prerequisites / NoticePrerequisite: Successful participation in the course "Geometry: Combinatorics & Algorithms" (takes place every HS) is required.
Semester Papers
There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again.
NumberTitleTypeECTSHoursLecturers
401-3750-01LSemester Paper Restricted registration - show details
No direct enrolment to this course unit in myStudies. Please fill in the online application form.
Requirements and application form under www.math.ethz.ch/intranet/students/study-administration/theses.html
(Afterwards the enrolment will be done by the Study Administration.)
W8 credits11AProfessors
AbstractSemester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing.
Objective
Prerequisites / NoticeThere are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again.
401-3750-02LSemester Paper Restricted registration - show details
No direct enrolment to this course unit in myStudies. Please fill in the online application form.
Requirements and application form under www.math.ethz.ch/intranet/students/study-administration/theses.html
(Afterwards the enrolment will be done by the Study Administration.)
W8 credits11AProfessors
AbstractSemester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing.
Objective
Prerequisites / NoticeThere are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again.
401-3750-03LSemester Paper Restricted registration - show details
No direct enrolment to this course unit in myStudies. Please fill in the online application form.
Requirements and application form under www.math.ethz.ch/intranet/students/study-administration/theses.html
(Afterwards the enrolment will be done by the Study Administration.)
W8 credits11AProfessors
AbstractSemester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing.
Objective
Prerequisites / NoticeThere are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again.
GESS Science in Perspective
» Recommended Science in Perspective (Type B) for D-MATH
» see Science in Perspective: Type A: Enhancement of Reflection Capability
» see Science in Perspective: Language Courses ETH/UZH
Master's Thesis
NumberTitleTypeECTSHoursLecturers
401-2000-00LScientific Works in Mathematics Information
Target audience:
Third year Bachelor students;
Master students who cannot document to have received an adequate training in working scientifically.

Mandatory for all Bachelor and Master students with matriculation in the autumn semester 2014 or later.
Directive Link
O0 creditsE. Kowalski
AbstractIntroduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)
ObjectiveLearn the basic standards of scientific works in mathematics.
Content- Types of mathematical works
- Publication standards in pure and applied mathematics
- Data handling
- Ethical issues
- Citation guidelines
Lecture notesMoodle of the Mathematics Library: https://moodle-app2.let.ethz.ch/course/view.php?id=519
Prerequisites / NoticeThis course is completed by the optional course "Recherchieren in der Mathematik" (held in German) by the Mathematics Library. For more details see: http://www.math.ethz.ch/library/services/schulungen

Directive Link
401-4990-00LMaster's Thesis Restricted registration - show details
Only students who fulfil the following criteria are allowed to begin with their master thesis:
a. successful completion of the bachelor programme;
b. fulfilling of any additional requirements necessary to gain admission to the master programme.

No direct enrolment to this course unit in myStudies. Please fill in the online application form.
Requirements and application form under www.math.ethz.ch/intranet/students/study-administration/theses.html
(Afterwards the enrolment will be done by the Study Administration.)
O30 credits57DProfessors
AbstractThe master's thesis concludes the study programme. Writing up the master's thesis allows students to independently produce a major piece of work on a mathematical topic. It generally involves consulting the literature, solving any ensuing problems, and putting together the results in writing.
Objective
Additional Courses
NumberTitleTypeECTSHoursLecturers
401-5000-00LZurich Colloquium in Mathematics Information E-0 creditsP. L. Bühlmann, M. Burger, S. Mishra, R. Pandharipande, University lecturers
AbstractThe lectures try to give an overview of "what is going on" in important areas of contemporary mathematics, to a wider non-specialised audience of mathematicians.
Objective
401-5990-00LZurich Graduate Colloquium Information E-0 credits1KA. Iozzi, University lecturers
AbstractThe Graduate Colloquium is an informal seminar aimed at graduate students and postdocs whose purpose is to provide a forum for communicating one's interests and thoughts in mathematics.
Objective
401-5110-00LNumber Theory Seminar Information E-0 credits1KÖ. Imamoglu, P. S. Jossen, E. Kowalski, P. D. Nelson, R. Pink, G. Wüstholz
AbstractResearch colloquium
ObjectiveTalks on various topics of current research.
ContentResearch seminar in algebra, number theory and geometry. This seminar is aimed in particular to members of the research groups in these areas and their graduate students.
401-5530-00LGeometry Seminar Information E-0 credits1KM. Burger, M. Einsiedler, A. Iozzi, U. Lang, A. Sisto, University lecturers
AbstractResearch colloquium
Objective
401-5350-00LAnalysis Seminar Information E-0 credits1KM. Struwe, A. Carlotto, F. Da Lio, A. Figalli, N. Hungerbühler, T. Kappeler, T. Rivière, D. A. Salamon
AbstractResearch colloquium
Objective
ContentResearch seminar in Analysis
401-5580-00LSymplectic Geometry Seminar Information E-0 credits2KD. A. Salamon, A. Cannas da Silva
AbstractResearch colloquium
Objective
401-5330-00LTalks in Mathematical Physics Information E-0 credits1KA. Cattaneo, G. Felder, M. Gaberdiel, G. M. Graf, C. A. Keller, H. Knörrer, T. H. Willwacher, University lecturers
AbstractResearch colloquium
Objective
ContentForschungsseminar mit wechselnden Themen aus dem Gebiet der mathematischen Physik.
401-5650-00LZurich Colloquium in Applied and Computational Mathematics Information E-0 credits2KR. Abgrall, R. Alaifari, H. Ammari, U. S. Fjordholm, A. Jentzen, S. Mishra, S. Sauter, C. Schwab
AbstractResearch colloquium
Objective
401-5600-00LSeminar on Stochastic Processes Information E-0 credits1KJ. Bertoin, A. Nikeghbali, P. Nolin, B. D. Schlein, A.‑S. Sznitman, V. Tassion, W. Werner
AbstractResearch colloquium
Objective
401-5620-00LResearch Seminar on Statistics Information E-0 credits2KP. L. Bühlmann, L. Held, T. Hothorn, D. Kozbur, M. H. Maathuis, N. Meinshausen, S. van de Geer, M. Wolf
AbstractResearch colloquium
Objective
401-5640-00LZüKoSt: Seminar on Applied Statistics Information E-0 credits1KM. Kalisch, P. L. Bühlmann, R. Furrer, L. Held, T. Hothorn, M. H. Maathuis, M. Mächler, L. Meier, N. Meinshausen, M. Robinson, C. Strobl, S. van de Geer
Abstract5 to 6 talks on applied statistics.
ObjectiveKennenlernen von statistischen Methoden in ihrer Anwendung in verschiedenen Gebieten, besonders in Naturwissenschaft, Technik und Medizin.
ContentIn 5-6 Einzelvorträgen pro Semester werden Methoden der Statistik einzeln oder überblicksartig vorgestellt, oder es werden Probleme und Problemtypen aus einzelnen Anwendungsgebieten besprochen.
3 bis 4 der Vorträge stehen in der Regel unter einem Semesterthema.
Lecture notesBei manchen Vorträgen werden Unterlagen verteilt.
Eine Zusammenfassung ist kurz vor den Vorträgen im Internet unter http://stat.ethz.ch/talks/zukost abrufbar.
Ankündigunen der Vorträge werden auf Wunsch zugesandt.
Prerequisites / NoticeDies ist keine Vorlesung. Es wird keine Prüfung durchgeführt, und es werden keine Kreditpunkte vergeben.
Nach besonderem Programm. Koordinator M. Kalisch, Tel. 044 632 3435
Lehrsprache ist Englisch oder Deutsch je nach ReferentIn.
Course language is English or German and may depend on the speaker.
401-5910-00LTalks in Financial and Insurance Mathematics Information E-0 credits1KP. Cheridito, P. Embrechts, M. Schweizer, M. Soner, J. Teichmann, M. V. Wüthrich
AbstractResearch colloquium
ObjectiveIntroduction to current research topics in "Insurance Mathematics and Stochastic Finance".
Contenthttps://www.math.ethz.ch/imsf/courses/talks-in-imsf.html
401-5900-00LOptimization Seminar Information E-0 credits1KR. Zenklusen
AbstractLectures on current topics in optimization.
ObjectiveThis lecture series introduces graduate students to ongoing research activities (including applications) in the domain of optimization.
ContentThis seminar is a forum for researchers interested in optimization theory and its applications. Speakers, invited from both academic and non-academic institutions, are expected to stimulate discussions on theoretical and applied aspects of optimization and related subjects. The focus is on efficient (or practical) algorithms for continuous and discrete optimization problems, complexity analysis of algorithms and associated decision problems, approximation algorithms, mathematical modeling and solution procedures for real-world optimization problems in science, engineering, industries, public sectors etc.
402-0101-00LThe Zurich Physics Colloquium Information E-0 credits1KR. Renner, G. Aeppli, C. Anastasiou, N. Beisert, G. Blatter, S. Cantalupo, M. Carollo, C. Degen, G. Dissertori, K. Ensslin, T. Esslinger, J. Faist, M. Gaberdiel, G. M. Graf, R. Grange, J. Home, S. Huber, A. Imamoglu, P. Jetzer, S. Johnson, U. Keller, K. S. Kirch, S. Lilly, L. M. Mayer, J. Mesot, B. Moore, D. Pescia, A. Refregier, A. Rubbia, K. Schawinski, T. C. Schulthess, M. Sigrist, A. Vaterlaus, R. Wallny, A. Wallraff, W. Wegscheider, A. Zheludev, O. Zilberberg
AbstractResearch colloquium
Objective
Prerequisites / NoticeOccasionally, talks may be delivered in German.
251-0100-00LComputer Science Colloquium Information E-0 credits2KLecturers
AbstractInvited talks on the entire spectrum of Computer Science. External guests are welcome. A detailed program is published at the beginning of every semester.
Objective
ContentEingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.
252-4202-00LSeminar in Theoretical Computer Science Information E-2 credits2SE. Welzl, B. Gärtner, M. Hoffmann, J. Lengler, A. Steger, B. Sudakov
AbstractPresentation of recent publications in theoretical computer science, including results by diploma, masters and doctoral candidates.
ObjectiveTo get an overview of current research in the areas covered by the involved research groups. To present results from the literature.
Course Units for Additional Admission Requirements
The courses below are only available for MSc students with additional admission requirements.
NumberTitleTypeECTSHoursLecturers
406-2004-AALAlgebra II
Enrolment 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 credits11RL. Halbeisen
AbstractGalois theory and Representations of finite groups, algebras.

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
ObjectiveIntroduction to fundamentals of Galois theory, and representation theory of finite groups and algebras
ContentFundamentals of Galois theory
Representation theory of finite groups and algebras
LiteratureS. Lang, Algebra, Springer Verlag
B.L. van der Waerden: Algebra I und II, Springer Verlag
I.R. Shafarevich, Basic notions of algebra, Springer verlag
G. Mislin: Algebra I, vdf Hochschulverlag
U. Stammbach: Algebra, in der Polybuchhandlung erhältlich
I. Stewart: Galois Theory, Chapman Hall (2008)
G. Wüstholz, Algebra, vieweg-Verlag, 2004
J-P. Serre, Linear representations of finite groups, Springer Verlag
Prerequisites / NoticeAlgebra I
406-2005-AALAlgebra I and II
Enrolment 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 credits26RL. Halbeisen
AbstractIntroduction 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.
Objective
ContentBasic 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
LiteratureS. Lang, Algebra, Springer Verlag
B.L. van der Waerden: Algebra I und II, Springer Verlag
I.R. Shafarevich, Basic notions of algebra, Springer verlag
G. Mislin: Algebra I, vdf Hochschulverlag
U. Stammbach: Algebra, in der Polybuchhandlung erhältlich
I. Stewart: Galois Theory, Chapman Hall (2008)
G. Wüstholz, Algebra, vieweg-Verlag, 2004
J-P. Serre, Linear representations of finite groups, Springer Verlag
406-2284-AALMeasure and Integration
Enrolment 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 credits13RM. Schweizer
AbstractIntroduction 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
ObjectiveBasic acquaintance with the abstract theory of measure and integration
ContentIntroduction 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 notesno lecture notes
Literature1. 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 / NoticeThe precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2303-AALComplex Analysis
Enrolment 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 credits13RR. Pandharipande
AbstractComplex 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.
Objective
LiteratureL. 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 / NoticeThe precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2554-AALTopology
Enrolment 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 credits13RW. Werner
AbstractTopological and metric spaces, continuity, connectedness, compactness, product and quotient spaces, separation axioms, quotient spaces, Baire category, homotopy, fundamental group, covering spaces.
ObjectiveCover the basic notions of set-theoretic topology.
Lecture notesSee lecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-2554-00L/
LiteratureKlaus Jänich: Topologie (Springer)
http://link.springer.com/book/10.1007/978-3-662-10575-7
Boto von Querenburg: Mengentheoretische Topologie (Springer)
http://link.springer.com/book/10.1007/978-3-642-56860-2
Lynn Arthur Steen, J. Arthur Seebach Jr.: Counterexamples in Topology (Springer)
http://link.springer.com/book/10.1007/978-1-4612-6290-9
Nicolas Bourbaki: Topologie Générale, chapitres 1 à 10 (Hermann, Paris) oder General Topology (Chapters 1-10) (Springer)
Ryszard Engelking: General topology. Heldermann Verlag, Berlin, 1989.
Prerequisites / NoticeThe precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2604-AALProbability and Statistics
Enrolment 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-7 credits15RS. van de Geer
AbstractIntroduction to probability and statistics with many examples, based on chapters from the books "Probability and Random Processes" by G. Grimmett and D. Stirzaker and "Mathematical Statistics and Data Analysis" by J. Rice.
ObjectiveThe goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. In addition to a mathematically rigorous treatment, also an intuitive understanding and familiarity with the ideas behind the definitions are emphasized. Measure theory is not used systematically, but it should become clear why and where measure theory is needed.
ContentProbability:
Chapters 1-5 (Probabilities and events, Discrete and continuous random variables, Generating functions) and Sections 7.1-7.5 (Convergence of random variables) from the book "Probability and Random Processes". Most of this material is also covered in Chap. 1-5 of "Mathematical Statistics and Data Analysis", on a slightly easier level.

Statistics:
Sections 8.1 - 8.5 (Estimation of parameters), 9.1 - 9.4 (Testing Hypotheses), 11.1 - 11.3 (Comparing two samples) from "Mathematical Statistics and Data Analysis".
LiteratureGeoffrey Grimmett and David Stirzaker, Probability and Random Processes.
3rd Edition. Oxford University Press, 2001.

John A. Rice, Mathematical Statistics and Data Analysis, 3rd edition.
Duxbury Press, 2006.
406-3461-AALFunctional Analysis I
Enrolment 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-10 credits21RM. Struwe
AbstractBaire category; Banach spaces and linear operators; Fundamental theorems: Open Mapping Theorem, Closed Range Theorem, Uniform Boundedness Principle, Hahn-Banach Theorem; Convexity; reflexive spaces; Spectral theory.
Objective
Prerequisites / NoticeThe precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-3621-AALFundamentals of Mathematical Statistics
Enrolment 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-10 credits21RF. Balabdaoui
AbstractThe course covers the basics of inferential statistics.
Objective