Suchergebnis: Katalogdaten im Frühjahrssemester 2021

Nummer	Titel	Typ	ECTS	Umfang	Dozierende
Mathematik Master
Kernfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Kernfächer aus Bereichen der reinen Mathematik
401-3002-12L	Algebraic Topology II	W	8 KP	4G	P. Biran
Kurzbeschreibung	This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality.
Lernziel
Literatur	1) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 2) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. The book can be downloaded for free at: http://www.math.cornell.edu/~hatcher/AT/ATpage.html 3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / Besonderes	General topology, linear algebra, singular homology 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.
401-3226-00L	Symmetric Spaces	W	8 KP	4G	A. Iozzi
Kurzbeschreibung	* Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples * Symmetric spaces of non-compact type: flats and rank, roots and root spaces * Iwasawa decomposition, Weyl group, Cartan decomposition * Hints of the geometry at infinity of SL(n,R)/SO(n).
Lernziel	Learn the basics of symmetric spaces
401-3532-08L	Differential Geometry II	W	10 KP	4V + 1U	W. Merry
Kurzbeschreibung	This is a continuation course of Differential Geometry I. Topics covered include: - Connections and curvature, - Riemannian geometry, - Gauge theory and Chern-Weil theory.
Lernziel
Skript	I will produce full lecture notes, available on my website: https://www.merry.io/courses/differential-geometry/
Literatur	There are many excellent textbooks on differential geometry. A friendly and readable book that contains everything covered in Differential Geometry I is: John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag. For Differential Geometry II, the textbooks: - S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley, - I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP, are both excellent. The monograph - A. L. Besse "Einstein Manifolds", (1987), Springer, gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.)
Voraussetzungen / Besonderes	Familiarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). Lecture notes for Differential Geometry I can be found on my website.
401-3462-00L	Functional Analysis II	W	10 KP	4V + 1U	A. Carlotto
Kurzbeschreibung	Sobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles.
Lernziel	Acquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods for studying elliptic boundary value problems.
Literatur	Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14. Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi. Lectures on elliptic partial differential equations. Springer - Edizioni della Normale, Pisa, 2018. David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001. Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011. Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003.
Voraussetzungen / Besonderes	Functional Analysis I plus a solid background in measure theory, Lebesgue integration and L^p spaces.
401-8142-21L	Algebraic Geometry II (University of Zurich) Der Kurs muss direkt an der UZH belegt werden. UZH Modulkürzel: MAT517 Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/deadlines.html	W	9 KP	4V + 1U	Uni-Dozierende
Kurzbeschreibung	We continue the development of scheme theory. Among the topics that will be discussed are: properties of schemes and their morphisms (flatness, smoothness), coherent modules, cohomology, etc.
Lernziel

Kernfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3052-10L	Graph Theory	W	10 KP	4V + 1U	B. Sudakov
Kurzbeschreibung	Basics, 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
Lernziel	The 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.
Skript	Lecture will be only at the blackboard.
Literatur	West, D.: "Introduction to Graph Theory" Diestel, R.: "Graph Theory" Further literature links will be provided in the lecture.
Voraussetzungen / Besonderes	Students are expected to have a mathematical background and should be able to write rigorous proofs.
401-3642-00L	Brownian Motion and Stochastic Calculus	W	10 KP	4V + 1U	W. Werner
Kurzbeschreibung	This 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.
Lernziel	This 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.
Skript	Lecture notes will be distributed in class.
Literatur	- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016). - 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).
Voraussetzungen / Besonderes	Familiarity 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).
401-3632-00L	Computational Statistics	W	8 KP	3V + 1U	M. Mächler
Kurzbeschreibung	We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R.
Lernziel	The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R.
Inhalt	See the class website
Voraussetzungen / Besonderes	At least one semester of (basic) probability and statistics. Programming experience is helpful but not required.
401-3602-00L	Applied Stochastic Processes	W	8 KP	3V + 1U	V. Tassion
Kurzbeschreibung	Poisson-Prozesse; Erneuerungsprozesse; Markovketten in diskreter und in stetiger Zeit; einige Beispiele und Anwendungen.
Lernziel	Stochastische Prozesse dienen zur Beschreibung der Entwicklung von Systemen, die sich in einer zufälligen Weise entwickeln. In dieser Vorlesung bezieht sich die Entwicklung auf einen skalaren Parameter, der als Zeit interpretiert wird, so dass wir die zeitliche Entwicklung des Systems studieren. Die Vorlesung präsentiert mehrere Klassen von stochastischen Prozessen, untersucht ihre Eigenschaften und ihr Verhalten und zeigt anhand von einigen Beispielen, wie diese Prozesse eingesetzt werden können. Die Hauptbetonung liegt auf der Theorie; "applied" ist also im Sinne von "applicable" zu verstehen.
Literatur	R. 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)
Voraussetzungen / Besonderes	Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).
401-3652-00L	Numerical Methods for Hyperbolic Partial Differential Equations	W	10 KP	4V + 1U	A. Ruf
Kurzbeschreibung	This 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.
Lernziel	The goal of this course is familiarity with the fundamental ideas and mathematical consideration underlying modern numerical methods for conservation laws and wave equations.
Inhalt	* 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.
Skript	Lecture slides will be made available to participants. However, additional material might be covered in the course.
Literatur	H. 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.
Voraussetzungen / Besonderes	Having 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"

Wahlfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Wahlfächer aus Bereichen der reinen Mathematik
Auswahl: Algebra, Zahlentheorie, Topologie, diskrete Mathematik, Logik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4116-12L	Lectures on Drinfeld Modules	W	6 KP	3V	R. Pink
Kurzbeschreibung	Drinfeld modules: Basic theory, analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations, endomorphism rings, etc.
Lernziel
Inhalt	A central role in the arithmetic of fields of positive characteristic p is played by the Frobenius map x ---> x^p. The theory of Drinfeld modules exploits this map in a systematic fashion. Drinfeld modules of rank 1 can be viewed as analogues of the multiplicative group and are used in the class field theory of global function fields. Drinfeld modules of arbitrary rank possess a rich theory which has many aspects in common with that of elliptic curves, including analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations. A full understanding of Drinfeld modules requires some knowledge in the arithmetic of function fields and, for comparison, the arithmetic of elliptic curves, which cannot all be presented in the framework of this course. Relevant results from these areas will be presented only cursorily when they are needed, but a fair amount of the theory can be developed without them.
Literatur	Drinfeld, V. G.: Elliptic modules (Russian), Mat. Sbornik 94 (1974), 594--627, translated in Math. USSR Sbornik 23 (1974), 561--592. Deligne, P., Husemöller, D: Survey of Drinfeld modules, Contemp. Math. 67, 1987, 25-91. Goss, D.: Basic structures in function field arithmetic. Springer-Verlag, 1996. Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in Alden-Biesen, September 9¿14, 1996. Edited by E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. Thakur, Dinesh S.: Function field arithmetic. World Scientific Publishing Co., Inc., River Edge, NJ, 2004. Further literature will be indicated during the course
401-3109-65L	Probabilistic Number Theory	W	8 KP	4G	E. Kowalski
Kurzbeschreibung	The course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.
Lernziel	The goal of the course is to present some results of probabilistic number theory in a unified manner.
Inhalt	The main concepts will be presented in parallel with the proof of a few main theorems: (1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions; (2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line; (3) the Chebychev bias for primes in arithmetic progressions; (4) functional limit theorems for the paths of partial sums of families of exponential sums.
Skript	The lecture notes for the class are available at https://www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf
Voraussetzungen / Besonderes	Prerequisites: Complex analysis, measure and integral, and at least the basic language of probability theory (the main concepts, such as convergence in law, will be recalled). Some knowledge of number theory is useful but the main results will also be summarized.
401-3362-21L	Spectral Theory of Eisenstein Series	W	4 KP	2V	P. D. Nelson
Kurzbeschreibung	We plan to discuss the basic theory of Eisenstein series and the spectral decomposition of the space of automorphic forms, with focus on the groups GL(2) and GL(n).
Lernziel
Voraussetzungen / Besonderes	Some familiarity with basics on Lie groups and functional analysis would be helpful, and some prior exposure to modular forms or homogeneous spaces may provide useful motivation.
401-3058-00L	Kombinatorik I	W	4 KP	2G	N. Hungerbühler
Kurzbeschreibung	Der Kurs Kombinatorik I und II ist eine Einführung in die abzählende Kombinatorik.
Lernziel	Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.
Inhalt	Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele.
Voraussetzungen / Besonderes	Wer 401-3052-00L Kombinatorik (letztmals im FS 2008 gelesen) für den Bachelor- oder Master-Studiengang Mathematik anrechnen lässt, darf 401-3058-00L Kombinatorik I nur noch fürs Mathematik Lehrdiplom oder fürs Didaktik-Zertifikat Mathematik anrechnen lassen.

Auswahl: Geometrie
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4118-21L	Spectral Theory of Hyperbolic Surfaces	W	4 KP	2V	C. Burrin
Kurzbeschreibung	The Laplacian plays a prominent role in many parts of mathematics. On a flat surface like the torus, understanding its spectrum is the topic of Fourier analysis, whose 19th century development allowed to solve the heat and wave equations. On the sphere, one studies spherical harmonics. In this course, we will study the spectrum of hyperbolic surfaces and its Maass forms (eigenfunctions).
Lernziel	We will start from scratch, with an overview of hyperbolic geometry and harmonic analysis on the hyperbolic plane. The objectives are to prove the spectral theorem and Selberg's trace formula, and explore applications in geometry and number theory.
Inhalt	Tentative syllabus: Hyperbolic geometry (the hyperbolic plane and Fuchsian groups) Construction of arithmetic hyperbolic surfaces Harmonic analysis on the hyperbolic plane The spectral theorem Selberg's trace formula Applications in geometry (isoperimetric inequalities, geodesic length spectrum) and number theory (links to the Riemann zeta function and Riemann hypothesis) Possible further topics (if time permits): Eisenstein series Explicit constructions of Maass forms (after Maass) A special case of the Jacquet-Langlands correspondence (after the exposition of Bergeron, see references)
Literatur	Nicolas Bergeron, The Spectrum of Hyperbolic Surfaces, Springer Universitext 2011. Armand Borel, Automorphic forms on SL(2,R), Cambridge University Press 1997. Peter Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser 1992. Henryk Iwaniec, Spectral methods of automorphic forms. Graduate studies in mathematics, AMS 2002.
Voraussetzungen / Besonderes	Knowledge of the material covered in the first two years of bachelor studies is assumed. Prior knowledge of differential geometry, functional analysis, or Riemann surfaces is not required.
401-4206-17L	Groups Acting on Trees	W	6 KP	3G	B. Brück
Kurzbeschreibung	As a main theme, we will see 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.
Lernziel	Learn basics of Bass-Serre theory; get to know concepts from geometric group theory.
Inhalt	As a mathematical object, a tree is a graph without any loops. It turns out that if a group acts on such an object, the algebraic structure of the group has a nice description in terms of the combinatorics of the graph. In particular, groups acting on trees can be decomposed in a certain way into simpler pieces.These decompositions can be described combinatorially, but are closely related to concepts from topology such as fundamental groups and covering spaces. This interplay between (elementary) concepts of algebra, combinatorics and geometry/topology is typical for geometric group theory. The course can also serve as an introduction to basic concepts of this field. Topics that will be covered in the lecture include: - Trees and their automorphisms - Different characterisations of free groups - Amalgamated products and HNN extensions - Graphs of groups - Kurosh's theorem on subgroups of free (amalgamated) products
Literatur	J.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9 O. Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+177 pp. ISBN: 978-3-03719-041-8 C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Voraussetzungen / Besonderes	Basic knowledge of group theory; being familiar with fundamental groups (e.g. the Seifert-van-Kampen Theorem) and covering theory is definitely helpful, although not strictly necessary. In particular, the standard material of the first two years of the Mathematics Bachelor is sufficient.
401-3056-00L	Endliche Geometrien I Findet dieses Semester nicht statt.	W	4 KP	2G	N. Hungerbühler
Kurzbeschreibung	Endliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate.
Lernziel	Endliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne.
Inhalt	Endliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne
Literatur	- Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries.
401-3574-61L	Introduction to Knot Theory Findet dieses Semester nicht statt.	W	6 KP	3G
Kurzbeschreibung	Introduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.
Lernziel	The aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.
Inhalt	Definition of a knot and of equivalent knots. Definition of a knot invariant and some elementary examples. Various operations on knots. Knot polynomials (Jones, ev. Alexander.....)
Literatur	An extensive bibliography will be handed out in the course.
Voraussetzungen / Besonderes	Prerequisites are some elementary knowledge of algebra and topology.

Auswahl: Analysis
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4422-21L	An Introduction to the Calculus of Variations	W	4 KP	2V	A. Figalli
Kurzbeschreibung	Calculus of variations is a fundamental tool in mathematical analysis, used to investigate the existence, uniqueness, and properties of minimizers to variational problems. Classic examples include, for instance, the existence of the shortest curve between two points, the equilibrium shape of an elastic membrane, and so on.
Lernziel
Inhalt	In the course, we will study both 1-dimensional and multi-dimensional problems.
Voraussetzungen / Besonderes	Basic knowledge of Sobolev spaces is important, so some extra additional readings would be required for those unfamiliar with the topic.
401-3378-19L	Entropy in Dynamics	W	8 KP	4G	M. Einsiedler
Kurzbeschreibung	Definition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points. Measure rigidity for commuting maps on the circle group.
Lernziel	The course will lead to a firm understanding of measure theoretic dynamical entropy and its applications within dynamics. We will start with the basic properties of (conditional) entropy, relate it to the question of effective coding techniques, discuss and prove the Shannon-McMillan-Breiman theorem that is also known as the ergodic theorem for entropy. Moreover, we will discuss a topological counter part and relate this topological entropy to the measure theoretic entropy by the variational principle. We will use these methods to classify certain natural homogeneous measures, prove equidistribution of periodic points on compact quotients of hyperbolic surfaces, and establish a measure rigidity theorem for commuting maps on the circle group.
Skript	Entropy book under construction, available online under https://tbward0.wixsite.com/books/entropy
Voraussetzungen / Besonderes	No prior knowledge of dynamical systems will be assumed but measure theory will be assumed and very important. Doctoral students are welcome to attend the course for 2KP.

Auswahl: Weitere Gebiete
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3502-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	2 KP	4A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	3 KP	6A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	4 KP	9A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel

Wahlfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Auswahl: Numerische Mathematik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4658-00L	Computational Methods for Quantitative Finance: PDE Methods	W	6 KP	3V + 1U	C. Marcati, A. Stein
Kurzbeschreibung	Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB and Python programming and knowledge of numerical mathematics at ETH BSc level.
Lernziel	Introduce 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 and Python. Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.
Inhalt	1. 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.
Skript	There will be english lecture notes as well as MATLAB or Python software for registered participants in the course.
Literatur	Main reference (course text): N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013. Supplementary texts: R. 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.
Voraussetzungen / Besonderes	Knowledge of Numerical Analysis/ Scientific Computing Techniques corresponding roughly to BSc MATH or BSc RW/CSE at ETH is expected. Basic programming skills in MATLAB or Python are required for the exercises, and are _not_ taught in this course.
401-4656-21L	Deep Learning in Scientific Computing Aimed at students in a Master's Programme in Mathematics, Engineering and Physics.	W	6 KP	2V + 1U	S. Mishra
Kurzbeschreibung	Machine Learning, particularly deep learning is being increasingly applied to perform, enhance and accelerate computer simulations of models in science and engineering. This course aims to present a highly topical selection of themes in the general area of deep learning in scientific computing, with an emphasis on the application of deep learning algorithms for systems, modeled by PDEs.
Lernziel	The objective of this course will be to introduce students to advanced applications of deep learning in scientific computing. The focus will be on the design and implementation of algorithms as well as on the underlying theory that guarantees reliability of the algorithms. We will provide several examples of applications in science and engineering where deep learning based algorithms outperform state of the art methods.
Inhalt	A selection of the following topics will be presented in the lectures. 1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with high-dimensional state and parameter spaces. 2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets. 3. Theoretical Foundations: Universal approximation properties of the Neural networks, Bias-Variance decomposition, Bounds on approximation and generalization errors. 4. Supervised deep learning for solutions fields and observables of high-dimensional parametric PDEs. Use of low-discrepancy sequences and multi-level training to reduce generalization error. 5. Uncertainty Quantification for PDEs with supervised learning algorithms. 6. Deep Neural Networks as Reduced order models and prediction of solution fields. 7. Active Learning algorithms for PDE constrained optimization. 8. Recurrent Neural Networks and prediction of time series for dynamical systems. 9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to high-dimensional PDEs. 10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation. All the algorithms will be illustrated on a variety of PDEs: diffusion models, Black-Scholes type PDEs from finance, wave equations, Euler and Navier-Stokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others.
Skript	Lecture notes will be provided at the end of the course.
Literatur	All the material in the course is based on research articles written in last 1-2 years. The relevant references will be provided.
Voraussetzungen / Besonderes	The students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications. Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful.
401-4652-21L	Nonlocal Inverse Problems	W	4 KP	2V	J. Railo
Kurzbeschreibung	This course is an introduction to the Calderón problem and nonlocal inverse problems for the fractional Schrödinger equation. These are examples of nonlinear inverse problems. The classical Calderón problem models electrical impedance tomography (EIT) and fractional operators appear, for example, in some mathematical models in finance.
Lernziel	Students become familiar with the Calderón problem and some nonlocal phenomena related to the fractional Laplacian. Advanced students should be able to read research articles on the fractional Calderón problems after the course.
Inhalt	In the beginning of the course, we will introduce some basic theory for the classical Calderón problem. The focus of the course will be in the study of nonlocal inverse problems for the fractional Schrödinger equation with lower order perturbations. We discuss necessary preliminaries on Sobolev spaces, Fourier analysis, functional analysis and theory of PDEs. Our scope will be in the uniqueness properties. Classical Calderón problem (about 1/3): Conductivity and Schrödinger equations, Dirichlet-to-Neumann maps, Cauchy data, and related boundary value inverse problems. The methods include, for example, complex geometric optics (CGO) solutions. Fractional Calderón problem (about 2/3): Nonlocal unique continuation principles (UCP), Runge approximation properties, and uniqueness for the fractional Calderón problem. The methods include, for example, Caffarelli-Silvestre extensions, the fractional Poincaré inequality and Riesz transforms.
Skript	Lecture notes and exercises
Literatur	1. M. Salo: Calderón problem. Lecture notes, University of Helsinki (2008). (Available at http://users.jyu.fi/~salomi/index.html.) 2. T. Ghosh, M. Salo, G. Uhlmann: The Calderón problem for the fractional Schrödinger equation. Analysis & PDE 13 (2020), no. 2, 455-475. 3. A. Rüland, M. Salo: The fractional Calderón problem: low regularity and stability. Nonlinear Analysis 193 (2020), special issue "Nonlocal and Fractional Phenomena", 111529. 4. Other literature will be specified in the course.
Voraussetzungen / Besonderes	Functional Analysis I & II or similar knowledge. Any additional knowledge of Fourier analysis, Sobolev spaces, distributions and PDEs will be an asset.
401-3426-21L	Time-Frequency Analysis	W	4 KP	2G	R. Alaifari
Kurzbeschreibung	This course gives a basic introduction to time-frequency analysis from the viewpoint of applied harmonic analysis.
Lernziel	By the end of the course students should be familiar with the concept of the short-time Fourier transform, the Bargmann transform, quadratic time-frequency representations (ambiguity function and Wigner distribution), Gabor frames and modulation spaces. The connection and comparison to time-scale representations will also be subject of this course.
Inhalt	Time-frequency analysis lies at the heart of many applications in signal processing and aims at capturing time and frequency information simultaneously (as opposed to the classical Fourier transform). This course gives a basic introduction that starts with studying the short-time Fourier transform and the special role of the Gauss window. We will visit quadratic representations and then focus on discrete time-frequency representations, where Gabor frames will be introduced. Later, we aim at a more quantitative analysis of time-frequency information through modulation spaces. At the end, we touch on wavelets (time-scale representation) as a counterpart to the short-time Fourier transform.
Literatur	Gröchenig, K. (2001). Foundations of time-frequency analysis. Springer Science & Business Media.
Voraussetzungen / Besonderes	Functional analysis, Fourier analysis, complex analysis, operator theory

Auswahl: Wahrscheinlichkeitstheorie, Statistik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4611-21L	Rough Path Theory	W	4 KP	2V	A. Allan, J. Teichmann
Kurzbeschreibung	The aim of this course is to provide an introduction to the theory of rough paths, with a particular focus on their integration theory and associated rough differential equations, and how the theory relates to and enhances the field of stochastic calculus.
Lernziel	Our first motivation will be to understand the limitations of classical notions of integration to handle paths of very low regularity, and to see how the rough integral succeeds where other notions fail. We will construct rough integrals and establish solutions of differential equations driven by rough paths, as well as the continuity of these objects with respect to the paths involved, and their consistency with stochastic integration and SDEs. Various applications and extensions of the theory will then be discussed.
Skript	Lecture notes will be provided by the lecturer.
Literatur	P. K. Friz and M. Hairer, A course on rough paths with an introduction to regularity structures, Springer (2014). P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths, Cambridge University Press (2010).
Voraussetzungen / Besonderes	The aim will be to make the course as self-contained as possible, but some knowledge of stochastic analysis is highly recommended. The course “Brownian Motion and Stochastic Calculus” would be ideal, but not strictly required.
401-4626-00L	Advanced Statistical Modelling: Mixed Models Findet dieses Semester nicht statt.	W	4 KP	2V	M. Mächler
Kurzbeschreibung	Mixed Models = (*\| generalized\| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms. In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences.
Lernziel	- Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models. - Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis: "fit -> interpret & diagnose -> modify the fit -> interpret & ...." - Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software.
Inhalt	The lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online. Inference (significance of factors, confidence intervals) will focus on the more realistic unbalanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference.
Skript	We will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples. These lecture notes and all R scripts are made available from https://github.com/mmaechler/MEMo
Literatur	(see web page and lecture notes)
Voraussetzungen / Besonderes	- We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..). Typically this means at least two classes of (math based) statistics, say 1. Intro to probability and statistics 2. (Applied) regression including Matrix-Vector notation Y = X b + E - Basic (1 semester) "Matrix calculus" / linear algebra is also assumed. - If familiarity with [R](https://www.r-project.org/) is not given, it should be acquired during the course (by the student on own initiative).
401-4627-00L	Empirical Process Theory and Applications	W	4 KP	2V	S. van de Geer
Kurzbeschreibung	Empirical process theory provides a rich toolbox for studying the properties of empirical risk minimizers, such as least squares and maximum likelihood estimators, support vector machines, etc.
Lernziel
Inhalt	In this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection.
401-4632-15L	Causality	W	4 KP	2G	C. Heinze-Deml
Kurzbeschreibung	In 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.
Lernziel	After 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
Voraussetzungen / Besonderes	Prerequisites: basic knowledge of probability theory and regression
401-6102-00L	Multivariate Statistics Findet dieses Semester nicht statt.	W	4 KP	2G	keine Angaben
Kurzbeschreibung	Multivariate 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.
Lernziel	After 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
Inhalt	Visualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis
Skript	None
Literatur	The course will be based on class notes and books that are available electronically via the ETH library.
Voraussetzungen / Besonderes	Target 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.
401-4637-67L	On Hypothesis Testing	W	4 KP	2V	F. Balabdaoui
Kurzbeschreibung	This course is a review of the main results in decision theory.
Lernziel	The goal of this course is to present a review for the most fundamental results in statistical testing. This entails reviewing the Neyman-Pearson Lemma for simple hypotheses and the Karlin-Rubin Theorem for monotone likelihood ratio parametric families. The students will also encounter the important concept of p-values and their use in some multiple testing situations. Further methods for constructing tests will be also presented including likelihood ratio and chi-square tests. Some non-parametric tests will be reviewed such as the Kolmogorov goodness-of-fit test and the two sample Wilcoxon rank test. The most important theoretical results will reproved and also illustrated via different examples. Four sessions of exercises will be scheduled (the students will be handed in an exercise sheet a week before discussing solutions in class).
Literatur	- Statistical Inference (Casella & Berger) - Testing Statistical Hypotheses (Lehmann and Romano)

Auswahl: Finanz- und Versicherungsmathematik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3629-00L	Quantitative Risk Management	W	4 KP	2V + 1U	P. Cheridito
Kurzbeschreibung	This course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, risk measures, extreme value theory, multivariate models, copulas, dependence structures and operational risk.
Lernziel	The goal is to learn the most important methods from probability theory and statistics used in financial risk modeling.
Inhalt	1. Introduction 2. Basic Concepts in Risk Management 3. Empirical Properties of Financial Data 4. Financial Time Series 5. Extreme Value Theory 6. Multivariate Models 7. Copulas and Dependence 8. Operational Risk
Skript	Course material is available on https://people.math.ethz.ch/~patrickc/qrm
Literatur	Quantitative 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
Voraussetzungen / Besonderes	The 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.
401-3923-00L	Selected Topics in Life Insurance Mathematics	W	4 KP	2V	M. Koller
Kurzbeschreibung	Stochastic 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
Lernziel
401-3917-00L	Stochastic Loss Reserving Methods	W	4 KP	2V	R. Dahms
Kurzbeschreibung	Loss 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.
Lernziel	Our 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.
Inhalt	We 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
Literatur	M. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Voraussetzungen / Besonderes	The 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.
401-3956-00L	Economic Theory of Financial Markets	W	4 KP	2V	M. V. Wüthrich
Kurzbeschreibung	This lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
Lernziel	This 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.
Inhalt	We 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
Voraussetzungen / Besonderes	The 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.
401-3936-00L	Data Analytics for Non-Life Insurance Pricing	W	4 KP	2V	C. M. Buser, M. V. Wüthrich
Kurzbeschreibung	We study statistical methods in supervised learning for non-life insurance pricing such as generalized linear models, generalized additive models, Bayesian models, neural networks, classification and regression trees, random forests and gradient boosting machines.
Lernziel	The student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction.
Inhalt	We present the following chapters: - generalized linear models (GLMs) - generalized additive models (GAMs) - neural networks - credibility theory - classification and regression trees (CARTs) - bagging, random forests and boosting
Skript	The lecture notes are available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2870308
Voraussetzungen / Besonderes	This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch Good knowledge in probability theory, stochastic processes and statistics is assumed.
401-4920-00L	Market-Consistent Actuarial Valuation Findet dieses Semester nicht statt.	W	4 KP	2V	M. V. Wüthrich
Kurzbeschreibung	Introduction 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.
Lernziel	Goal 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.
Inhalt	In 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)
Literatur	Market-Consistent Actuarial Valuation, 3rd edition. Wüthrich, M.V. EAA Series, Springer 2016. ISBN: 978-3-319-46635-4 Wüthrich, M.V., Merz, M. Claims run-off uncertainty: the full picture. SSRN Manuscript ID 2524352 (2015). England, P.D, Verrall, R.J., Wüthrich, M.V. On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins. Insurance: Mathematics and Economics 85 (2019), 74-88. 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 Cheridito, P., Ery, J., Wüthrich, M.V. Assessing asset-liability risk with neural networks. Risks 8/1 (2020), article 16.
Voraussetzungen / Besonderes	The 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.
401-3888-00L	Introduction to Mathematical Finance Ein verwandter Kurs ist 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS-KP). Obwohl beide Kurse unabhängig voneinander belegt werden können, darf nur einer ans gesamte Mathematik-Studium (Bachelor und Master) angerechnet werden.	W	10 KP	4V + 1U	D. Possamaï
Kurzbeschreibung	This 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.
Lernziel	This 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.
Inhalt	This course focuses on discrete-time financial markets. It presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course is offered every year in the Spring semester. This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II. For an overview of courses offered in the area of mathematical finance, see Link.
Skript	The course is based on different parts from different textbooks as well as on original research literature. Lecture notes will not be available.
Literatur	Literature: Michael U. Dothan, "Prices in Financial Markets", Oxford University Press Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer
Voraussetzungen / Besonderes	A 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" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II. For an overview of courses offered in the area of mathematical finance, see Link.
401-3932-19L	Machine Learning in Finance	W	6 KP	3V + 1U	J. Teichmann
Kurzbeschreibung	The course will deal with the following topics with rigorous proofs and many coding excursions: Universal approximation theorems, Stochastic gradient Descent, Deep networks and wavelet analysis, Deep Hedging, Deep calibration, Different network architectures, Reservoir Computing, Time series analysis by machine learning, Reinforcement learning, generative adversersial networks, Economic games.
Lernziel
Voraussetzungen / Besonderes	Bachelor in mathematics, physics, economics or computer science.

Auswahl: Mathematische Physik, Theoretische Physik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4816-21L	Mathematical Aspects of Classical and Quantum Field Theory	W	8 KP	4V	M. Schiavina, Uni-Dozierende
Kurzbeschreibung	The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint. Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided.
Lernziel	The objective of this course is to expose master and graduate students in mathematics and physics to the mathematical foundations of classical and quantum field theory. The course will provide a solid mathematical foundation to essential topics in classical and quantum field theories, both useful to mathematics master and graduate students with an interest but no previous background in QFT, as well as for physics master and graduate students who want to focus on more formal aspects of field theory.
Inhalt	Abstract (long version) The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint. Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided. The course will feature relevant instances of field theories and sigma models, and it will provide a first introduction the the concepts of quantisation, from mechanics to field theory. Using scalar field theory and quantum electrodynamics as guideline, the course will present an overview of quantum field theory, focusing on its more mathematical aspects, including, if time permits, a modern approach to gauge theories and the renormalisation group. Content The course will start with an overview of geometric concepts that will be used throughout, such as graded differential geometry, as well as fiber and vector bundles. After brief review of classical mechanics, interpreted as a first example of a field theory, a thorough discussion of classical, local, Lagrangian field theory will follow, covering topics such as Noether’s Theorems, local and global symmetries. We will then present and discuss a number of examples from gauge theory. In the second part of the course, quantisation will be discussed. The main examples of the scalar field and electrodynamics will be used as a guideline for more general considerations on the quantisation of more general and involved field theories. In the last part of the course, we plan the discussion of modern approaches to quantisation of field theories with symmetries, renormalisation, and of the open challenges that arise.
402-0206-00L	Quantum Mechanics II Fachstudierende UZH müssen das Modul PHY351 direkt an der UZH buchen.	W	10 KP	3V + 2U	P. Jetzer
Kurzbeschreibung	Many-body quantum physics rests on symmetry considerations that lead to two kinds of particles, fermions and bosons. Formal techniques include Hartree-Fock theory and second-quantization techniques, as well as quantum statistics with ensembles. Few- and many-body systems include atoms, molecules, the Fermi sea, elastic chains, radiation and its interaction with matter, and ideal quantum gases.
Lernziel	Basic command of few- and many-particle physics for fermions and bosons, including second quantisation and quantum statistical techniques. Understanding of elementary many-body systems such as atoms, molecules, the Fermi sea, electromagnetic radiation and its interaction with matter, ideal quantum gases and relativistic theories.
Inhalt	The description of indistinguishable particles leads us to (exchange-) symmetrized wave functions for fermions and bosons. We discuss simple few-body problems (Helium atoms, hydrogen molecule) und proceed with a systematic description of fermionic many body problems (Hartree-Fock approximation, screening, correlations with applications on atomes and the Fermi sea). The second quantisation formalism allows for the compact description of the Fermi gas, of elastic strings (phonons), and the radiation field (photons). We study the interaction of radiation and matter and the associated phenomena of radiative decay, light scattering, and the Lamb shift. Quantum statistical description of ideal Bose and Fermi gases at finite temperatures concludes the program. If time permits, we will touch upon of relativistic one particle physics, the Klein-Gordon equation for spin-0 bosons and the Dirac equation describing spin-1/2 fermions.
Literatur	G. Baym, Lectures on Quantum Mechanics (Benjamin, Menlo Park, California, 1969) L.I. Schiff, Quantum Mechanics (Mc-Graw-Hill, New York, 1955) A. Messiah, Quantum Mechanics I & II (North-Holland, Amsterdam, 1976) E. Merzbacher, Quantum Mechanics (John Wiley, New York, 1998) C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics I & II (John Wiley, New York, 1977) P.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (Mc Graw-Hill, New York, 1965) A.L. Fetter and J.D. Walecka, Theoretical Mechanics of Particles and Continua (Mc Graw-Hill, New York, 1980) J.J. Sakurai, Modern Quantum Mechanics (Addison Wesley, Reading, 1994) J.J. Sakurai, Advanced Quantum mechanics (Addison Wesley) F. Gross, Relativistic Quantum Mechanics and Field Theory (John Wiley, New York, 1993)
Voraussetzungen / Besonderes	Basic knowledge of single-particle Quantum Mechanics
402-0844-00L	Quantum Field Theory II Studierende der UZH dürfen diese Lerneinheit nicht an der ETH belegen, sondern müssen das entsprechende Modul direkt an der UZH buchen.	W	10 KP	3V + 2U	N. Beisert
Kurzbeschreibung	The subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.
Lernziel	The goal of this course is to lay down the path integral formulation of quantum field theories and in particular to provide a solid basis for the study of non-abelian gauge theories and of the Standard Model
Inhalt	The 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 - the Goldstone theorem and the Higgs mechanism - gauge theories with spontaneous symmetry breaking and their quantization - renormalization of spontaneously broken gauge theories and quantum effective actions
Literatur	M.E. Peskin and D.V. Schroeder, "An introduction to Quantum Field Theory", Perseus (1995). S. Pokorski, "Gauge Field Theories" (2nd Edition), Cambridge Univ. Press (2000) P. Ramond, "Field Theory: A Modern Primer" (2nd Edition), Westview Press (1990) S. Weinberg, "The Quantum Theory of Fields" (Volume 2), CUP (1996).

Auswahl: Mathematische Optimierung, Diskrete Mathematik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3902-21L	Network & Integer Optimization: From Theory to Application	W	6 KP	3G	R. Zenklusen
Kurzbeschreibung	This course covers various topics in Network and (Mixed-)Integer Optimization. It starts with a rigorous study of algorithmic techniques for some network optimization problems (with a focus on matching problems) and moves to key aspects of how to attack various optimization settings through well-designed (Mixed-)Integer Programming formulations.
Lernziel	Our goal is for students to both get a good foundational understanding of some key network algorithms and also to learn how to effectively employ (Mixed-)Integer Programming formulations, techniques, and solvers, to tackle a wide range of discrete optimization problems.
Inhalt	Key topics include: - Matching problems; - Integer Programming techniques and models; - Extended formulations and strong problem formulations; - Solver techniques for (Mixed-)Integer Programs; - Decomposition approaches.
Literatur	- Bernhard Korte, Jens Vygen: Combinatorial Optimization. 6th edition, Springer, 2018. - Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. This work has 3 volumes. - Vanderbeck François, Wolsey Laurence: Reformulations and Decomposition of Integer Programs. Chapter 13 in: 50 Years of Integer Programming 1958-2008. Springer, 2010. - Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986.
Voraussetzungen / Besonderes	Solid background in linear algebra. Preliminary knowledge of Linear Programming is ideal but not a strict requirement. Prior attendance of the course Mathematical Optimization is a plus.
401-3908-21L	Polynomial Optimization	W	6 KP	3G	A. A. Kurpisz
Kurzbeschreibung	Introduction to Polynomial Optimization and methods to solve its convex relaxations.
Lernziel	The goal of this course is to provide a treatment of non-convex Polynomial Optimization problems through the lens of various techniques to solve its convex relaxations. Part of the course will be focused on learning how to apply these techniques to practical examples in finance, robotics and control.
Inhalt	Key topics include: - Polynomial Optimization as a non-convex optimization problem and its connection to certifying non-negativity of polynomials - Optimization-free and Linear Programming based techniques to approach Polynomial Optimization problems. - Introduction of Second-Order Cone Programming, Semidefinite Programming and Relative Entropy Programming as a tool to solve relaxations of Polynomial Optimization problems. - Applications to optimization problems in finance, robotics and control.
Skript	A script will be provided.
Literatur	Other helpful materials include: - Jean Bernard Lasserre, An Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge University Press, February 2015 - Pablo Parrilo. 6.972 Algebraic Techniques and Semidefinite Optimization. Spring 2006. Massachusetts Institute of Technology: MIT OpenCourseWare, . License: .
Voraussetzungen / Besonderes	Background in Linear and Integer Programming is recommended.

Auswahl: Theoretische Informatik, diskrete Mathematik Im Master-Studiengang Mathematik ist auch 401-3052-05L Graph Theory als Wahlfach anrechenbar, aber nur unter der Bedingung, dass 401-3052-10L Graph Theory nicht angerechnet wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (www.math.ethz.ch/studiensekretariat).
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
252-0408-00L	Cryptographic Protocols	W	6 KP	2V + 2U + 1A	M. Hirt, U. Maurer
Kurzbeschreibung	The 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.
Lernziel	Indroduction to a very active research area with many gems and paradoxical results. Spark interest in fundamental problems.
Inhalt	The 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.
Skript	the lecture notes are in German, but they are not required as the entire course material is documented also in other course material (in english).
Voraussetzungen / Besonderes	A basic understanding of fundamental cryptographic concepts (as taught for example in the course Information Security or in the course Cryptography Foundations) is useful, but not required.
263-4660-00L	Applied Cryptography Number of participants limited to 150.	W	8 KP	3V + 2U + 2P	K. Paterson
Kurzbeschreibung	This course will introduce the basic primitives of cryptography, using rigorous syntax and game-based security definitions. The course will show how these primitives can be combined to build cryptographic protocols and systems.
Lernziel	The goal of the course is to put students' understanding of cryptography on sound foundations, to enable them to start to build well-designed cryptographic systems, and to expose them to some of the pitfalls that arise when doing so.
Inhalt	Basic symmetric primitives (block ciphers, modes, hash functions); generic composition; AEAD; basic secure channels; basic public key primitives (encryption,signature, DH key exchange); ECC; randomness; applications.
Literatur	Textbook: Boneh and Shoup, “A Graduate Course in Applied Cryptography”, https://crypto.stanford.edu/~dabo/cryptobook/BonehShoup_0_4.pdf.
Voraussetzungen / Besonderes	Students should have taken the D-INFK Bachelor's course “Information Security" (252-0211-00) or an alternative first course covering cryptography at a similar level. / In this course, we will use Moodle for content delivery: https://moodle-app2.let.ethz.ch/course/view.php?id=14558.
263-4400-00L	Advanced Graph Algorithms and Optimization	W	8 KP	3V + 1U + 3A	R. Kyng, M. Probst
Kurzbeschreibung	This course will cover a number of advanced topics in optimization and graph algorithms.
Lernziel	The course will take students on a deep dive into modern approaches to graph algorithms using convex optimization techniques. By studying convex optimization through the lens of graph algorithms, students should develop a deeper understanding of fundamental phenomena in optimization. The course will cover some traditional discrete approaches to various graph problems, especially flow problems, and then contrast these approaches with modern, asymptotically faster methods based on combining convex optimization with spectral and combinatorial graph theory.
Inhalt	Students should leave the course understanding key concepts in optimization such as first and second-order optimization, convex duality, multiplicative weights and dual-based methods, acceleration, preconditioning, and non-Euclidean optimization. Students will also be familiarized with central techniques in the development of graph algorithms in the past 15 years, including graph decomposition techniques, sparsification, oblivious routing, and spectral and combinatorial preconditioning.
Voraussetzungen / Besonderes	This course is targeted toward masters and doctoral students with an interest in theoretical computer science. Students should be comfortable with design and analysis of algorithms, probability, and linear algebra. Having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, but not formally required. If you are not sure whether you're ready for this class or not, please consult the instructor.

Auswahl: Weitere Gebiete
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4944-20L	Mathematics of Data Science Findet dieses Semester nicht statt.	W	8 KP	4G	A. Bandeira
Kurzbeschreibung	Mostly self-contained, but fast-paced, introductory masters level course on various theoretical aspects of algorithms that aim to extract information from data.
Lernziel	Introduction to various mathematical aspects of Data Science.
Inhalt	These topics lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature a couple of Mathematical Open Problem(s) related to Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others.
Skript	https://people.math.ethz.ch/~abandeira/BandeiraSingerStrohmer-MDS-draft.pdf
Voraussetzungen / Besonderes	The main mathematical tools used will be Probability, Linear Algebra (and real analysis), and a working knowledge of these subjects is required. In addition to these prerequisites, this class requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs. We encourage students who are interested in mathematical data science to take both this course and ``227-0434-10L Mathematics of Information'' taught by Prof. H. Bölcskei. The two courses are designed to be complementary. A. Bandeira and H. Bölcskei
227-0434-10L	Mathematics of Information	W	8 KP	3V + 2U + 2A	H. Bölcskei
Kurzbeschreibung	The class focuses on mathematical aspects of 1. Information science: Sampling theorems, frame theory, compressed sensing, sparsity, super-resolution, spectrum-blind sampling, subspace algorithms, dimensionality reduction 2. Learning theory: Approximation theory, greedy algorithms, uniform laws of large numbers, Rademacher complexity, Vapnik-Chervonenkis dimension
Lernziel	The aim of the class is to familiarize the students with the most commonly used mathematical theories in data science, high-dimensional data analysis, and learning theory. The class consists of the lecture, exercise sessions with homework problems, and of a research project, which can be carried out either individually or in groups. The research project consists of either 1. software development for the solution of a practical signal processing or machine learning problem or 2. the analysis of a research paper or 3. a theoretical research problem of suitable complexity. Students are welcome to propose their own project at the beginning of the semester. The outcomes of all projects have to be presented to the entire class at the end of the semester.
Inhalt	Mathematics of Information 1. Signal representations: Frame theory, wavelets, Gabor expansions, sampling theorems, density theorems 2. Sparsity and compressed sensing: Sparse linear models, uncertainty relations in sparse signal recovery, super-resolution, spectrum-blind sampling, subspace algorithms (ESPRIT), estimation in the high-dimensional noisy case, Lasso 3. Dimensionality reduction: Random projections, the Johnson-Lindenstrauss Lemma Mathematics of Learning 4. Approximation theory: Nonlinear approximation theory, best M-term approximation, greedy algorithms, fundamental limits on compressibility of signal classes, Kolmogorov-Tikhomirov epsilon-entropy of signal classes, optimal compression of signal classes 5. Uniform laws of large numbers: Rademacher complexity, Vapnik-Chervonenkis dimension, classes with polynomial discrimination
Skript	Detailed lecture notes will be provided at the beginning of the semester.
Voraussetzungen / Besonderes	This course is aimed at students with a background in basic linear algebra, analysis, statistics, and probability. We encourage students who are interested in mathematical data science to take both this course and "401-4944-20L Mathematics of Data Science" by Prof. A. Bandeira. The two courses are designed to be complementary. H. Bölcskei and A. Bandeira
263-5300-00L	Guarantees for Machine Learning Number of participants limited to 30. Last cancellation/deregistration date for this graded semester performance: 17 March 2021! Please note that after that date no deregistration will be accepted and a "no show" will appear on your transcript.	W	7 KP	3G + 3A	F. Yang
Kurzbeschreibung	This course is aimed at advanced master and doctorate students who want to conduct independent research on theory for modern machine learning (ML). It teaches classical and recent methods in statistical learning theory commonly used to prove theoretical guarantees for ML algorithms. The knowledge is then applied in independent project work that focuses on understanding modern ML phenomena.
Lernziel	Learning objectives: - acquire enough mathematical background to understand a good fraction of theory papers published in the typical ML venues. For this purpose, students will learn common mathematical techniques from statistics and optimization in the first part of the course and apply this knowledge in the project work - critically examine recently published work in terms of relevance and determine impactful (novel) research problems. This will be an integral part of the project work and involves experimental as well as theoretical questions - find and outline an approach (some subproblem) to prove a conjectured theorem. This will be practiced in lectures / exercise and homeworks and potentially in the final project. - effectively communicate and present the problem motivation, new insights and results to a technical audience. This will be primarily learned via the final presentation and report as well as during peer-grading of peer talks.
Inhalt	This course touches upon foundational methods in statistical learning theory aimed at proving theoretical guarantees for machine learning algorithms, touching on the following topics - concentration bounds - uniform convergence and empirical process theory - high-dimensional statistics (e.g. sparsity) - regularization for non-parametric statistics (e.g. in RKHS, neural networks) - implicit regularization via gradient descent (e.g. margins, early stopping) - minimax lower bounds The project work focuses on current theoretical ML research that aims to understand modern phenomena in machine learning, including but not limited to - how overparameterization could help generalization ( RKHS, NN ) - how overparameterization could help optimization ( non-convex optimization, loss landscape ) - complexity measures and approximation theoretic properties of randomly initialized and trained NN - generalization of robust learning ( adversarial robustness, standard and robust error tradeoff, distribution shift)
Voraussetzungen / Besonderes	It’s absolutely necessary for students to have a strong mathematical background (basic real analysis, probability theory, linear algebra) and good knowledge of core concepts in machine learning taught in courses such as “Introduction to Machine Learning”, “Regression”/ “Statistical Modelling”. In addition to these prerequisites, this class requires a high degree of mathematical maturity—including abstract thinking and the ability to understand and write proofs. Students have usually taken a subset of Fundamentals of Mathematical Statistics, Probabilistic AI, Neural Network Theory, Optimization for Data Science, Advanced ML, Statistical Learning Theory, Probability Theory (D-MATH)
227-0432-00L	Learning, Classification and Compression	W	4 KP	2V + 1U	E. Riegler
Kurzbeschreibung	The focus of the course is aligned to a theoretical approach of learning theory and classification and an introduction to lossy and lossless compression for general sets and measures. We will mainly focus on a probabilistic approach, where an underlying distribution must be learned/compressed. The concepts acquired in the course are of broad and general interest in data sciences.
Lernziel	After attending this lecture and participating in the exercise sessions, students will have acquired a working knowledge of learning theory, classification, and compression.
Inhalt	1. Learning Theory (a) Framework of Learning (b) Hypothesis Spaces and Target Functions (c) Reproducing Kernel Hilbert Spaces (d) Bias-Variance Tradeoff (e) Estimation of Sample and Approximation Error 2. Classification (a) Binary Classifier (b) Support Vector Machines (separable case) (c) Support Vector Machines (nonseparable case) (d) Kernel Trick 3. Lossy and Lossless Compression (a) Basics of Compression (b) Compressed Sensing for General Sets and Measures (c) Quantization and Rate Distortion Theory for General Sets and Measures
Skript	Detailed lecture notes will be provided.
Voraussetzungen / Besonderes	This course is aimed at students with a solid background in measure theory and linear algebra and basic knowledge in functional analysis.
401-3502-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	2 KP	4A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	3 KP	6A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	4 KP	9A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel

Anwendungsgebiet Nur für das Master-Diplom in Angewandter Mathematik erforderlich und anrechenbar. In der Kategorie Anwendungsgebiet für den Master in Angewandter Mathematik muss eines der zur Auswahl stehenden Anwendungsgebiete gewählt werden. Im gewählten Anwendungsgebiet müssen mindestens 8 KP erworben werden.
Atmospherical Physics
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
701-1216-00L	Numerical Modelling of Weather and Climate	W	4 KP	3G	C. Schär, J. Vergara Temprado, M. Wild
Kurzbeschreibung	The course provides an introduction to weather and climate models. It discusses how these models are built addressing both the dynamical core and the physical parameterizations, and it provides an overview of how these models are used in numerical weather prediction and climate research. As a tutorial, students conduct a term project and build a simple atmospheric model using the language PYTHON.
Lernziel	At the end of this course, students understand how weather and climate models are formulated from the governing physical principles, and how they are used for climate and weather prediction purposes.
Inhalt	The 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.
Skript	Slides and lecture notes will be made available at Link
Literatur	List of literature will be provided.
Voraussetzungen / Besonderes	Prerequisites: 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. Previous experience with PYTHON is useful but not required.

Biology
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
551-0016-00L	Biologie II	W	2 KP	2V	M. Stoffel, E. Hafen, K. Köhler
Kurzbeschreibung	Gegenstand der Vorlesung Biologie II ist zusammen mit der Vorlesung Biologie I des vorangegangenen Wintersemesters eine Einführung in die Grundlagen der Biologie für Studierende der Materialwissenschaften sowie der Chemie und der Chemieingenieurwissenschaften.
Lernziel	Ziel der Vorlesung Biologie II ist das Verständnis der Form, Funktion und Entwicklung von Tieren und der zu Grunde liegenden Mechanismen.
Inhalt	Die folgenden Kapitelnummern beziehen sich auf das der Vorlesung zugrundeliegende Lehrbuch "Biology" (Campbell & Rees, 10th edition, 2015) Kapitel 1-4 des Lehrbuchs werden als Grundwissen vorausgesetzt. Die Abschnitte "Aufbau der Zelle" (Kap. 5-10, 12, 17) und "Allgemeine Genetik" (Kap. 13-16, 18, 46) sind Inhalt der Vorlesung Biologie I. 1. Genome, DNA-Technologie, Genetische Grundlage der Entwicklung Kapitel 19: Eukaryotische Genome: Organisation, Regulation und Evolution Kapitel 20: DNA Technologie und Genomik Kapitel 21: Genetische Grundlagen der Entwicklung 2. Form, Funktion und Entwicklung von Tieren I Kapitel 40: Grundlagen der Struktur und Funktion von Tieren Kapitel 41: Ernährung bei Tieren Kapitel 44: Osmoregulation und Exkretion Kapitel 47: Entwicklung der Tiere 3. Form, Funktion und Entwicklung von Tieren II Kapitel 42: Kreislauf und Gasaustausch Kapitel 43: Das Immunsystem Kapitel 45: Hormone und das Endokrine Sytem Kapitel 48: Nervensysteme Kapitel 49: Sensorik und Motorik
Skript	Der Vorlesungsstoff ist sehr nahe am empfohlenen Lehrbuch gehalten. Ergänzende Unterlagen werden ggf. durch die Dozenten zur Verfügung gestellt.
Literatur	Das folgende Lehrbuch ist Grundlage für die Vorlesungen Biologie I und II: Biology, Campbell and Rees, 10th Edition, 2015, Pearson/Benjamin Cummings, ISBN 978-3-8632-6725-4
Voraussetzungen / Besonderes	Voraussetzung: Vorlesung Biologie I des Herbstsemestr
262-0200-00L	Bayesian Phylodynamics – Taming the BEAST	W	4 KP	2G + 2A	T. Stadler, T. Vaughan
Kurzbeschreibung	How fast is COVID-19 spreading at the moment? How fast was Ebola spreading in West Africa? Where and when did these epidemic outbreak start? How can we construct the phylogenetic tree of great apes, and did gene flow occur between different apes? At the end of the course, students will have designed, performed, presented, and discussed their own phylodynamic data analysis to answer such questions.
Lernziel	Attendees will extend their knowledge of Bayesian phylodynamics obtained in the “Computational Biology” class (636-0017-00L) and will learn how to apply this theory to real world data. The main theoretical concepts introduced are: * Bayesian statistics * Phylogenetic and phylodynamic models * Markov Chain Monte Carlo methods Attendees will apply these concepts to a number of applications yielding biological insight into: * Epidemiology * Pathogen evolution * Macroevolution of species
Inhalt	During the first part of the block course, the theoretical concepts of Bayesian phylodynamics will be presented by us as well as leading international researchers in that area. The presentations will be followed by attendees using the software package BEAST v2 to apply these theoretical concepts to empirical data. We will use previously published datasets on e.g. COVID-19, Ebola, Zika, Yellow Fever, Apes, and Penguins for analysis. Examples of these practical tutorials are available on https://taming-the-beast.org/. In the second part of the block course, students choose an empirical dataset of genetic sequencing data and possibly some non-genetic metadata. They then design and conduct a research project in which they perform Bayesian phylogenetic analyses of their dataset. A final written report on the research project has to be submitted after the block course for grading.
Skript	All material will be available on https://taming-the-beast.org/.
Literatur	The following books provide excellent background material: • Drummond, A. & Bouckaert, R. 2015. Bayesian evolutionary analysis with BEAST. • Yang, Z. 2014. Molecular Evolution: A Statistical Approach. • Felsenstein, J. 2003. Inferring Phylogenies. More detailed information is available on https://taming-the-beast.org/.
Voraussetzungen / Besonderes	This class builds upon the content which we teach in the Computational Biology class (636-0017-00L). Attendees must have either taken the Computational Biology class or acquired the content elsewhere.

Control and Automation
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
151-0660-00L	Model Predictive Control	W	4 KP	2V + 1U	M. Zeilinger, A. Carron
Kurzbeschreibung	Model 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.
Lernziel	Design and implement Model Predictive Controllers (MPC) for various system classes to provide high performance controllers with desired properties (stability, tracking, robustness,..) for constrained systems.
Inhalt	- 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
Skript	Script / lecture notes will be provided.
Voraussetzungen / Besonderes	One 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).
227-0207-00L	Nonlinear Systems and Control Voraussetzung: Control Systems (227-0103-00L)	W	6 KP	4G	E. Gallestey Alvarez, P. F. Al Hokayem
Kurzbeschreibung	Introduction to the area of nonlinear systems and their control. Familiarization with tools for analysis of nonlinear systems. Discussion of the various nonlinear controller design methods and their applicability to real life problems.
Lernziel	On completion of the course, students understand the difference between linear and nonlinear systems, know the mathematical techniques for analysing these systems, and have learnt various methods for designing controllers accounting for their characteristics. Course puts the student in the position to deploy nonlinear control techniques in real applications. Theory and exercises are combined for better understanding of the virtues and drawbacks present in the different methods.
Inhalt	Virtually all practical control problems are of nonlinear nature. In some cases application of linear control methods leads to satisfactory controller performance. In many other cases however, only application of nonlinear analysis and control synthesis methods will guarantee achievement of the desired objectives. During the past decades mature nonlinear controller design methods have been developed and have proven themselves in applications. After an introduction of the basic methods for analysing nonlinear systems, these methods will be introduced together with a critical discussion of their pros and cons. Along the course 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 and mathematical analysis.
Skript	An english manuscript will be made available on the course homepage during the course.
Literatur	H.K. Khalil: Nonlinear Systems, Prentice Hall, 2001.
Voraussetzungen / Besonderes	Prerequisites: Linear Control Systems, or equivalent.
227-0224-00L	Stochastic Systems Findet dieses Semester nicht statt.	W	4 KP	2V + 1U	Noch nicht bekannt
Kurzbeschreibung	Probability. Stochastic processes. Stochastic differential equations. Ito. Kalman filters. St Stochastic optimal control. Applications in financial engineering.
Lernziel	Stochastic dynamic systems. Optimal control and filtering of stochastic systems. Examples in technology and finance.
Inhalt	- 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
Skript	H. P. Geering et al., Stochastic Systems, Measurement and Control Laboratory, 2007 and handouts
151-0530-00L	Nonlinear Dynamics and Chaos II	W	4 KP	4G	G. Haller
Kurzbeschreibung	The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems
Lernziel	The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.
Inhalt	I. 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
Skript	Students have to prepare their own lecture notes
Literatur	Books will be recommended in class
Voraussetzungen / Besonderes	Nonlinear Dynamics I (151-0532-00) or equivalent
151-0566-00L	Recursive Estimation	W	4 KP	2V + 1U	R. D'Andrea
Kurzbeschreibung	Estimation of the state of a dynamic system based on a model and observations in a computationally efficient way.
Lernziel	Learn the basic recursive estimation methods and their underlying principles.
Inhalt	Introduction 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.
Skript	Lecture notes available on course website: http://www.idsc.ethz.ch/education/lectures/recursive-estimation.html
Voraussetzungen / Besonderes	Requirements: Introductory probability theory and matrix-vector algebra.

Economics
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
363-0552-00L	Economic Growth and Resource Use	W	3 KP	2G	C. Karydas
Kurzbeschreibung	The course deals with the factors that contribute to economic development. Throughout the course theoretical economic modelling will be used to discuss the effects of factors – such as land, human/physical capital, technology, fossil energy resources, and climate change – on economic growth and to draw conclusions for the future.
Lernziel	The general objective of the course is to provide students tools and intuition to: i) think in a structured way – though economic modelling – about the factors that have lead to the different growth experiences among countries, and still shape our contemporary situation; ii) assess and design policies on the basis of economic development; iii) draw conclusions for the future of economic development, that take into account prevalent issues such as the scarcity of fossil energy resources and climate change.
Inhalt	Why is economic growth worth studying? Which are the factors behind economic growth? What is the role of natural resources in shaping economic development? Is our finite planet able to support sustainable long-term economic growth? Economics aims at explaining human behaviour; how do we model it and how can we steer it for the better? How do you design an efficient economic policy for a sustainable future? What is sustainable anyway? These are some of the questions you will learn to answer in this course. After spending the first lecture on overviewing the course, and the second lecture on building our mathematical and economic foundation, we begin with the three main modules that comprise this course. The first module – called “Land and Economic Growth” – deals with the historical evolution of the factors behind economic development from the pre-industrial times to our modern growth experiences. By studying the history of economic growth, we understand change and how the society we live in came to be. In this module we will develop economic models that capture the transition from an era of miniscule economic growth that persisted for millennia before the industrial revolution – with land and human labour as the main inputs to economic activity, to our modern growth experience where the continuous improvement in technology and services is our status quo. The second module – called “Non-Renewable Resources and Growth” – deals with the problem of optimal exploitation of non-renewable resources, as well as with the issue of “Resource Curse” – i.e., the observed negative relationship between economic development and resource abundance. Emerging in the 1970s due to two oil crises, the problem of the economy’s extreme dependence on fossil and depletable energy resources sparked a great deal of research to guide our way forward. Some important questions we will formally answer in this module are the following. How do we optimally exploit a given stock of a non-renewable resource? What affects the prices of non-renewable resources? If fossil energy sources – a (so far) important input to production – are getting ever depleted, is long-term growth possible? How do we explain the “Resource Curse” and what are the policies that allow a sustainable future in countries that suffer from such a curse? The third module – called “Climate Change and Growth” – deals with the pressing problem of our changing climate. Greenhouse gas emissions – so far essential for economic activity – accumulate in the atmosphere and alter environmental patterns. This phenomenon – commonly known as climate change – is responsible for the increase in the frequency and the intensity of natural disasters, which damage our stocks of capital and put a drag on economic growth. To derive appropriate policies for a sustainable future, we will incorporate these aspects in workhorse models of the economics and finance literature. Students will learn how to derive and set the “correct” price on the use of polluting energy resources from the perspective of policy-makers. Additionally, pricing of climate change risks for financial markets is important, both for individual investors and central banks, as it is they who provide liquidity to firms to pursue their long-term growth targets. Accordingly, we will close the lecture with the pricing of climate change risks from an investor’s perspective. After the last lecture of each of the three modules students will be handed out an exercise set which will be submitted by the beginning of the following week’s lecture. That lecture will be an exercise session where we will discuss the solutions in class. Each exercise set will be graded. The average grade from the best two exercise sets will account for 25% of the final grade; the rest 75% will be determined by a written exam.
Skript	Lecture Notes of the course will be sent by email to officially subscribed students.
Literatur	The 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.
Voraussetzungen / Besonderes	Knowledge of basic calculus (differentiation - integration) and basic statistics (e.g. what is an expectation; variance-covariance) is considered as a prerequisite. Elementary knowledge of dynamic systems analysis, optimal control theory and economic theory is a plus but not a prerequisite.
363-0514-00L	Energy 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.	W	3 KP	2G	M. Filippini, S. Srinivasan
Kurzbeschreibung	An introduction to energy economics and policy that covers the following topics: energy demand, investment in energy efficiency, investment in renewables, energy markets, market failures and behavioral anomalies, market-based and non-market based energy and climate policy instruments in industrialized and developing countries.
Lernziel	The students will develop an understanding of economic principles and tools necessary to analyze energy issues and to understand energy and climate policy instruments. Emphasis will be put on empirical analysis of energy demand and supply, market failures, behavioral anomalies, energy and climate policy instruments in industrialized and developing countries, and investments in renewables and in energy-efficient technologies.
Inhalt	The course provides an introduction to energy economics principles and policy applications. The first part of the course will introduce the microeconomic foundation of energy demand and supply as well as market failures and behavioral anomalies. In a second part, we introduce the concept of investment analysis (such as the NPV) in the context of renewable and energy-efficient technologies. In the last part, we use the previously introduced concepts to analyze energy policies: from a government perspective, we discuss the mechanisms and implications of market oriented and non-market oriented policy instruments as well as applications in developing countries. Throughout the entire course, we combine the material with insights from current research in energy economics. This combination will enable students to understand standard scientific literature in the field of energy economics and policy. Moreover, the class aims to show students how to relate current issues in the energy and climate spheres that influence industrialized and developing countries to insights from energy economics and policy. Course evaluation: at the end of the course, there will be a written exam covering the topics of the course.
Voraussetzungen / Besonderes	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.
364-0576-00L	Advanced Sustainability Economics PhD course, open for MSc students	W	3 KP	3G	L. Bretschger, A. Pattakou
Kurzbeschreibung	The 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.
Lernziel	Understanding 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.
363-0575-00L	Economic Growth, Cycles and Policy	W	3 KP	2G	H. Gersbach
Kurzbeschreibung	This intermediate course focuses on the core thinking devices and foundations in macroeconomics and monetary economics, and uses these devices to understand economic growth, business cycles, crises as well as how to conduct monetary and fiscal policies and policies to foster the stability of financial and economic systems.
Lernziel	- Fundamental knowledge about the drivers of economic growth in the short and long run, key macroeconomic variables and observed patterns in developed countries - Comprehensive understanding of core macroeconomic frameworks and thinking devices
Inhalt	This intermediate course focuses on the core thinking devices and foundations in macroeconomics and monetary economics, and uses these devices to understand economic growth, business cycles, crises as well as how to conduct monetary and fiscal policies and policies to foster the stability of financial and economic systems. The course is structured in the following way: Part I: Basics - Introduction - IS-LM Model in Closed Economy (Repetition) - Schools of Thought - Consumption and Investment - The Solow Growth Model Part II: Special Themes - Money Holding, Inflation, and Monetary Policy - Crises in Market Economies - IS-LM Model and Open Economy - Theories of exchange rate determination - Technical Appendix
Skript	Copies of the slides will be made available.
Literatur	Chapters in Manfred Gärtner (2009), Macroeconomics, Third Edition, Prentice Hall. and selected chapters in other books and/or papers
Voraussetzungen / Besonderes	It is required that participants have attended the lecture "Principles of Macroeconomics" (363-0565-00L).
363-0515-00L	Decisions and Markets	W	3 KP	2V	A. Bommier
Kurzbeschreibung	This course provides an introduction to microeconomics. The course emphasizes the conceptual foundations of microeconomics and contains concrete examples of their application.
Lernziel	The purpose of this course is to provide 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. After completing this course: - Students will be able to understand and use existing models to make predictions of consumer and firm behavior. - Students understand the fundamental welfare theorems and will be able to analyze equilibria of markets with perfect and imperfect competition. - Students will be able to analyze under which conditions market allocations are not efficient (market failure).
Inhalt	Microeconomics is the branch of economics which studies the decision-making by an individual, household, firm, industry or level of government. The economic equilibrium is the result of agents' interactions. Microeconomics is an element of nearly every subfield in economic analysis today. This course introduces the fundamental frameworks which form the basis of many economic models. Theory of the consumer: - Consumer preferences and utility - Budget sets and optimal choice - Demand functions - Labor supply and intertemporal choice - Welfare economics Theory of the producer: - Technological constraints and the production function - Cost minimization - Profit maximization Market structure: - Perfectly competitive markets - Monopoly behavior - Duopoly behavior General equilibrium analysis: - Market equilibrium in an exchange economy
Skript	The lecture will be based on lecture slides, which will be made available on Moodle.
Literatur	The course is mostly based on the textbook by R. Serrano and A. Feldman: "A Short Course in Intermediate Microeconomics with Calculus" (Cambridge University Press, 2013). Another textbook of interest is "Intermediate Microeconomics: A Modern Approach" by H. Varian (Norton, 2014). Exercises are available in the textbook by R. Serrano and A. Feldman ("A Short Course in Intermediate Microeconomics 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).
Voraussetzungen / Besonderes	The course is open to students who have completed an undergraduate course in economics principles and an undergraduate course in multivariate calculus.
363-1017-00L	Risk and Insurance Economics	W	3 KP	2G	I. Gemmo
Kurzbeschreibung	The course covers the economics of risk and insurance, in particular the following topics will be discussed: 2) individual decision making under risk 3) fundamentals of insurance 4) information asymmetries in insurance markets 5) the macroeconomic role of insurers
Lernziel	The goal is to introduce students to basic concepts of risk, risk management and economics of insurance.
Inhalt	“The ability to define what may happen in the future and to choose among alternatives lies at the heart of contemporary societies. Risk management guides us over a vast range of decision-making from allocation of wealth to safeguarding public health, from waging war to planning a family, from paying insurance premiums to wearing a seatbelt, from planting corn to marketing cornflakes.” (Peter L. Bernstein) Every member of society faces various decisions under uncertainty on a daily basis. Many individuals apply measures to manage these risks without even thinking about it; many are subject to behavioral biases when making these decisions. In the first part of this lecture, we discuss normative decision concepts, such as Expected Utility Theory, and contrast them with empirically observed behavior. Students learn about the rationale for individuals to purchase insurance as part of a risk management strategy. In a theoretical framework, we then derive the optimal level of insurance demand and discuss how this result depends on the underlying assumptions. After learning the basics for understanding the specifications, particularities, and mechanisms of insurance markets, we discuss the consequences of information asymmetries in these markets. Insurance companies do not only provide individuals with a way to decrease uncertainty of wealth, they also play a vital role for businesses that want to manage business risk, for the real economy by providing funds and pooling risks, and for the financial market by being important counterparties in numerous financial transactions. In the last part of this lecture, we shed light on these different roles of insurance companies. We compare the implications for different stakeholders and (insurance) markets in general. Finally, course participants familiarize themselves with selected research papers that analyze individuals’ decision-making under risk or examine specific details about the different roles of insurance companies.
Literatur	Main literature: - Eeckhoudt, L., Gollier, C., & Schlesinger, H. (2005). Economic and Financial Decisions under Risk. Princeton University Press. - Zweifel, P., & Eisen, R. (2012). Insurance Economics. Springer. Further readings: - Dionne, G. (Ed.). (2013). Handbook of Insurance (2nd ed.). Springer. - Hufeld, F., Koijen, R. S., & Thimann, C. (Eds.). (2017). The Economics, Regulation, and Systemic Risk of Insurance Markets. Oxford University Press. - Niehaus, H., & Harrington, S. (2003). Risk Management and Insurance (2nd ed.). McGraw Hill. - Rees, R., & Wambach, A. (2008). The Microeconomics of Insurance, Foundations and Trends® in Microeconomics, 4(1–2), 1-163.

Finance
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-8916-00L	Advanced Corporate Finance II (University of Zurich) Der Kurs muss direkt an der UZH belegt werden. UZH Modulkürzel: MFOEC144 Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/deadlines.html	W	3 KP	2V	Uni-Dozierende
Kurzbeschreibung	To provide the students with good understanding of the problems and issues in corporate finance.
Lernziel	To provide the students with good understanding of the problems and issues in corporate finance.
Inhalt	The 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.
Skript	See: http://www.isb.uzh.ch/institut/staff/habib.michel/teaching/
Literatur	See: http://www.isb.uzh.ch/institut/staff/habib.michel/teaching/
401-8915-00L	Advanced Financial Economics (University of Zurich) Der Kurs muss direkt an der UZH belegt werden. UZH Modulkürzel: MFOEC206 Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/deadlines.html	W	6 KP	4G	Uni-Dozierende
Kurzbeschreibung	Portfolio Theory, CAPM, Financial Derivatives, Incomplete Markets, Corporate Finance, Behavioural Finance, Evolutionary Finance
Lernziel	Students should get familiar with the cornerstones of modern financial economics.
Voraussetzungen / Besonderes	This course replaces "Advanced Financial Economics" (MFOEC105), which will be discontinued. Students who have taken "Advanced Financial Economics" (MFOEC105) in the past, are not allowed to book this course "Advanced Financial Economics" (MFOEC206). There will be a podcast for this lecture.

Image Processing and Computer Vision
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
102-0617-01L	Methodologies for Image Processing of Remote Sensing Data	W	3 KP	2G	I. Hajnsek, O. Frey, S. Li
Kurzbeschreibung	The aim of this course is to get an overview of several methodologies/algorithms for analysis of different sensor specific information products. It is focused at students that like to deepen their knowledge and understanding of remote sensing for environmental applications.
Lernziel	The course is divided into two main parts, starting with a brief introduction to remote sensing imaging (4 lectures), and is followed by an introduction to different methodologies (8 lectures) for the quantitative estimation of bio-/geo-physical parameters. The main idea is to deepen the knowledge in remote sensing tools in order to be able to understand the information products, with respect to quality and accuracy.
Inhalt	Each lecture will be composed of two parts: Theory: During the first hour, we go trough the main concepts needed to understand the specific algorithm. Practice: During the second hour, the student will test/develop the actual algorithm over some real datasets using Matlab. The student will not be asked to write all the code from scratch (especially during the first lectures), but we will provide some script with missing parts or pseudo-code. However, in the later lectures the student is supposed to build up some working libraries.
Skript	Handouts for each topic will be provided.
Literatur	Suggested readings: T. M. Lillesand, R.W. Kiefer, J.W. Chipman, Remote Sensing and Image Interpretation, John Wiley & Sons Verlag, 2008 J. R. Jensen, Remote Sensing of the Environment: An Earth Resource Perspective, Prentice Hall Series in Geograpic Information Science, 2000
227-0391-00L	Medical Image Analysis Basic knowledge of computer vision would be helpful.	W	3 KP	2G	E. Konukoglu, M. A. Reyes Aguirre
Kurzbeschreibung	It is the objective of this lecture to introduce the basic concepts used in Medical Image Analysis. In particular the lecture focuses on shape representation schemes, segmentation techniques, machine learning based predictive models and various image registration methods commonly used in Medical Image Analysis applications.
Lernziel	This lecture aims to give an overview of the basic concepts of Medical Image Analysis and its application areas.
Voraussetzungen / Besonderes	Prerequisites: Basic concepts of mathematical analysis and linear algebra. Preferred: Basic knowledge of computer vision and machine learning would be helpful. The course will be held in English.
227-0396-00L	EXCITE Interdisciplinary Summer School on Bio-Medical Imaging The school admits 60 MSc or PhD students with backgrounds in biology, chemistry, mathematics, physics, computer science or engineering based on a selection process. Students have to apply for acceptance. To apply a curriculum vitae and an application letter need to be submitted. Further information can be found at: www.excite.ethz.ch.	W	4 KP	6G	S. Kozerke, G. Csúcs, J. Klohs-Füchtemeier, S. F. Noerrelykke, M. P. Wolf
Kurzbeschreibung	Two-week summer school organized by EXCITE (Center for EXperimental & Clinical Imaging TEchnologies Zurich) on biological and medical imaging. The course covers X-ray imaging, magnetic resonance imaging, nuclear imaging, ultrasound imaging, optoacoustic imaging, infrared and optical microscopy, electron microscopy, image processing and analysis.
Lernziel	Students understand basic concepts and implementations of biological and medical imaging. Based on relative advantages and limitations of each method they can identify preferred procedures and applications. Common foundations and conceptual differences of the methods can be explained.
Inhalt	Two-week summer school on biological and medical imaging. The course covers concepts and implementations of X-ray imaging, magnetic resonance imaging, nuclear imaging, ultrasound imaging, optoacoustic imaging, infrared and optical microscopy and electron microscopy. Multi-modal and multi-scale imaging and supporting technologies such as image analysis and modeling are discussed. Dedicated modules for physical and life scientists taking into account the various backgrounds are offered.
Skript	Presentation slides, Web links
Voraussetzungen / Besonderes	The school admits 60 MSc or PhD students with backgrounds in biology, chemistry, mathematics, physics, computer science or engineering based on a selection process. To apply a curriculum vitae, a statement of purpose and applicants references need to be submitted. Further information can be found at: http://www.excite.ethz.ch/education/summer-school.html

Information and Communication Technology
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
227-0420-00L	Information Theory II	W	6 KP	4G	A. Lapidoth, S. M. Moser
Kurzbeschreibung	This course builds on Information Theory I. It introduces additional topics in single-user communication, connections between Information Theory and Statistics, and Network Information Theory.
Lernziel	The course's objective is to introduce the students to additional information measures and to equip them with the tools that are needed to conduct research in Information Theory as it relates to Communication Networks and to Statistics.
Inhalt	Sanov's Theorem, Rényi entropy and guessing, differential entropy, maximum entropy, the Gaussian channel, the entropy-power inequality, the broadcast channel, the multiple-access channel, Slepian-Wolf coding, the Gelfand-Pinsker problem, and Fisher information.
Skript	n/a
Literatur	T.M. Cover and J.A. Thomas, Elements of Information Theory, second edition, Wiley 2006
Voraussetzungen / Besonderes	Basic introductory course on Information Theory.
227-0427-10L	Advanced Signal Analysis, Modeling, and Machine Learning	W	6 KP	4G	H.‑A. Loeliger
Kurzbeschreibung	The course develops a selection of topics pivoting around graphical models (factor graphs), state space methods, sparsity, and pertinent algorithms.
Lernziel	The course develops a selection of topics pivoting around factor graphs, state space methods, and pertinent algorithms: - factor graphs and message passing algorithms - hidden-Markov models - linear state space models, Kalman filtering, and recursive least squares - Gaussian message passing - Gibbs sampling, particle filter - recursive local polynomial fitting & applications - parameter learning by expectation maximization - sparsity and spikes - binary control and digital-to-analog conversion - duality and factor graph transforms
Skript	Lecture notes
Voraussetzungen / Besonderes	Solid mathematical foundations (especially in probability, estimation, and linear algebra) as provided by the course "Introduction to Estimation and Machine Learning".

Material Modelling and Simulation
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
327-2201-00L	Transport Phenomena II	W	5 KP	4G	J. Vermant
Kurzbeschreibung	Numerical and analytical methods for real-world "Transport Phenomena"; atomistic understanding of transport properties based on kinetic theory and mesoscopic models; fundamentals, applications, and simulations
Lernziel	The 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, ...
Inhalt	Thermodynamics of Interfaces Interfacial Balance Equations Interfacial Force-Flux Relations Polymer Processing Transport Around a Sphere Refreshing Topics in Equilibrium Statistical Mechanics Kinetic Theory of Gases Kinetic Theory of Polymeric Liquids Transport in Biological Systems Dynamic Light Scattering
Skript	The course is based on the book D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018)
Literatur	1. D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018) 2. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd Ed. (Wiley, 2001) 3. Deen,W. Analysis of Transport Phenomena, Oxford University Press, 2012 4. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287
Voraussetzungen / Besonderes	Complex 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).
151-0515-00L	Continuum Mechanics 2	W	4 KP	2V + 1U	E. Mazza, R. Hopf
Kurzbeschreibung	An 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.
Lernziel	To provide a modern introduction to the foundations of continuum mechanics and prepare students for further studies in solid mechanics and related disciplines.
Inhalt	1. 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
Skript	None.
Literatur	Recommended texts: (1) Nonlinear solid mechanics, G.A. Holzapfel (2000). (2) An introduction to continuum mechanics, M.B. Rubin (2003).

Quantum Chemistry
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
529-0474-00L	Quantenchemie	W	6 KP	3G	M. Reiher, T. Weymuth
Kurzbeschreibung	Einführung in Konzepte der Elektronenstruktur-Theorie und in die Methoden der numerischen Quantenchemie; begleitende Übungen mit Papier und Bleistift, sowie Anleitungen zu praktischen Berechnungen mit Quantenchemie-Programmen am Computer.
Lernziel	Chemie kann inzwischen vollständig am Computer betrieben werden, eine intellektuelle Leistung, für die 1998 der Nobelpreis an Pople und Kohn verliehen wurde. Diese Vorlesung zeigt, wie das geht. Erarbeitet wird dabei die Vielteilchen-Quantentheorie von Mehrelektronensystemen (Atome und Moleküle) und ihre Implementierung in Computerprogramme. Es soll ein vollständiges Bild der Quantenchemie vermittelt werden, das alles Rüstzeug zur Verfügung stellt, um selbst solche Berechnungen durchführen zu können (sei es begleitend zum Experiment oder als Start in eine Vertiefung dieser Theorie).
Inhalt	Grundlegende Konzepte der Vielteilchen-Quantenmechanik. Entwicklung der Mehrelektronentheorie für Atome und Moleküle; beginnend bei der harmonischen Näherung für das Kern-Problem und bei der Hartree-Fock-Theorie für das elektronische Problem über Moeller-Plesset-Störungstheorie und Konfigurationswechselwirkung zu Coupled-Cluster und Multikonfigurationsverfahren. Dichtefunktionaltheorie. Verwendung quantenchemischer Software und Problemlösungen mit dem Computer.
Skript	Ein Skript zu allen Vorlesungsstunden wird zur Verfügung gestellt (die aufgearbeitete Theorie wird durch praktische Beispiele kontinuierlich begleitet). Sämtliche Informationen zur Vorlesung, sowie die links zum Online-Streaming werden auf dieser Webseite bekanntgegeben: https://reiher.ethz.ch/courses-and-seminars/exercises/QC_2021.html
Literatur	Lehrbücher: F.L. Pilar, Elementary Quantum Chemistry, Dover Publications I.N. Levine, Quantum Chemistry, Prentice Hall Hartree-Fock in Basisdarstellung: A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill Bücher zur Computerchemie: F. Jensen, Introduction to Computational Chemistry, John Wiley & Sons C.J. Cramer, Essentials of Computational Chemistry, John Wiley & Sons
Voraussetzungen / Besonderes	Voraussetzungen: einführende Vorlesung in Quantenmechanik (z.B. Physikalische Chemie III: Quantenmechanik)

Simulation of Semiconductor Devices
Simulation of Semiconductor Devices (Kreditpunkte nicht anrechenbar)
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
227-0056-00L	Halbleiterbauelemente	E-	4 KP	2V + 2U	C. Bolognesi
Kurzbeschreibung	The 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.
Lernziel	Understanding of the basic principles of semiconductor devices in micro-, opto-, and power electronics.
Inhalt	Brief 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.
Skript	Lecture slides.
Literatur	The lecture course follows the book Neamen, Semiconductor Physics and Devices, ISBN 978-007-108902-9, Fr. 89.00
Voraussetzungen / Besonderes	Qualifications: Physics I+II

Systems Design
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
151-0530-00L	Nonlinear Dynamics and Chaos II	W	4 KP	4G	G. Haller
Kurzbeschreibung	The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems
Lernziel	The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.
Inhalt	I. 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
Skript	Students have to prepare their own lecture notes
Literatur	Books will be recommended in class
Voraussetzungen / Besonderes	Nonlinear Dynamics I (151-0532-00) or equivalent
363-0588-00L	Complex Networks	W	4 KP	2V + 1U	F. Schweitzer
Kurzbeschreibung	The 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.
Lernziel	* 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
Inhalt	Networks 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.
Skript	The 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=12428
Literatur	See handouts. Specific literature is provided for download - for registered students, only.
Voraussetzungen / Besonderes	There 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.
363-0543-00L	Agent-Based Modelling of Social Systems	W	3 KP	2V + 1U	F. Schweitzer, G. Vaccario
Kurzbeschreibung	Agent-based modeling is introduced as a bottom-up approach to understand the complex dynamics of social systems. The course is based on formal models of agents and their interactions. Computer simulations using Python allow the quantitative analysis of a wide range of social phenomena, e.g. cooperation and competition, opinion dynamics, spatial interactions and behaviour in social networks.
Lernziel	A successful participant of this course is able to - understand the rationale of agent-based models of social systems - understand the relation between rules implemented at the individual level and the emerging behavior at the global level - learn to choose appropriate model classes to characterize different social systems - grasp the influence of agent heterogeneity on the model output - efficiently implement agent-based models using Python and visualize the output
Inhalt	This full-featured course on agent-based modeling (ABM) allows participants with no prior expertise to understand concepts, methods and tools of ABM, to apply them in their master or doctoral thesis. We focus on a formal description of agents and their interactions, to allow for a suitable implementation in computer simulations. Given certain rules for the agents, we are interested to model their collective dynamics on the systemic level. Agent-based modeling is introduced as a bottom-up approach to understand the complex dynamics of social systems. Agents represent the basic constituents of such systems. The are described by 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, which we want to understand and to analyze. 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 modeling, weekly exercise classes provide practical skills. Using the Python programming language, the participants implement agent-based models in guided and in self-chosen projects, which they present and jointly discuss.
Skript	The lecture slides will be available on the Moodle platform, for registered students only.
Literatur	See handouts. Specific literature is provided for download, for registered students only.
Voraussetzungen / Besonderes	Participants of the course should have some background in mathematics and an interest in formal modeling and in 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 students. The examination will account for 70% of the grade and will be conducted electronically. The "closed book" rule applies: no books, no summaries, no lecture materials. The exam questions and answers will be only in English. The use of a paper-based dictionary is permitted. The group project to be handed in at the beginning of July will count 30% to the final grade.

Theoretical Physics Im Master-Studiengang Angewandte Mathematik ist auch 402-0204-00L Elektrodynamik als Fach im Vertiefungsgebiet Theoretical Physics anrechenbar, aber nur unter der Bedingung, dass 402-0224-00L Theoretische Physik nicht angerechnet wurde oder wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (www.math.ethz.ch/studiensekretariat).
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
402-0812-00L	Computational Statistical Physics	W	8 KP	2V + 2U	M. Krstic Marinkovic
Kurzbeschreibung	Simulationsmethoden in der statistischen Physik. Klassische Monte-Carlo-Simulationen: finite-size scaling, Clusteralgorithmen, Histogramm-Methoden, Renormierungsgruppe. Anwendung auf Boltzmann-Maschinen. Simulation von Nichtgleichgewichtssystemen. Molekulardynamik-Simulationen: langreichweitige Wechselwirkungen, Ewald-Summation, diskrete Elemente, Parallelisierung.
Lernziel	Die Vorlesung ist eine Vertiefung von Simulationsmethoden in der statistischen Physik, und daher ideal als Fortführung der Veranstaltung "Introduction to Computational Physics" des Herbstsemesters. Im ersten Teil lernen Studenten die folgenden Methoden anzuwenden: Klassische Monte-Carlo-Simulationen, finite-size scaling, Clusteralgorithmen, Histogramm-Methoden, Renormierungsgruppe. Ausserdem lernen Studenten die Anwendung der Methoden aus der Statistischen Physik auf Boltzmann-Maschinen kennen und lernen wie Nichtgleichgewichtssysteme simuliert werden. Im zweiten Teil wenden die Studenten Methoden zur Simulation von Molekulardynamiken an. Das beinhaltet unter anderem auch langreichweitige Wechselwirkungen, Ewald-Summation und diskrete Elemente.
Inhalt	Simulationsmethoden in der statistischen Physik. Klassische Monte-Carlo-Simulationen: finite-size scaling, Clusteralgorithmen, Histogramm-Methoden, Renormierungsgruppe. Anwendung auf Boltzmann-Maschinen. Simulation von Nichtgleichgewichtssystemen. Molekulardynamik-Simulationen: langreichweitige Wechselwirkungen, Ewald-Summation, diskrete Elemente, Parallelisierung.
Skript	Skript und Folien sind online verfügbar und werden bei Bedarf verteilt.
Literatur	Literaturempfehlungen und Referenzen sind im Skript enthalten.
Voraussetzungen / Besonderes	Grundlagenwissen in der Statistischen Physik, Klassischen Mechanik und im Bereich der Rechnergestützten Methoden ist empfohlen.
402-0810-00L	Computational Quantum Physics Fachstudierende UZH müssen das Modul PHY522 direkt an der UZH buchen.	W	8 KP	2V + 2U	M. H. Fischer
Kurzbeschreibung	This course provides an introduction to simulation methods for quantum systems. Starting from the one-body problem, a special emphasis is on quantum many-body problems, where we cover both approximate methods (Hartree-Fock, density functional theory) and exact methods (exact diagonalization, matrix product states, and quantum Monte Carlo methods).
Lernziel	Through lectures and practical programming exercises, after this course: Students are able to describe the difficulties of quantum mechanical simulations. Students are able to explain the strengths and weaknesses of the methods covered. Students are able to select an appropriate method for a given problem. Students are able to implement basic versions of all algorithms discussed.
Skript	A script for this lecture will be provided.
Literatur	A list of additional references will be provided in the script.
Voraussetzungen / Besonderes	A basic knowledge of quantum mechanics, numerical tools (numerical differentiation and integration, linear solvers, eigensolvers, root solvers, optimization), and a programming language (for the teaching assignments, you are free to choose your preferred one).
402-0206-00L	Quantum Mechanics II Fachstudierende UZH müssen das Modul PHY351 direkt an der UZH buchen.	W	10 KP	3V + 2U	P. Jetzer
Kurzbeschreibung	Many-body quantum physics rests on symmetry considerations that lead to two kinds of particles, fermions and bosons. Formal techniques include Hartree-Fock theory and second-quantization techniques, as well as quantum statistics with ensembles. Few- and many-body systems include atoms, molecules, the Fermi sea, elastic chains, radiation and its interaction with matter, and ideal quantum gases.
Lernziel	Basic command of few- and many-particle physics for fermions and bosons, including second quantisation and quantum statistical techniques. Understanding of elementary many-body systems such as atoms, molecules, the Fermi sea, electromagnetic radiation and its interaction with matter, ideal quantum gases and relativistic theories.
Inhalt	The description of indistinguishable particles leads us to (exchange-) symmetrized wave functions for fermions and bosons. We discuss simple few-body problems (Helium atoms, hydrogen molecule) und proceed with a systematic description of fermionic many body problems (Hartree-Fock approximation, screening, correlations with applications on atomes and the Fermi sea). The second quantisation formalism allows for the compact description of the Fermi gas, of elastic strings (phonons), and the radiation field (photons). We study the interaction of radiation and matter and the associated phenomena of radiative decay, light scattering, and the Lamb shift. Quantum statistical description of ideal Bose and Fermi gases at finite temperatures concludes the program. If time permits, we will touch upon of relativistic one particle physics, the Klein-Gordon equation for spin-0 bosons and the Dirac equation describing spin-1/2 fermions.
Literatur	G. Baym, Lectures on Quantum Mechanics (Benjamin, Menlo Park, California, 1969) L.I. Schiff, Quantum Mechanics (Mc-Graw-Hill, New York, 1955) A. Messiah, Quantum Mechanics I & II (North-Holland, Amsterdam, 1976) E. Merzbacher, Quantum Mechanics (John Wiley, New York, 1998) C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics I & II (John Wiley, New York, 1977) P.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (Mc Graw-Hill, New York, 1965) A.L. Fetter and J.D. Walecka, Theoretical Mechanics of Particles and Continua (Mc Graw-Hill, New York, 1980) J.J. Sakurai, Modern Quantum Mechanics (Addison Wesley, Reading, 1994) J.J. Sakurai, Advanced Quantum mechanics (Addison Wesley) F. Gross, Relativistic Quantum Mechanics and Field Theory (John Wiley, New York, 1993)
Voraussetzungen / Besonderes	Basic knowledge of single-particle Quantum Mechanics
402-0871-00L	Solid State Theory Studierende der UZH dürfen diese Lerneinheit nicht an der ETH belegen, sondern müssen das Modul PHY411 direkt an der UZH buchen.	W	10 KP	4V + 1U	V. Geshkenbein
Kurzbeschreibung	Diese Vorlesung richtet sich an Studierende der Experimentalphysik und der theoretischen Physik. Sie bietet eine Einführung in wichtige theoretische Konzepte der Festkörperphysik.
Lernziel	Ziel der Vorlesung ist die Entwicklung eines theoretischen Rahmens zum Verständnis grundlegender Phänomene der Festkörperphysik. Dazu gehören Symmetrien, Bandstrukturen, Teilchen-Teilchen Wechselwirkung, Landau Fermi-Flüssigkeiten, sowie spezifische Themen wie Transport, Quanten-Hall-Effekt und Magnetismus. Die Übungen unterstützen und illustrieren die Vorlesung durch handwerkliches Lösen spezifischer Probleme. Der Student versteht grundlegende theoretische Konzepte der Festkörperphysik und kann Probleme selbständig lösen. Es werden keine diagrammatischen Techniken verwendet.
Inhalt	Diese Vorlesung richtet sich an Studierende der Experimentalphysik und der theoretischen Physik. Sie bietet eine Einführung in wichtige theoretische Konzepte der Festkörperphysik. Es werden folgende Themen abgedeckt: Symmetrien und Gruppentheorie, Elektronenstruktur in Kristallen, Isolatoren-Halbleiter-Metalle, Phononen, Wechselwirkungseffekte, (un-)geladene Fermi-Flüssigkeiten, lineare Antworttheorie, kollektive Moden, Abschirmung, Transport in Halbleitern und Metallen, Magnetismus, Mott-Isolatoren, Quanten-Hall-Effekt.
Skript	in Englisch
402-0844-00L	Quantum Field Theory II Studierende der UZH dürfen diese Lerneinheit nicht an der ETH belegen, sondern müssen das entsprechende Modul direkt an der UZH buchen.	W	10 KP	3V + 2U	N. Beisert
Kurzbeschreibung	The subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.
Lernziel	The goal of this course is to lay down the path integral formulation of quantum field theories and in particular to provide a solid basis for the study of non-abelian gauge theories and of the Standard Model
Inhalt	The 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 - the Goldstone theorem and the Higgs mechanism - gauge theories with spontaneous symmetry breaking and their quantization - renormalization of spontaneously broken gauge theories and quantum effective actions
Literatur	M.E. Peskin and D.V. Schroeder, "An introduction to Quantum Field Theory", Perseus (1995). S. Pokorski, "Gauge Field Theories" (2nd Edition), Cambridge Univ. Press (2000) P. Ramond, "Field Theory: A Modern Primer" (2nd Edition), Westview Press (1990) S. Weinberg, "The Quantum Theory of Fields" (Volume 2), CUP (1996).
402-0394-00L	Theoretical Cosmology Fachstudierende UZH müssen das Modul AST513 direkt an der UZH buchen.	W	10 KP	4V + 2U	L. M. Mayer, J. Yoo
Kurzbeschreibung	This 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.
Lernziel	Learning the fundamentals of modern physical cosmology. This entails understanding the physical principles behind the description of the homogeneous Universe on large scales in the first part of the course, and moving on to the inhomogeneous Universe model where perturbation theory is used to study the development of structure through gravitational instability in the second part of the course. Modern notions of dark matter and dark energy will also be introduced and discussed.
Inhalt	The 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
Skript	In 2021, the lectures will be live-streamed online at ETH from the Room HPV G5 at the lecture hours. The recordings will be available at the ETH website. The detailed information will be provided by the course website and the SLACK channel.
Literatur	Suggested 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
Voraussetzungen / Besonderes	Knowledge of General Relativity is recommended.
» Wahlfächer Theoretische Physik

Transportation Science
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
101-0478-00L	Measurement and Modelling of Travel Behaviour	W	6 KP	4G	K. W. Axhausen
Kurzbeschreibung	Comprehensive introduction to survey methods in transport planning and modeling of travel behavior, using advanced discrete choice models.
Lernziel	Enabling the student to understand and apply the various measurement approaches and models of modelling travel behaviour.
Inhalt	Behavioral 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
Skript	Various papers and notes are distributed during the course.

Seminare und Semesterarbeiten
Seminare Dieses Semester haben viele Seminare eine Warteliste mit speziellem Auswahlverfahren. Falls keine anderen Auswahlkriterien vorliegen, werden bei der definitiven Belegung zuerst jene Studierenden berücksichtigt, welche noch keine andere Seminarbelegung haben. Wenn Sie sich in zwei Wartelisten eintragen, so tun Sie dies am besten so: wählen Sie zuerst das Seminar aus, das Sie bevorzugen, und wählen Sie ein paar Minuten später eine Ausweichmöglichkeit aus. WICHTIG: Schreiben Sie sich in höchstens zwei Wartelisten ein!
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3110-21L	Student Seminar in Number Theory: Modular Forms Number of participants limited to 26.	W	4 KP	2S	M. Schwagenscheidt
Kurzbeschreibung	Seminar on the basic theory of classical elliptic modular forms
Lernziel	In the seminar we will learn about the basic theory of classical elliptic modular forms. We start with the action of the modular group on the complex upper half-plane by Moebius transformations and describe its fundamental domain. As first examples of modular forms, we will investigate Eisenstein series, Ramanujan's Delta function, the Dedekind eta function, and the modular j-invariant. We will show that the space of modular forms of a fixed weight is finite dimensional, and determine its dimension. We will also study Hecke operators and the Petersson inner product on spaces of modular forms, and the L-functions associated with modular forms. Towards the end of the seminar we will discuss some more advanced topics, such as differential operators and quasimodular forms, the CM values of the j-function, and the periods of modular forms.
Skript	Link
Literatur	Cohen, Strömberg: Modular Forms: A Classical Approach Diamond, Shurman: A first course in modular forms Koblitz: Introduction to elliptic curves and modular forms Koecher, Krieg: Elliptische Funktionen und Modulformen Lang: Introduction to modular forms Miyake: Modular forms Serre: A course in arithmetic Zagier: The 1-2-3 of modular forms Lecture notes on modular forms, available online: Link
Voraussetzungen / Besonderes	We will need the fundamental results from complex analysis, and some elementary group theory. The website of the seminar can be found at https://people.math.ethz.ch/~mschwagen/modularforms
401-3140-21L	Monstrous Moonshine Number of participants limited to 12.	W	4 KP	2S	T.‑H. Bülles, R. Pandharipande
Kurzbeschreibung	We study Monstrous Moonshine, the surprising connection between modular forms and the Monster group.
Lernziel	To understand the equation 196884 = 196883 + 1.
Voraussetzungen / Besonderes	Algebra I and II. Some familiarity with modular forms and Lie algebras is helpful, but not crucial: all necessary concepts will be introduced in the early talks.
401-3520-21L	Sphere Packings, Lattices and Codes Number of participants limited to 12.	W	4 KP	2S	D. Radchenko
Kurzbeschreibung	Seminar on Sphere Packings, Lattices and Codes
Lernziel	To learn about the sphere packing problem and its connection to various other topics such as error-correcting codes, combinatorial and spherical designs, and modular forms.
Inhalt	Some of the tentative topics include: sphere packing problem; the kissing number problem; error-correcting codes; Shannon capacity; finite projective planes; binary Golay code; spherical designs; theta functions of lattices; linear programming bounds for spherical codes and sphere packings.
Literatur	J.H. Conway, N.J. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer-Verlag New York, 2017. W. Ebeling, Lattices and Codes: A Course Partially Based on Lectures by Friedrich Hirzebruch, third edition, Springer Spektrum, Wiesbaden, 2013. D. Zagier, Elliptic modular forms and their applications, in "The 1-2-3 of Modular Forms" (K. Ranestad, ed.), Universitext, Springer, Berlin, 2008. C. Zong, Sphere Packings, Universitext, Springer-Verlag New York, 1999.
Voraussetzungen / Besonderes	Many of the topics are self-contained and require only basic knowledge of linear algebra and analysis. Some of the later talks require basic knowledge of complex analysis. Some degree of familiarity with modular forms is also helpful, but not strictly necessary.
401-3350-21L	Classical Theory of Elliptic Partial Differential Equations Number of participants limited to 12.	W	4 KP	2S	J. Serra
Kurzbeschreibung	Following the book "Elliptic Partial Differential Equations" of Qing Han and Fanhua Lin, the seminar will cover ---from an introductory perspective--- some important classical tools and results in the standard theory of Elliptic PDE
Lernziel	To present some of the most useful classical tools and results in nonlinear Elliptic PDE (weak and viscosity solutions and their maximum principles, moving plane method, Bernstein's technique, De Giorgi-Nash-Moser Harnack Inequality, etc.)
Inhalt	(flexible depending on the background of the students) -Review of harmonic functions -Weak and viscosity solutions -Maximum principles and barriers -Moving plane method -Bernstein's technique -Schauder estimates (review) -De Giorgi-Nash-Moser and Hölder continuity of gradients
Literatur	Elliptic Partial Differential Equations: Second Edition Qing Han and Fanghua Lin Publication Year: 2011 ISBN-10: 0-8218-5313-9 ISBN-13: 978-0-8218-5313-9 Courant Lecture Notes, vol. 1.R
Voraussetzungen / Besonderes	Although many parts of the book are rather self-contained, it would be advisable to have followed before the bachelor course Functional Analysis II
401-4350-21L	Topics in Non-Collisional Kinetic Theory Number of participants limited to 12.	W	4 KP	2S	M. Iacobelli, L. Cesbron
Kurzbeschreibung
Lernziel
401-3830-21L	Wave Equations on Black Hole Spacetimes Number of participants limited to 12.	W	4 KP	2S	C. Kehle
Kurzbeschreibung	Introduction to Lorentzian geometry, to the notion of a black hole, and to the study of linear wave equations on such spacetimes.
Lernziel	We will learn about the basics of Lorentzian geometry, the geometric framework which incorporates space and time as one geometric entity---spacetime. Then, we will briefly introduce the Einstein equations of General Relativity and study the Schwarzschild and Reissner--Nordström black holes solutions. We will further discuss tools to study linear wave equations on black holes and other spacetimes.
Inhalt	Black holes are among the central theoretical predictions of general relativity which is governed by the celebrated Einstein's equations. The notion of a black hole has a clean mathematical definition, and the concept is already exhibited by the simplest non-trivial solution of the Einstein vacuum equation: the Schwarzschild solution. These “black hole spacetimes” give rise to many natural mathematical problems in the analysis of (hyperbolic) PDE which in turn describe physical phenomena related to black holes. More specifically we will cover the following topics: Basic Lorentzian geometry, the Schwarzschild and Reissner-Nordström black hole, the wave equation on general Lorentzian manifolds, the wave equation on black hole backgrounds. We will also adapt the content to the prior knowledge of the students.
Literatur	Main reference: Lecture Notes of Mihalis Dafermos: https://www.dpmms.cam.ac.uk/~md384/ETH-Nachdiplom-temp.pdf Further references (going beyond the scope of the seminar): - Dafermos, Mihalis, and Igor Rodnianski. "Lectures on black holes and linear waves." Clay Math. Proc 17 (2013): 97-205. (see also arXiv:0811.0354) - Aretakis, Stefanos. "General Relativity". https://www.math.toronto.edu/aretakis/General%20Relativity-Aretakis.pdf - Christodoulou, Demetrios. Mathematical problems of general relativity I. Vol. 1. European Mathematical Society, 2008.
Voraussetzungen / Besonderes	Ideally, participants have some familiarity with the basics of differential manifolds (definition of smooth manifolds, tangent space, vector fields, as well as the formal apparatus of Riemannian geometry: connections, curvature, geodesics) and basic functional analysis (Sobolev spaces, etc.).
401-3650-19L	Numerical Analysis Seminar: Deep Neural Network Approximation Findet dieses Semester nicht statt.	W	4 KP	2S	C. Schwab
Kurzbeschreibung	This seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions.
Lernziel
Inhalt	Presentation of the Seminar: Deep Neural Networks (DNNs) have recently attracted substantial interest and attention due to outperforming the best established techniques in a number of tasks (Chess, Go, Shogi, autonomous driving, language translation, image classification, etc.). In many cases, these successes have been achieved by heuristic implementations combined with massive compute power and training data. The seminar will address mathematical results on the approximation/ expressive power of DNNs. For a (bird's eye) overview, see https://arxiv.org/abs/1901.05639 and, more mathematical and closer to the seminar theme, https://arxiv.org/abs/1901.02220 Specifically, this seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. Mathematical results support that DNNs can equalize or outperform the best mathematical results known to date. Particular cases comprise: high-dimensional parametric maps, analytic and holomorphic maps, maps containing multi-scale features which arise as solution classes from PDEs, classes of maps which are invariant under group actions.
Voraussetzungen / Besonderes	Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. The seminar format will be oral student presentations in the first half of May 2021, combined with a written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester (end of February). Disclaimer: The seminar will _not_ address recent developments in DNN software, such as training heuristics, or programming techniques for DNN training in various specific applications.
401-3940-21L	Student Seminar in Mathematics and Data: Optimal Transport Number of participants limited to 12.	W	4 KP	2S	A. Bandeira, G. Chinot
Kurzbeschreibung	The Seminar starts with a basic introduction to Optimal Transport (including but not limited to: Monge and Kantorovich formulations, duality, Wassertstein distance). After the introductory material, each week will be devoted to either a research article in the topic or a more advanced concept. Particular emphasis will be given to applications to statistics and data science.
Lernziel
Skript	More information, including list of papers, will be available at Link
Literatur	More information, including list of papers, will be available at Link
Voraussetzungen / Besonderes	This seminar requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs. Probability theory and Linear Algebra is a required pre-requisite. Some basic familiarity with Optimization and Functional Analysis is beneficial.
401-3600-21L	Student Seminar in Probability Theory Limited number of participants. Registration to the seminar will only be effective once confirmed by email from the organizers.	W	4 KP	2S	W. Werner, J. Bertoin, V. Tassion
Kurzbeschreibung
Lernziel
401-3620-21L	Student Seminar in Statistics: Statistical Network Modeling Maximale Teilnehmerzahl: 48 Hauptsächlich für Studierende der Bachelor- und Master-Studiengänge Mathematik, welche nach der einführenden Lerneinheit 401-2604-00L Wahrscheinlichkeit und Statistik (Probability and Statistics) mindestens ein Kernfach oder Wahlfach in Statistik besucht haben. Das Seminar wird auch für Studierende der Master-Studiengänge Statistik bzw. Data Science angeboten.	W	4 KP	2S	P. L. Bühlmann, M. Azadkia
Kurzbeschreibung	Network models can be used to analyze non-iid data because their structure incorporates interconnectedness between the individuals. We introduce networks, describe them mathematically, and consider applications.
Lernziel	Network models can be used to analyze non-iid data because their structure incorporates interconnectedness between the individuals. The participants of the seminar acquire knowledge to formulate and analyze network models and to apply them in examples.
Literatur	E. D. Kolaczyk and G. Csárdi. Statistical analysis of network data with R. Springer, Cham, Switzerland, second edition, 2020. Tianxi Li, Elizaveta Levina, and Ji Zhu. Network cross-validation by edge sampling, 2020. Preprint arXiv:1612.04717. Tianxi Li, Elizaveta Levina, and Ji Zhu. Community models for partially observed networks from surveys, 2020. Preprint arXiv:2008.03652. Tianxi Li, Elizaveta Levina, and Ji Zhu. Prediction Models for Network-Linked Data, 2018. Preprint arXiv:1602.01192.
Voraussetzungen / Besonderes	Every class will consist of an oral presentation highlighting key ideas of selected book chapters by a pair of students. Another two students will be responsible for asking questions during the presentation and providing a discussion of the the presented concepts and ideas, including pros+cons, at the end. Finally, an additional two students are responsible for giving an evaluation on the quality of the presentations/discussions and provide constructive feedback for improvement.
401-3620-20L	Student Seminar in Statistics: Inference in Non-Classical Regression Models Findet dieses Semester nicht statt. Maximale Teilnehmerzahl: 24 Hauptsächlich für Studierende der Bachelor- und Master-Studiengänge Mathematik, welche nach der einführenden Lerneinheit 401-2604-00L Wahrscheinlichkeit und Statistik (Probability and Statistics) mindestens ein Kernfach oder Wahlfach in Statistik besucht haben. Das Seminar wird auch für Studierende der Master-Studiengänge Statistik bzw. Data Science angeboten.	W	4 KP	2S	F. Balabdaoui
Kurzbeschreibung	Review of some non-standard regression models and the statistical properties of estimation methods in such models.
Lernziel	The main goal is the students get to discover some less known regression models which either generalize the well-known linear model (for example monotone regression) or violate some of the most fundamental assumptions (as in shuffled or unlinked regression models).
Inhalt	Linear regression is one of the most used models for prediction and hence one of the most understood in statistical literature. However, linearity might too simplistic to capture the actual relationship between some response and given covariates. Also, there are many real data problems where linearity is plausible but the actual pairing between the observed covariates and responses is completely lost or at partially. In this seminar, we review some of the non-classical regression models and the statistical properties of the estimation methods considered by well-known statisticians and machine learners. This will encompass: 1. Monotone regression 2. Single index model 3. Unlinked regression 4. Partially unlinked regression
Skript	No script is necessary for this seminar
Literatur	In the following is the material that will read and studied by each pair of students (all the items listed below are available through the ETH electronic library or arXiv): 1. Chapter 2 from the book "Nonparametric estimation under shape constraints" by P. Groeneboom and G. Jongbloed, 2014, Cambridge University Press 2. "Nonparametric shape-restricted regression" by A. Guntuoyina and B. Sen, 2018, Statistical Science, Volume 33, 568-594 3. "Asymptotic distributions for two estimators of the single index model" by Y. Xia, 2006, Econometric Theory, Volume 22, 1112-1137 4. "Least squares estimation in the monotone single index model" by F. Balabdaoui, C. Durot and H. K. Jankowski, Journal of Bernoulli, 2019, Volume 4B, 3276-3310 5. "Least angle regression" by B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, 2004, Annals of Statsitics, Volume 32, 407-499. 6. "Sharp thresholds for high dimensional and noisy sparsity recovery using l1-constrained quadratic programming (Lasso)" by M. Wainwright, 2009, IEEE transactions in Information Theory, Volume 55, 1-19 7."Denoising linear models with permuted data" by A. Pananjady, M. Wainwright and T. A. Courtade and , 2017, IEEE International Symposium on Information Theory, 446-450. 8. "Linear regression with shuffled data: statistical and computation limits of permutation recovery" by A. Pananjady, M. Wainwright and T. A. Courtade , 2018, IEEE transactions in Information Theory, Volume 64, 3286-3300 9. "Linear regression without correspondence" by D. Hsu, K. Shi and X. Sun, 2017, NIPS 10. "A pseudo-likelihood approach to linear regression with partially shuffled data" by M. Slawski, G. Diao, E. Ben-David, 2019, arXiv. 11. "Uncoupled isotonic regression via minimum Wasserstein deconvolution" by P. Rigollet and J. Weed, 2019, Information and Inference, Volume 00, 1-27
401-3900-16L	Advanced Topics in Discrete Optimization Number of participants limited to 12.	W	4 KP	2S	R. Zenklusen, R. Santiago Torres, V. Traub
Kurzbeschreibung	In this seminar we will discuss selected topics in discrete optimization. The main focus is on mostly recent research papers in the field of Combinatorial Optimization.
Lernziel	The goal of the seminar is twofold. First, we aim at improving students' presentation and communication skills. In particular, students are to present a research paper to their peers and the instructors in a clear and understandable way. Second, students learn a selection of recent cutting-edge approaches in the field of Combinatorial Optimization by attending the other students' talks. A very active participation in the seminar helps students to build up the necessary skills for parsing and digesting advanced technical texts on a significantly higher complexity level than usual textbooks. A key goal is that students prepare their presentations in a concise and accessible way to make sure that other participants get a clear idea of the presented results and techniques. Students intending to do a project in optimization are strongly encouraged to participate.
Inhalt	The selected topics will cover various classical and modern results in Combinatorial Optimization. Contrary to prior years, a very significant component of the seminar will be interactive discussions where active participation of the students is required.
Literatur	The learning material will be in the form of scientific papers.
Voraussetzungen / Besonderes	Requirements: We expect students to have a thorough understanding of topics covered in the course "Mathematical Optimization".
252-4102-00L	Seminar on Randomized Algorithms and Probabilistic Methods Findet dieses Semester nicht statt. The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar. Number of participants limited to 24.	W	2 KP	2S	A. Steger
Kurzbeschreibung	The 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 (SODA18).
Lernziel	Read papers from the forefront of today's research; learn how to give a scientific talk.
Voraussetzungen / Besonderes	The 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).
263-4203-00L	Geometry: Combinatorics and Algorithms The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar.	W	2 KP	2S	B. Gärtner, M. Hoffmann, E. Welzl, M. Wettstein
Kurzbeschreibung	This seminar complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent.
Lernziel	Each 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.
Inhalt	This 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.
Voraussetzungen / Besonderes	Prerequisite: Successful participation in the course "Geometry: Combinatorics & Algorithms" (takes place every HS) is required.

Semesterarbeiten Es gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3750-01L	Semesterarbeit Voraussetzung: erfolgreicher Abschluss der Lerneinheit 401-2000-00L Scientific Works in Mathematics Weitere Angaben unter www.math.ethz.ch/intranet/students/study-administration/theses.html	W	8 KP	11A	Betreuer/innen
Kurzbeschreibung	Semesterarbeiten dienen der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Semesterarbeiten sollen die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern.
Lernziel
Voraussetzungen / Besonderes	Es gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
401-3750-02L	Semesterarbeit (Nr. 2) Voraussetzung: erfolgreicher Abschluss der Lerneinheit 401-2000-00L Scientific Works in Mathematics Weitere Angaben unter www.math.ethz.ch/intranet/students/study-administration/theses.html	W	8 KP	11A	Betreuer/innen
Kurzbeschreibung	Semesterarbeiten dienen der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Semesterarbeiten sollen die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern.
Lernziel
Voraussetzungen / Besonderes	Es gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
401-3750-03L	Semesterarbeit (Nr. 3) Voraussetzung: erfolgreicher Abschluss der Lerneinheit 401-2000-00L Scientific Works in Mathematics Weitere Angaben unter www.math.ethz.ch/intranet/students/study-administration/theses.html	W	8 KP	11A	Betreuer/innen
Kurzbeschreibung	Semesterarbeiten dienen der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Semesterarbeiten sollen die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern.
Lernziel
Voraussetzungen / Besonderes	Es gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.

GESS Wissenschaft im Kontext Wer für den Bachelor-Abschluss bereits 3 KP an Sprachkursen anrechnen liess, benötigt auf Master-Stufe 2 KP aus dem "Wissenschaft im Kontext"-Programm ohne Sprachkurse. vgl. Link (Aus dem Kursprogramm müssen grundsätzlich acht Kreditpunkte (KP) erworben werden – im Rahmen des Bachelor-Studiums in der Regel sechs KP, im Rahmen des Master-Studiums in der Regel zwei KP. Sprachkurse des Sprachenzentrums UZH-ETH können im Umfang von maximal drei KP angerechnet werden. Es gelten überdies folgende Einschränkungen: Im Falle der europäischen Sprachen Englisch, Französisch, Italienisch und Spanisch werden nur fortgeschrittene Sprachkurse ab Niveau B2 angerechnet. Deutsche Sprachkurse werden ab Niveau C2 angerechnet.)
» siehe Studiengang Wissenschaft im Kontext: Typ A: Förderung allgemeiner Reflexionsfähigkeiten
» Empfehlungen aus dem Bereich Wissenschaft im Kontext (Typ B) für das D-MATH
» siehe Studiengang Wissenschaft im Kontext: Sprachkurse ETH/UZH

Master-Arbeit
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-2000-00L	Scientific Works in Mathematics Zielpublikum: Bachelor-Studierende im dritten Jahr; Master-Studierende, welche noch keine entsprechende Ausbildung vorweisen können.	O	0 KP		M. Burger
Kurzbeschreibung	Introduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)
Lernziel	Learn the basic standards of scientific works in mathematics.
Inhalt	- Types of mathematical works - Publication standards in pure and applied mathematics - Data handling - Ethical issues - Citation guidelines
Skript	Moodle of the Mathematics Library: https://moodle-app2.let.ethz.ch/course/view.php?id=519
Voraussetzungen / Besonderes	Directive Link
401-2000-01L	Lunch Sessions – Thesis Basics für Mathematik-Studierende Für Details und zur Registrierung für den freiwilligen MathBib-Schulungskurs: https://www.math.ethz.ch/mathbib-schulungen	Z	0 KP		Referent/innen
Kurzbeschreibung	Freiwilliger Kurs "Recherchieren in der Mathematik" angeboten von der Mathematikbibliothek.
Lernziel
401-4990-00L	Master's Thesis Zur Master-Arbeit wird nur zugelassen, wer: a. das Bachelor-Studium erfolgreich abgeschlossen hat; b. allfällige Auflagen für die Zulassung zum Master-Studiengang erfüllt hat. Voraussetzung: erfolgreicher Abschluss der Lerneinheit 401-2000-00L Scientific Works in Mathematics Weitere Angaben unter www.math.ethz.ch/intranet/students/study-administration/theses.html	O	30 KP	57D	Betreuer/innen
Kurzbeschreibung	Die Master-Arbeit bildet den Abschluss des Studiengangs. In der Master-Arbeit wird eine grössere mathematische Aufgabe selbständig behandelt. Sie umfasst in der Regel das Studium vorhandener Fachliteratur, die Lösung weiterer damit verbundener Fragen sowie die schriftliche Darstellung der Ergebnisse.
Lernziel

Zusätzliche Veranstaltungen
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-5000-00L	Zurich Colloquium in Mathematics	E-	0 KP		R. Abgrall, A. Bandeira, M. Iacobelli, A. Iozzi, S. Mishra, R. Pandharipande, weitere Dozierende
Kurzbeschreibung	The 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.
Lernziel
401-5990-00L	Zurich Graduate Colloquium	E-	0 KP	1K	A. Iozzi, weitere Referent/innen
Kurzbeschreibung	The 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.
Lernziel
401-4530-00L	Geometry Graduate Colloquium	E-	0 KP	1K	Referent/innen
Kurzbeschreibung
Lernziel
401-5110-00L	Number Theory Seminar	E-	0 KP	1K	Ö. Imamoglu, P. S. Jossen, E. Kowalski, P. D. Nelson, R. Pink, G. Wüstholz
Kurzbeschreibung	Forschungskolloquium
Lernziel	Vorträge über neue Themen aus der Forschung.
Inhalt	Forschungsseminar in Algebra, Zahlentheorie und Geometrie, richtet sich insbesondere an Mitarbeiterinnen und Mitarbeiter sowie Doktorandinnen und Doktoranden.
401-5350-00L	Analysis Seminar	E-	0 KP	1K	A. Carlotto, F. Da Lio, A. Figalli, N. Hungerbühler, M. Iacobelli, L. Kobel-Keller, T. Rivière, J. Serra, Uni-Dozierende
Kurzbeschreibung	Forschungskolloquium
Lernziel
Inhalt	Research seminar in Analysis
401-5370-00L	Ergodic Theory and Dynamical Systems	E-	0 KP	1K	M. Akka Ginosar, M. Einsiedler, Uni-Dozierende
Kurzbeschreibung	Research colloquium
Lernziel
401-5530-00L	Geometry Seminar	E-	0 KP	1K	M. Burger, M. Einsiedler, P. Feller, A. Iozzi, U. Lang, Uni-Dozierende
Kurzbeschreibung	Forschungskolloquium
Lernziel
401-5580-00L	Symplectic Geometry Seminar	E-	0 KP		P. Biran, A. Cannas da Silva
Kurzbeschreibung	Forschungskolloquium
Lernziel
401-5330-00L	Talks in Mathematical Physics	E-	0 KP	1K	A. Cattaneo, G. Felder, M. Gaberdiel, G. M. Graf, T. H. Willwacher, Uni-Dozierende
Kurzbeschreibung	Research colloquium
Lernziel
Inhalt	Forschungsseminar mit wechselnden Themen aus dem Gebiet der mathematischen Physik.
401-5650-00L	Zurich Colloquium in Applied and Computational Mathematics	E-	0 KP	1K	R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, S. Mishra, S. Sauter, C. Schwab
Kurzbeschreibung	Forschungskolloquium
Lernziel
401-5600-00L	Seminar on Stochastic Processes	E-	0 KP		J. Bertoin, A. Nikeghbali, B. D. Schlein, V. Tassion, W. Werner
Kurzbeschreibung	Forschungskolloquium
Lernziel
401-5620-00L	Research Seminar on Statistics	E-	0 KP	1K	P. L. Bühlmann, M. H. Maathuis, N. Meinshausen, S. van de Geer, A. Bandeira, R. Furrer, L. Held, T. Hothorn, D. Kozbur, M. Wolf
Kurzbeschreibung	Forschungskolloquium
Lernziel
401-5640-00L	ZüKoSt: Seminar on Applied Statistics	E-	0 KP	1K	M. Kalisch, F. Balabdaoui, A. Bandeira, 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
Kurzbeschreibung	5 bis 6 Vorträge zur angewandten Statistik.
Lernziel	Kennenlernen von statistischen Methoden in ihrer Anwendung in verschiedenen Gebieten, besonders in Naturwissenschaft, Technik und Medizin.
Inhalt	In 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.
Skript	Bei 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.
Voraussetzungen / Besonderes	Dies 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.
401-5680-00L	Foundations of Data Science Seminar	E-	0 KP		P. L. Bühlmann, A. Bandeira, H. Bölcskei, J. M. Buhmann, T. Hofmann, A. Krause, A. Lapidoth, H.‑A. Loeliger, M. H. Maathuis, N. Meinshausen, G. Rätsch, S. van de Geer, F. Yang
Kurzbeschreibung	Research colloquium
Lernziel
401-5660-00L	Math and Data (MAD+)	E-	0 KP		A. Bandeira, externe Veranstalter
Kurzbeschreibung	Research colloquium
Lernziel
401-5910-00L	Talks in Financial and Insurance Mathematics	E-	0 KP	1K	B. Acciaio, P. Cheridito, D. Possamaï, M. Schweizer, J. Teichmann, M. V. Wüthrich
Kurzbeschreibung	Forschungskolloquium
Lernziel	Einfuehrung in aktuelle Forschungsthemen aus dem Bereich "Insurance Mathematics and Stochastic Finance".
Inhalt	https://www.math.ethz.ch/imsf/courses/talks-in-imsf.html
401-5900-00L	Optimization Seminar	E-	0 KP	1K	A. Bandeira, R. Weismantel, R. Zenklusen
Kurzbeschreibung	Lectures on current topics in optimization.
Lernziel	This lecture series introduces graduate students to ongoing research activities (including applications) in the domain of optimization.
Inhalt	This 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.
402-0101-00L	The Zurich Physics Colloquium	E-	0 KP	1K	S. Huber, A. Refregier, Uni-Dozierende
Kurzbeschreibung	Research colloquium
Lernziel
251-0100-00L	Kolloquium für Informatik	E-	0 KP	2K	Dozent/innen
Kurzbeschreibung	Eingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.
Lernziel
Inhalt	Eingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.
252-4202-00L	Seminar in Theoretical Computer Science	E-	2 KP	2S	E. Welzl, B. Gärtner, M. Ghaffari, M. Hoffmann, J. Lengler, D. Steurer, B. Sudakov
Kurzbeschreibung	Presentation of recent publications in theoretical computer science, including results by diploma, masters and doctoral candidates.
Lernziel	To get an overview of current research in the areas covered by the involved research groups. To present results from the literature.
Voraussetzungen / Besonderes	This seminar takes place as part of the joint research seminar of several theory groups. Intended participation is for students with excellent performance only. Formal restriction is: prior successful participation in a master level seminar in theoretical computer science.

Auflagen-Lerneinheiten Das untenstehende Lehrangebot gilt nur für MSc Studierende mit Zulassungsauflagen.
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
406-2004-AAL	Algebra II Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben. Alle anderen Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.	E-	5 KP	11R	M. Burger
Kurzbeschreibung	Galois theory and related topics. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
Lernziel	Introduction to fundamentals of field extensions, Galois theory, and related topics.
Inhalt	The main topic is Galois Theory. Starting point is the problem of solvability of algebraic equations by radicals. Galois theory solves this problem by making a connection between field extensions and group theory. Galois theory will enable us to prove the theorem of Abel-Ruffini, that there are polynomials of degree 5 that are not solvable by radicals, as well as Galois' theorem characterizing those polynomials which are solvable by radicals.
Literatur	Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society Galois Theory is the topic treated in Chapter A5.
Voraussetzungen / Besonderes	Algebra I, in Rotman's book this corresponds to the topics treated in the Chapters A3 and A4.
406-2005-AAL	Algebra I and II Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben. Alle anderen Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.	E-	12 KP	26R	M. Burger, M. Einsiedler
Kurzbeschreibung	Introduction and development of some basic algebraic structures - groups, rings, fields including Galois theory, representations of finite groups, algebras. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
Lernziel
Inhalt	Basic notions and examples of groups; Subgroups, Quotient groups and Homomorphisms, Group actions and applications Basic notions and examples of rings; Ring Homomorphisms, ideals, and quotient rings, rings of fractions Euclidean domains, Principal ideal domains, Unique factorization domains Basic notions and examples of fields; Field extensions, Algebraic extensions, Classical straight edge and compass constructions Fundamentals of Galois theory Representation theory of finite groups and algebras
Literatur	Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society
406-2284-AAL	Measure and Integration Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben. Alle anderen Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.	E-	6 KP	13R	F. Da Lio
Kurzbeschreibung	Introduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces
Lernziel	Basic acquaintance with the abstract theory of measure and integration
Inhalt	Introduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces
Skript	no lecture notes
Literatur	1. P.R. Halmos, "Measure Theory", Springer 2. Extra material: Lecture Notes by Emmanuel Kowalski and Josef Teichmann from spring semester 2012, http://www.math.ethz.ch/~jteichma/measure-integral_120615.pdf 3. Extra material: P. Cannarsa & T. D'Aprile, "Lecture Notes on Measure Theory and Functional Analysis", http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf
Voraussetzungen / Besonderes	The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2303-AAL	Complex Analysis Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben. Alle anderen Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.	E-	6 KP	13R	A. Bandeira
Kurzbeschreibung	Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, conformal mappings, Riemann mapping theorem.
Lernziel
Literatur	L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co. B. Palka: "An introduction to complex function theory." Undergraduate Texts in Mathematics. Springer-Verlag, 1991. R.Remmert: Theory of Complex Functions.. Springer Verlag E.Hille: Analytic Function Theory. AMS Chelsea Publication
Voraussetzungen / Besonderes	The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2554-AAL	Topology Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben. Alle anderen Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.	E-	6 KP	13R	P. Feller
Kurzbeschreibung	Topics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces.
Lernziel	An introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures.
Skript	See lecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-2554-00L/
Literatur	James Munkres: Topology
Voraussetzungen / Besonderes	The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2604-AAL	Probability and Statistics Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben. Alle anderen Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.	E-	7 KP	15R	J. Teichmann
Kurzbeschreibung	- Statistical models - Methods of moments - Maximum likelihood estimation - Hypothesis testing - Confidence intervals - Introductory Bayesian statistics - Linear regression model - Rudiments of high-dimensional statistics
Lernziel	The goal of this part of the course is to provide a solid introduction into statistics. It offers of a wide overview of the main tools used in statistical inference. The course will start with an introduction to statistical models and end with some notions of high-dimensional statistics. Some time will be spent on proving certain important results. Tools from probability and measure theory will be assumed to be known and hence will be only and occasionally recalled.
Skript	Script of Prof. Dr. S. van de Geer
Literatur	These references could be use complementary sources: R. Berger and G. Casella, Statistical Inference J. A. Rice, Mathematical Statistics and Data Analysis L. Wasserman, All of Statistics

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