# Search result: Catalogue data in Autumn Semester 2018

Mathematics Master | ||||||

Core Courses For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | ||||||

Core Courses: Pure Mathematics | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3225-00L | Introduction to Lie Groups | W | 8 credits | 4G | M. Burger | |

Abstract | Topological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups. | |||||

Learning objective | The goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it. | |||||

Literature | A. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser) A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73) F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer) H. Samelson: "Notes on Lie algebras" (Springer, '90) S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78) A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press) | |||||

Prerequisites / Notice | Topology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester. Course webpage: https://metaphor.ethz.ch/x/2018/hs/401-3225-00L/ | |||||

401-3001-61L | Algebraic Topology I | W | 8 credits | 4G | P. Biran | |

Abstract | This is an introductory course in algebraic topology. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms, cohomology. Along the way we will introduce the basics of homological algebra and category theory. | |||||

Learning objective | ||||||

Literature | 1) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 2) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: http://www.math.cornell.edu/~hatcher/AT/ATpage.html See also: http://www.math.cornell.edu/~hatcher/#anchor1772800 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||

Prerequisites / Notice | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level usually covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not absolutely necessary. Some (elementary) group theory and algebra will also be needed. | |||||

401-3132-00L | Commutative Algebra | W | 10 credits | 4V + 1U | P. D. Nelson | |

Abstract | This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. | |||||

Learning objective | We shall cover approximately the material from --- most of the textbook by Atiyah-MacDonald, or --- the first half of the textbook by Bosch. Topics include: * Basics about rings, ideals and modules * Localization * Primary decomposition * Integral dependence and valuations * Noetherian rings * Completions * Basic dimension theory | |||||

Literature | Primary Reference: 1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969) Secondary Reference: 2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013) Tertiary References: 3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995) 4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989) 5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer) | |||||

Prerequisites / Notice | Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory). | |||||

Core Courses: Applied Mathematics and Further Appl.-Oriented Fields ¬ | ||||||

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

401-3651-00L | Numerical Methods for Elliptic and Parabolic Partial Differential Equations (University of Zurich)Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students. Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester. No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT802 Mind the enrolment deadlines at UZH: https://www.uzh.ch/cmsssl/en/studies/application/mobilitaet.html | W | 9 credits | 4V + 2U | S. Sauter | |

Abstract | This course gives a comprehensive introduction into the numerical treatment of linear and non-linear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods. | |||||

Learning objective | Participants of the course should become familiar with * concepts underlying the discretization of elliptic and parabolic boundary value problems * analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems * methods for the efficient solution of discrete boundary value problems * implementational aspects of the finite element method | |||||

Content | A selection of the following topics will be covered: * Elliptic boundary value problems * Galerkin discretization of linear variational problems * The primal finite element method * Mixed finite element methods * Discontinuous Galerkin Methods * Boundary element methods * Spectral methods * Adaptive finite element schemes * Singularly perturbed problems * Sparse grids * Galerkin discretization of elliptic eigenproblems * Non-linear elliptic boundary value problems * Discretization of parabolic initial boundary value problems | |||||

Lecture notes | Course slides will be made available to the audience. | |||||

Literature | S. C. Brenner and L. Ridgway Scott: The mathematical theory of Finite Element Methods. New York, Berlin [etc]: Springer-Verl, cop.1994. A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 Additional Literature: D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications, Springer, 2012 [DOI: 10.1007/978-3-642-22980-0] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006). | |||||

Prerequisites / Notice | Practical exercises based on MATLAB | |||||

401-3621-00L | Fundamentals of Mathematical Statistics | W | 10 credits | 4V + 1U | S. van de Geer | |

Abstract | The course covers the basics of inferential statistics. | |||||

Learning objective | ||||||

401-4889-00L | Mathematical Finance | W | 11 credits | 4V + 2U | M. Schweizer | |

Abstract | Advanced course on mathematical finance: - semimartingales and general stochastic integration - absence of arbitrage and martingale measures - fundamental theorem of asset pricing - option pricing and hedging - hedging duality - optimal investment problems - additional topics | |||||

Learning objective | Advanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes) | |||||

Content | This is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models. Topics include - semimartingales and general stochastic integration - absence of arbitrage and martingale measures - fundamental theorem of asset pricing - option pricing and hedging - hedging duality - optimal investment problems - and probably others | |||||

Lecture notes | The course is based on different parts from different books as well as on original research literature. Lecture notes will not be available. | |||||

Literature | (will be updated later) | |||||

Prerequisites / Notice | Prerequisites are the standard courses - Probability Theory (for which lecture notes are available) - Brownian Motion and Stochastic Calculus (for which lecture notes are available) Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts. This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II. For an overview of courses offered in the area of mathematical finance, see Link. | |||||

401-3901-00L | Mathematical Optimization | W | 11 credits | 4V + 2U | R. Weismantel | |

Abstract | Mathematical treatment of diverse optimization techniques. | |||||

Learning objective | Advanced optimization theory and algorithms. | |||||

Content | 1) Linear optimization: The geometry of linear programming, the simplex method for solving linear programming problems, Farkas' Lemma and infeasibility certificates, duality theory of linear programming. 2) Nonlinear optimization: Lagrange relaxation techniques, Newton method and gradient schemes for convex optimization. 3) Integer optimization: Ties between linear and integer optimization, total unimodularity, complexity theory, cutting plane theory. 4) Combinatorial optimization: Network flow problems, structural results and algorithms for matroids, matchings, and, more generally, independence systems. | |||||

Literature | 1) D. Bertsimas & R. Weismantel, "Optimization over Integers". Dynamic Ideas, 2005. 2) A. Schrijver, "Theory of Linear and Integer Programming". John Wiley, 1986. 3) D. Bertsimas & J.N. Tsitsiklis, "Introduction to Linear Optimization". Athena Scientific, 1997. 4) Y. Nesterov, "Introductory Lectures on Convex Optimization: a Basic Course". Kluwer Academic Publishers, 2003. 5) C.H. Papadimitriou, "Combinatorial Optimization". Prentice-Hall Inc., 1982. | |||||

Prerequisites / Notice | Linear algebra. | |||||

Bachelor Core Courses: Pure Mathematics Further restrictions apply, but in particular: 401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme. Analogously for: 401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II 401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II 401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

401-3461-00L | Functional Analysis I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | E- | 10 credits | 4V + 1U | M. Einsiedler | |

Abstract | Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces; Fourier transform and applications. | |||||

Learning objective | Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps. | |||||

Literature | We will be using the book Functional Analysis, Spectral Theory, and Applications by Manfred Einsiedler and Thomas Ward and available by SpringerLink. Other useful, and recommended references include the following: Lecture Notes on "Funktionalanalysis I" by Michael Struwe Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. | |||||

Prerequisites / Notice | Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with measure theory, Lebesgue integration and L^p spaces). | |||||

401-3531-00L | Differential Geometry I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | E- | 10 credits | 4V + 1U | W. Merry | |

Abstract | This will be an introductory course in differential geometry. Topics covered include: - Smooth manifolds, submanifolds, vector fields, - Lie groups, homogeneous spaces, - Vector bundles, tensor fields, differential forms, - Integration on manifolds and the de Rham theorem, - Principal bundles. | |||||

Learning objective | ||||||

Lecture notes | I will produce full lecture notes, available on my website at www.merry.io/differential-geometry | |||||

Literature | There are many excellent textbooks on differential geometry. A friendly and readable book that covers everything in Differential Geometry I is: John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag. A more advanced (and far less friendly) series of books that covers everything in both Differential Geometry I and II is: S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volumes I and II (1963, 1969) Wiley. | |||||

Bachelor Core Courses: Applied Mathematics ... Further restrictions apply, but in particular: 401-3601-00L Probability Theory can only be recognised for the Master Programme if neither 401-3642-00L Brownian Motion and Stochastic Calculus nor 401-3602-00L Applied Stochastic Processes has been recognised for the Bachelor Programme. 402-0205-00L Quantum Mechanics I is eligible as an applied core course, but only if 402-0224-00L Theoretical Physics (offered for the last time in FS 2016) isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

401-3601-00L | Probability Theory At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | E- | 10 credits | 4V + 1U | A.‑S. Sznitman | |

Abstract | Basics of probability theory and the theory of stochastic processes in discrete time | |||||

Learning objective | This course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned: Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains. | |||||

Content | This course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned: Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains. | |||||

Lecture notes | available, will be sold in the course | |||||

Literature | R. Durrett, Probability: Theory and examples, Duxbury Press 1996 H. Bauer, Probability Theory, de Gruyter 1996 J. Jacod and P. Protter, Probability essentials, Springer 2004 A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006 D. Williams, Probability with martingales, Cambridge University Press 1991 | |||||

402-0205-00L | Quantum Mechanics I | W | 10 credits | 3V + 2U | M. Gaberdiel | |

Abstract | Introduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, wave functions, the description of observables as operators on a Hilbert space, as well as the formulation of symmetries, will be discussed. Basic phenomena will be analysed and illustrated by generic examples. | |||||

Learning objective | Introduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, angular momentum, perturbation theory) and generic examples and applications (bound states, tunneling, hydrogen atom, harmonic oscillator). Ability to solve simple problems. | |||||

Content | Keywords: Schrödinger equation, basic formalism of quantum mechanics (states, operators, commutators, measuring process), symmetries (translations, rotations, discrete symmetries), quantum mechanics in one dimension, spherically symmetric problems in three dimensions, hydrogen atom, harmonic oscillator, angular momentum, spin, addition of angular momenta, relation between QM and classical physics. | |||||

Literature | J.J. Sakurai: Modern Quantum Mechanics A. Messiah: Quantum Mechanics I S. Weinberg: Lectures on Quantum Mechanics | |||||

Electives For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | ||||||

Electives: Pure Mathematics | ||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||

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

401-3113-68L | Exponential Sums over Finite Fields | W | 8 credits | 4G | E. Kowalski | |

Abstract | Exponential sums over finite fields arise in many problems of number theory. We will discuss the elementary aspects of the theory (centered on the Riemann Hypothesis for curves, following Stepanov's method) and survey the formalism arising from Deligne's general form of the Riemann Hypothesis over finite fields. We will then discuss various applications, especially in analytic number theory. | |||||

Learning objective | The goal is to understand both the basic results on exponential sums in one variable, and the general formalism of Deligne and Katz that underlies estimates for much more general types of exponential sums, including the "trace functions" over finite fields. | |||||

Content | Examples of elementary exponential sums The Riemann Hypothesis for curves and its applications Definition of trace functions over finite fields The formalism of the Riemann Hypothesis of Deligne Selected applications | |||||

Lecture notes | Lectures notes from various sources will be provided | |||||

Literature | Kowalski, "Exponential sums over finite fields, I: elementary methods: Iwaniec-Kowalski, "Analytic number theory", chapter 11 Fouvry, Kowalski and Michel, "Trace functions over finite fields and their applications" | |||||

401-3100-68L | Introduction to Analytic Number Theory | W | 8 credits | 4G | I. N. Petrow | |

Abstract | This course is an introduction to classical multiplicative analytic number theory. The main object of study is the distribution of the prime numbers in the integers. We will study arithmetic functions and learn the basic tools for manipulating and calculating their averages. We will make use of generating series and tools from complex analysis. | |||||

Learning objective | The main goal for the course is to prove the prime number theorem in arithmetic progressions: If gcd(a,q)=1, then the number of primes p = a mod q with p<x is approximately (1/phi(q))*(x/log x), as x tends to infinity, where phi(q) is the Euler totient function. | |||||

Content | Developing the necessary techniques and theory to prove the prime number theorem in arithmetic progressions will lead us to the study of prime numbers by Chebyshev's method, to study techniques for summing arithmetic functions by Dirichlet series, multiplicative functions, L-series, characters of a finite abelian group, theory of integral functions, and a detailed study of the Riemann zeta function and Dirichlet's L-functions. | |||||

Lecture notes | Lecture notes will be provided for the course. | |||||

Literature | Multiplicative Number Theory by Harold Davenport Multiplicative Number Theory I. Classical Theory by Hugh L. Montgomery and Robert C. Vaughan Analytic Number Theory by Henryk Iwaniec and Emmanuel Kowalski | |||||

Prerequisites / Notice | Complex analysis Group theory Linear algebra Familiarity with the Fourier transform and Fourier series preferable but not required. | |||||

401-3059-00L | Combinatorics IIDoes not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | |||||

Learning objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | |||||

Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | |||||

Selection: Geometry | ||||||

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

401-3057-00L | Finite Geometries II | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares. | |||||

Learning objective | Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design. | |||||

Content | Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design | |||||

Literature | - Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries. | |||||

401-3111-68L | Elliptic Curves and Cryptography | W | 8 credits | 3V + 1U | L. Halbeisen | |

Abstract | Im ersten Teil der Vorlesung wird die algebraische Struktur von elliptischen Kurven behandelt. Insbesondere wird der Satz von Mordell bewiesen. Im zweiten Teil der Vorlesung werden dann Anwendungen elliptischer Kurven in der Kryptographie gezeigt, wie z.B. der Diffie-Hellman-Schluesselaustausch. | |||||

Learning objective | Rationale Punkte auf elliptischen Kurven, insbesondere Arithmetik auf elliptischen Kurven, Satz von Mordell, Kongruente Zahlen Anwendungen der elliptischen Kurven in der Kryptographie, wie zum Beispiel Diffie-Hellman-Schluesselaustausch, Pollard-Rho-Methode | |||||

Content | Im ersten Teil der Vorlesung wird die algebraische Struktur von elliptischen Kurven behandelt und die Menge der rationalen Punkte auf elliptischen Kurven untersucht. Insbesondere wird mit Hilfe von Saetzen aus der Algebra wie auch aus der projektiven Geometrie gezeigt, dass die Menge der rationalen Punkte auf einer elliptischen Kurven unter einer bestimmten Operation eine endlich erzeugte abelsche Gruppe bildet. Zudem werden elliptische Kurven untersucht, welche mit rationalen, rechtwinkligen Dreiecken mit ganzzahligem Flaecheninhalt zusammenhaengen. Im zweiten Teil der Vorlesung werden dann Anwendungen elliptischer Kurven in der Kryptographie gezeigt. Solche Anwendungen sind zum Beispiel ein auf elliptischen Kurven basierendes Kryptosystem oder ein Algorithmus zur Faktorisierung grosser Zahlen. | |||||

Literature | Joseph Silverman, John Tate: "Rational Points on Elliptic Curves", Undergraduate Texts in Mathematics, Springer-Verlag (1992) Ian Blake, Gadiel Seroussi, Nigel Smart: "Elliptic Curves in Cryptography", Lecture Notes Series 265, Cambridge University Press (2004) | |||||

Prerequisites / Notice | Voraussgesetzt werden Algebra I und Grundbegriffe der projektiven Geometrie. | |||||

Selection: Analysis | ||||||

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

401-4115-00L | Introduction to Geometric Measure Theory | W | 6 credits | 3V | U. Lang | |

Abstract | Introduction to Geometric Measure Theory from a metric viewpoint. Contents: Lipschitz maps, differentiability, area and coarea formula, rectifiable sets, introduction to the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, normal currents, relation to BV functions, slicing, compactness theorem for integral currents and applications. | |||||

Learning objective | ||||||

Content | Extendability and differentiability of Lipschitz maps, metric differentiability, rectifiable sets, approximate tangent spaces, area and coarea formula, brief survey of the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, currents with finite mass and normal currents, relation to BV functions, rectifiable and integral currents, slicing, compactness theorem for integral currents and applications. | |||||

Literature | - Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, 1995 - Herbert Federer, Geometric Measure Theory, 1969 - Leon Simon, Introduction to Geometric Measure Theory, 2014, web.stanford.edu/class/math285/ts-gmt.pdf - Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta math. 185 (2000), 1-80 - Urs Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683-742 | |||||

401-4463-62L | Fourier Analysis in Function Space Theory | W | 4 credits | 2V | T. Rivière | |

Abstract | In the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators. | |||||

Learning objective | ||||||

Content | During the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces. In the second part of the course we will study fundamental properties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occasion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings. In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators. This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course. If time permits we shall present the notion of Paraproduct, Paracompositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE. | |||||

Literature | 1) Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions" (PMS-30) Princeton University Press. 2) Javier Duoandikoetxea, "Fourier Analysis" AMS. 3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer. 4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer. | |||||

Prerequisites / Notice | Notions from ETH courses in Measure Theory, Functional Analysis I and II (Fundamental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions) | |||||

Selection: Further Realms | ||||||

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

401-3502-68L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 2 credits | 4A | Professors | |

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

Learning objective | ||||||

401-3503-68L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 3 credits | 6A | Professors | |

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

Learning objective | ||||||

401-3504-68L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 4 credits | 9A | Professors | |

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

Learning objective | ||||||

401-0000-00L | Communication in Mathematics | W | 1 credit | 1V | W. Merry | |

Abstract | Don't hide your Next Great Theorem behind bad writing. This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. | |||||

Learning objective | Knowing how to present written mathematics in a structured and clear manner. | |||||

Content | Topics covered include: - How to write a thesis (more generally, a mathematics paper). - Elementary LaTeX skills and language conventions. - How to write a personal statement for Masters and PhD applications. | |||||

Lecture notes | Full lecture notes will be made available on my website: www.merry.io/communication-in-mathematics | |||||

Prerequisites / Notice | There are no formal mathematical prerequisites. | |||||

Electives: Applied Mathematics and Further Application-Oriented Fields ¬ | ||||||

Selection: Numerical Analysis | ||||||

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

401-4657-00L | Numerical Analysis of Stochastic Ordinary Differential Equations Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods" | W | 6 credits | 3V + 1U | A. Jentzen, L. Yaroslavtseva | |

Abstract | Course on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables. | |||||

Learning objective | The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues. | |||||

Content | Generation of random numbers Monte Carlo methods for the numerical integration of random variables Stochastic processes and Brownian motion Stochastic ordinary differential equations (SODEs) Numerical approximations of SODEs Applications to computational finance: Option valuation | |||||

Lecture notes | Lecture notes are available as a PDF file: see Learning materials. | |||||

Literature | P. Glassermann: Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, 2004. P. E. Kloeden and E. Platen: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992. | |||||

Prerequisites / Notice | Prerequisites: Mandatory: Probability and measure theory, basic numerical analysis and basics of MATLAB programming. a) mandatory courses: Elementary Probability, Probability Theory I. b) recommended courses: Stochastic Processes. Start of lectures: Wednesday, September 19, 2018. Date of the End-of-Semester examination: Wednesday, December 19, 2018, 13:00-15:00; students must arrive before 12:30 at ETH HG E 19. Room for the End-of-Semester examination: ETH HG E 19. Exam inspection: Tuesday, February 26, 2019, 12:00-13:00 at HG D 7.2. Please bring your legi. | |||||

401-4785-00L | Mathematical and Computational Methods in Photonics | W | 8 credits | 4G | H. Ammari | |

Abstract | The aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces | |||||

Learning objective | The field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications. The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength. Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures. The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions. In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials. | |||||

401-4357-68L | On Deep Artificial Neural Networks and Partial Differential Equations | W | 4 credits | 2G | A. Jentzen | |

Abstract | In this lecture we rigorously analyse approximation capacities of deep artificial neural networks and prove that deep artificial neural networks do overcome the curse of dimensionality in the numerical approximation of solutions of partial differential equations (PDEs). | |||||

Learning objective | The aim of this course is to teach the students a decent knowledge on deep artificial neural networks and their approximation capacities. | |||||

Content | In recent years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a series of computational problems ranging from computer vision, image classification, speech recognition, and natural language processing to computational advertisement. Such numerical simulations indicate that deep artificial neural networks seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named applications. In this lecture we rigorously analyse approximation capacities of deep artificial neural networks and prove that deep artificial neural networks do overcome the curse of dimensionality in the numerical approximation of solutions of partial differential equations (PDEs). In particular, this course includes (i) a rigorous mathematical introduction to artificial neural networks, (ii) an introduction to some partial differential equations, and (iii) results on approximation capacities of deep artificial neural networks. | |||||

Lecture notes | Lecture notes will be available as a PDF file. | |||||

Literature | Related literature: * Arnulf Jentzen, Diyora Salimova, and Timo Welti, A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients. arXiv:1809.07321 (2018), 48 pages. Available online at [https://arxiv.org/abs/1809.07321]. * Philipp Grohs, Fabian Hornung, Arnulf Jentzen, and Philippe von Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv:1809.02362 (2018), 124 pages. Available online at [https://arxiv.org/abs/1809.02362]. * Andrew R. Barron, Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39 (1993), no. 3, 930--945. | |||||

Prerequisites / Notice | Prerequisites: Analysis I and II, Elementary Probability Theory, and Measure Theory | |||||

401-4503-68L | Reading Course: Reduced Basis Methods | W | 4 credits | 2G | R. Hiptmair | |

Abstract | Reduced Basis Methods (RBM) allow the efficient repeated numerical soluton of parameter depedent differential equations, which arise, e.g., in PDE-constrained optimization, optimal control, inverse problems, and uncertainty quantification. This course introduces the mathematical foundations of RBM and discusses algorithmic and implementation aspects. | |||||

Learning objective | * Knowledge about the main principles underlying RBMs * Familiarity with algorithms for the construction of reduced bases * Knowledge about the role of and techniques for a posteriori error estimation. * Familiarity with some applications of RBMs. | |||||

Literature | Main reference: Hesthaven, Jan S.; Rozza, Gianluigi; Stamm, Benjamin, Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics, 2016 Supplementary reference: Quarteroni, Alfio; Manzoni, Andrea; Negri, Federico, Reduced basis methods for partial differential equations. An introduction. Unitext 92, Springer, Cham, 2016. | |||||

Prerequisites / Notice | This is a reading course, which will closely follow the book by J. Hesthaven, G. Rozza and B. Stamm. Participants are expected to study particular sections of the book every week, which will then be discussed during the course sessions. | |||||

Selection: Probability Theory, Statistics | ||||||

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

401-4607-68L | Topics on the Gaussian Free Field | W | 4 credits | 2V | W. Werner | |

Abstract | We will discuss various aspects and properties of the Gaussian Free Field. | |||||

Learning objective | ||||||

Content | Topics discussed will include: - Discrete and continuous Gaussian Free Field - Local sets. - Relation to loop-soups. - Uniform spanning trees. | |||||

401-4611-68L | Regularity Structures | W | 6 credits | 3V | J. Teichmann | |

Abstract | We develop the main tools of Martin Hairer's theory of regularity structures to solve singular stochastic partial differential equations in a pathwise way or addtionally by re-normalization techniques. | |||||

Learning objective | ||||||

401-4619-67L | Advanced Topics in Computational StatisticsDoes not take place this semester. | W | 4 credits | 2V | N. Meinshausen | |

Abstract | This lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling. | |||||

Learning objective | Students learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes. | |||||

Content | The main focus will be on graphical models in various forms: Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models | |||||

Prerequisites / Notice | We assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics. | |||||

401-3628-14L | Bayesian StatisticsDoes not take place this semester. | W | 4 credits | 2V | ||

Abstract | Introduction to the Bayesian approach to statistics: Decision theory, prior distributions, hierarchical Bayes models, Bayesian tests and model selection, empirical Bayes, computational methods, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods. | |||||

Learning objective | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||

Content | Topics that we will discuss are: Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, Jeffreys priors), tests and model selection (Bayes factors, hyper-g priors in regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods) | |||||

Lecture notes | A script will be available in English. | |||||

Literature | Christian Robert, The Bayesian Choice, 2nd edition, Springer 2007. A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013). Additional references will be given in the course. | |||||

Prerequisites / Notice | Familiarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed. | |||||

401-0625-01L | Applied Analysis of Variance and Experimental Design | W | 5 credits | 2V + 1U | L. Meier | |

Abstract | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||

Learning objective | Participants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R. | |||||

Content | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||

Literature | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||

Prerequisites / Notice | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held. | |||||

401-0649-00L | Applied Statistical Regression | W | 5 credits | 2V + 1U | M. Dettling | |

Abstract | This course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis. | |||||

Learning objective | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||

Content | The course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies. The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data. | |||||

Lecture notes | A script will be available. | |||||

Literature | Faraway (2005): Linear Models with R Faraway (2006): Extending the Linear Model with R Draper & Smith (1998): Applied Regression Analysis Fox (2008): Applied Regression Analysis and GLMs Montgomery et al. (2006): Introduction to Linear Regression Analysis | |||||

Prerequisites / Notice | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held. In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Regression" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit. | |||||

401-4637-67L | On Hypothesis Testing | W | 4 credits | 2V | F. Balabdaoui | |

Abstract | This course is a review of the main results in decision theory. | |||||

Learning objective | 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). | |||||

Literature | - Statistical Inference (Casella & Berger) - Testing Statistical Hypotheses (Lehmann and Romano) | |||||

401-3627-00L | High-Dimensional StatisticsDoes not take place this semester. | W | 4 credits | 2V | P. L. Bühlmann | |

Abstract | "High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed. | |||||

Learning objective | Knowledge of methods and basic theory for high-dimensional statistical inference | |||||

Content | Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling | |||||

Literature | Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. ISBN 978-3-642-20191-2. | |||||

Prerequisites / Notice | Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics). | |||||

401-4623-00L | Time Series Analysis | W | 6 credits | 3G | N. Meinshausen | |

Abstract | Statistical analysis and modeling of observations in temporal order, which exhibit dependence. Stationarity, trend estimation, seasonal decomposition, autocorrelations, spectral and wavelet analysis, ARIMA-, GARCH- and state space models. Implementations in the software R. | |||||

Learning objective | Understanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R. | |||||

Content | This course deals with modeling and analysis of variables which change randomly in time. Their essential feature is the dependence between successive observations. Applications occur in geophysics, engineering, economics and finance. Topics covered: Stationarity, trend estimation, seasonal decomposition, autocorrelations, spectral and wavelet analysis, ARIMA-, GARCH- and state space models. The models and techniques are illustrated using the statistical software R. | |||||

Lecture notes | Not available | |||||

Literature | A list of references will be distributed during the course. | |||||

Prerequisites / Notice | Basic knowledge in probability and statistics | |||||

401-3612-00L | Stochastic Simulation | W | 5 credits | 3G | F. Sigrist | |

Abstract | This course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo. | |||||

Learning objective | Stochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R. | |||||

Content | Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). | |||||

Lecture notes | A script will be available in English. | |||||

Literature | P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer 2004. B. D. Ripley. Stochastic Simulation. Wiley, 1987. Ch. Robert, G. Casella. Monte Carlo Statistical Methods. Springer 2004 (2nd edition). | |||||

Prerequisites / Notice | Familiarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed. | |||||

401-3822-17L | Ising Model and Its Geometric Representations | W | 4 credits | 2V | V. Tassion | |

Abstract | ||||||

Learning objective | ||||||

Prerequisites / Notice | - Probability Theory. | |||||

Selection: Financial and Insurance Mathematics In the Master's programmes in Mathematics resp. Applied Mathematics 401-3913-01L Mathematical Foundations for Finance is eligible as an elective course, but only if 401-3888-00L Introduction to Mathematical Finance isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

401-3925-00L | Non-Life Insurance: Mathematics and Statistics | W | 8 credits | 4V + 1U | M. V. Wüthrich | |

Abstract | The lecture aims at providing a basis in non-life insurance mathematics which forms a core subject of actuarial sciences. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models, credibility theory, claims reserving and solvency. | |||||

Learning objective | The student is familiar with the basics in non-life insurance mathematics and statistics. This includes the basic mathematical models for insurance liability modeling, pricing concepts, stochastic claims reserving models and ruin and solvency considerations. | |||||

Content | The following topics are treated: Collective Risk Modeling Individual Claim Size Modeling Approximations for Compound Distributions Ruin Theory in Discrete Time Premium Calculation Principles Tariffication and Generalized Linear Models Bayesian Models and Credibility Theory Claims Reserving Solvency Considerations | |||||

Lecture notes | M. V. Wüthrich, Non-Life Insurance: Mathematics & Statistics http://ssrn.com/abstract=2319328 | |||||

Prerequisites / Notice | 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. Prerequisites: knowledge of probability theory, statistics and applied stochastic processes. | |||||

401-3922-00L | Life Insurance Mathematics | W | 4 credits | 2V | M. Koller | |

Abstract | The classical life insurance model is presented together with the important insurance types (insurance on one and two lives, term and endowment insurance and disability). Besides that the most important terms such as mathematical reserves are introduced and calculated. The profit and loss account and the balance sheet of a life insurance company is explained and illustrated. | |||||

Learning objective | ||||||

401-3928-00L | Reinsurance Analytics | W | 4 credits | 2V | P. Antal, P. Arbenz | |

Abstract | This course provides an actuarial introduction to reinsurance. The objective is to understand the fundamentals of risk transfer through reinsurance, and the mathematical models for extreme events such as natural or man-made catastrophes. The lecture covers reinsurance contracts, Experience and Exposure pricing, natural catastrophe modelling, solvency regulation, and alternative risk transfer | |||||

Learning objective | This course provides an introduction to reinsurance from an actuarial point of view. The objective is to understand the fundamentals of risk transfer through reinsurance, and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes. Topics covered include: - Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business. - Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models - Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks - Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context - Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2 - Alternative Risk Transfer: Alternatives to traditional reinsurance such as insurance linked securities and catastrophe bonds | |||||

Content | This course provides an introduction to reinsurance from an actuarial point of view. The objective is to understand the fundamentals of risk transfer through reinsurance, and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes. Topics covered include: - Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business. - Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models - Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks - Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context - Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2 - Alternative Risk Transfer: Alternatives to traditional reinsurance such as insurance linked securities and catastrophe bonds | |||||

Lecture notes | Slides, lecture notes, and references to literature will be made available. | |||||

Prerequisites / Notice | Basic knowledge in statistics, probability theory, and actuarial techniques | |||||

401-3927-00L | Mathematical Modelling in Life Insurance | W | 4 credits | 2V | T. J. Peter | |

Abstract | In Life insurance, it is essential to have adequate mortality tables, be it for reserving or pricing purposes. We learn to create mortality tables from scratch. Additionally, we study various guarantees embedded in life insurace products and learn to price them with the help of stochastic models. | |||||

Learning objective | The course's objective is to provide the students with the understanding and the tools to create mortality tables on their own. Additionally, students should learn to price embedded options in Life insurance. Aside of the mere application of specific models, they should develop an intuition for the various drivers of the value of these options. | |||||

Content | Following main topics are covered: 1. Overview on guarantees & options in life insurance with a real-world example demonstrating their risks 2. Mortality tables - Determining raw mortality rates - Smoothing of raw mortality rates - Trends in mortality rates - Lee-Carter model - Integration of safety margins 3. Primer on Financial Mathematics - Ito integral - Black-Scholes and Hull-White model 4. Valuation of Unit linked contracts with embedded options 5. Valuation of Participating contracts | |||||

Lecture notes | Lectures notes and slides will be provided | |||||

Prerequisites / Notice | The exams ONLY take place during the official ETH examination period. The course counts towards the diploma of "Aktuar SAV". Good knowledge in probability theory and stochastic processes is assumed. Some knowledge in financial mathematics is useful. | |||||

401-4912-11L | Trends in Stochastic Portfolio Theory | W | 4 credits | 2V | M. Larsson | |

Abstract | This course presents an introduction to Stochastic Portfolio Theory, which provides a mathematical framework for studying and exploiting empirically observed regularities of large equity markets. A central goal of the theory is to describe certain forms of arbitrage that arise over sufficiently long time horizons. | |||||

Learning objective | ||||||

Content | This course presents an introduction to Stochastic Portfolio Theory, which provides a mathematical framework for studying and exploiting empirically observed regularities of large equity markets. A central goal of the theory is to describe certain forms of arbitrage that arise over sufficiently long time horizons. Since it was first introduced by Robert Fernholz almost 20 years ago, the theory has experienced rapid developments. This course will cover the foundations of Stochastic Portfolio Theory, including topics like relative arbitrage, functional portfolio generation, and capital distribution curves, as well as more recent developments. | |||||

Prerequisites / Notice | Prerequisites: Familiarity with Ito calculus at the level of Brownian Motion and Stochastic Calculus. Some background in mathematical finance is helpful. A course with similar content was offered in HS 2015 under the title "New Trends in Stochastic Portfolio Theory". | |||||

401-3905-68L | Convex Optimization in Machine Learning and Computational Finance | W | 4 credits | 2V | P. Cheridito, M. Baes | |

Abstract | ||||||

Learning objective | ||||||

Content | Part 1: Convex Analysis Lecture 1: General introduction, convex sets and functions Lecture 2: Semidefinite cone, Separation theorems (Application to the Fundamental Theorem of Asset Pricing) Lecture 3: Analytic properties of convex functions, duality (Application to Support Vector Machines) Lecture 4: Lagrangian duality, conjugate functions, support functions Lecture 5: Subgradients and subgradient calculus (Application to Automatic Differentiation and Lexicographic Differentiation) Lecture 6: Karush-Kuhn-Tucker Conditions (Application to Markowitz portfolio optimization) Part 2: Applications Lecture 7: Approximation, Lasso optimization, Covariance matrix estimation (Application: a politically optimal splitting of Switzerland) Lecture 8: Clustering and MaxCut problems, Optimal coalitions and Shapley Value Part 3: Algorithms Lecture 9: Intractability of Optimization, Gradient Method for convex optimization, Stochastic Gradient Method (Application to Neural Networks) Lecture 10: Fundamental flaws of Gradient Methods, Mirror Descent Method (Application to Multiplicative Weight Method and Adaboost) Lecture 11: Accelerated Gradient Method, Smoothing Technique (Application to large-scale Lasso optimization) Lecture 12: Newton Method and its fundamental drawbacks, Self-Concordant Functions Lecture 13: Interior-Point Methods | |||||

Selection: Mathematical Physics, Theoretical Physics | ||||||

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

402-0843-00L | Quantum Field Theory I | W | 10 credits | 4V + 2U | A. Gehrmann-De Ridder | |

Abstract | This course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity. Topics include: - Relativistic quantum mechanics - Quantisation of bosonic and fermionic fields - Interactions in perturbation theory - Scattering processes and decays - Elementary processes in QED - Radiative corrections | |||||

Learning objective | The goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics. | |||||

Literature | as provided in the entity Lernmaterialien | |||||

402-0861-00L | Statistical Physics | W | 10 credits | 4V + 2U | G. Blatter | |

Abstract | The lecture focuses on classical and quantum statistical physics. Various techniques, cumulant expansion, path integrals, and specific systems are discussed: Fermions, photons/phonons, Bosons, magnetism, van der Waals gas. Phase transitions are studied in mean field theory (Weiss, Landau). Including fluctuations leads to critical phenomena, scaling, and the renormalization group. | |||||

Learning objective | This lecture gives an introduction into the the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education. | |||||

Content | Thermodynamics, three laws of thermodynamics, thermodynamic potentials, phenomenology of phase transitions. Classical statistical physics: micro-canonical-, canonical-, and grandcanonical ensembles, applications to simple systems. Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons, statistical interaction. Techniques: variational approach, cumulant expansion, path integral formulation. Degenerate fermions: Fermi gas, electrons in magnetic field. Bosons: photons and phonons, Bose-Einstein condensation. Magnetism: Ising-, XY-, Heisenberg models, Weiss mean-field theory. Van der Waals gas-liquid transition. Landau theory of phase transitions, first- and second order, tricritical. Fluctuations: field theory approach, Gauss theory, self-consistent field, Ginzburg criterion. Critical phenomena: scaling theory, universality. Renormalization group: general theory and applications to spin models (real space RG), phi^4 theory (k-space RG), Kosterlitz-Thouless theory. | |||||

Lecture notes | Lecture notes available in English. | |||||

Literature | No specific book is used for the course. Relevant literature will be given in the course. | |||||

402-0830-00L | General Relativity | W | 10 credits | 4V + 2U | R. Renner | |

Abstract | Manifold, Riemannian metric, connection, curvature; Special Relativity; Lorentzian metric; Equivalence principle; Tidal force and spacetime curvature; Energy-momentum tensor, field equations, Newtonian limit; Post-Newtonian approximation; Schwarzschild solution; Mercury's perihelion precession, light deflection. | |||||

Learning objective | Basic understanding of general relativity, its mathematical foundations, and some of the interesting phenomena it predicts. | |||||

Literature | Suggested textbooks: C. Misner, K, Thorne and J. Wheeler: Gravitation S. Carroll - Spacetime and Geometry: An Introduction to General Relativity R. Wald - General Relativity S. Weinberg - Gravitation and Cosmology N. Straumann - General Relativity with applications to Astrophysics | |||||

402-0897-00L | Introduction to String Theory | W | 6 credits | 2V + 1U | B. Hoare | |

Abstract | This course gives an introduction to string theory. The first half of the course will cover the bosonic string and its quantization in flat space, concluding with the introduction of D-branes and T-duality. The second half will cover various advanced topics selected from those listed below. | |||||

Learning objective | The aim of this course is to motivate the subject of string theory, exploring the important role it has played in the development of modern theoretical and mathematical physics. The goal of the first half of the course is to give a pedagogical introduction to the bosonic string in flat space. Building on this foundation, the goal of the second half of the course is to give a flavour of various more advanced topics. | |||||

Content | I. Introduction II. The relativistic point particle III. The classical closed string IV. Quantizing the closed string V. The open string and D-branes VI. T-duality in flat space Possible advanced topics include: VII. Conformal field theory VIII. The Polyakov path integral IX. String interactions X. Low energy effective actions XI. Superstring theory | |||||

Literature | Lecture notes: String Theory - D. Tong http://www.damtp.cam.ac.uk/user/tong/string.html Lectures on String Theory - G. Arutyunov http://stringworld.ru/files/Arutyunov_G._Lectures_on_string_theory.pdf Books: Superstring Theory - M. Green, J. Schwarz and E. Witten (two volumes, CUP, 1988) Volume 1: Introduction Volume 2: Loop Amplitudes, Anomalies and Phenomenology String Theory - J. Polchinski (two volumes, CUP, 1998) Volume 1: An Introduction to the Bosonic String Volume 2: Superstring Theory and Beyond Errata: http://www.kitp.ucsb.edu/~joep/errata.html Basic Concepts of String Theory - R. Blumenhagen, D. Lüst and S. Theisen (Springer-Verlag, 2013) | |||||

Selection: Mathematical Optimization, Discrete Mathematics | ||||||

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

401-3054-14L | Probabilistic Methods in Combinatorics | W | 6 credits | 2V + 1U | B. Sudakov | |

Abstract | This course provides a gentle introduction to the Probabilistic Method, with an emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems. | |||||

Learning objective | ||||||

Content | The topics covered in the class will include (but are not limited to): linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities and pseudo-randomness. | |||||

Literature | - The Probabilistic Method, by N. Alon and J. H. Spencer, 3rd Edition, Wiley, 2008. - Random Graphs, by B. Bollobás, 2nd Edition, Cambridge University Press, 2001. - Random Graphs, by S. Janson, T. Luczak and A. Rucinski, Wiley, 2000. - Graph Coloring and the Probabilistic Method, by M. Molloy and B. Reed, Springer, 2002. | |||||

Auswahl: Theoretical Computer Science, Discrete Mathematics | ||||||

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

263-4500-00L | Advanced Algorithms | W | 6 credits | 2V + 2U + 1A | M. Ghaffari, A. Krause | |

Abstract | This is an advanced course on the design and analysis of algorithms, covering a range of topics and techniques not studied in typical introductory courses on algorithms. | |||||

Learning objective | This course is intended to familiarize students with (some of) the main tools and techniques developed over the last 15-20 years in algorithm design, which are by now among the key ingredients used in developing efficient algorithms. | |||||

Content | the lectures will cover a range of topics, including the following: graph sparsifications while preserving cuts or distances, various approximation algorithms techniques and concepts, metric embeddings and probabilistic tree embeddings, online algorithms, multiplicative weight updates, streaming algorithms, sketching algorithms, and a bried glance at MapReduce algorithms. | |||||

Prerequisites / Notice | This course is designed for masters and doctoral students and it especially targets those interested in theoretical computer science, but it should also be accessible to last-year bachelor students. Sufficient comfort with both (A) Algorithm Design & Analysis and (B) Probability & Concentrations. E.g., having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, though not required formally. If you are not sure whether you're ready for this class or not, please consulte the instructor. | |||||

252-1425-00L | Geometry: Combinatorics and Algorithms | W | 6 credits | 2V + 2U + 1A | E. Welzl, L. F. Barba Flores, M. Hoffmann | |

Abstract | Geometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?) | |||||

Learning objective | The goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve theoretical and practical problems in the area and in various application domains. In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project. | |||||

Content | Planar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar triangulations. | |||||

Lecture notes | yes | |||||

Literature | Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008. Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004. Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002. Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004. | |||||

Prerequisites / Notice | Prerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH. Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area. | |||||

252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 8 credits | 3V + 2U + 2A | A. Steger | |

Abstract | Las Vegas & Monte Carlo algorithms; inequalities of Markov, Chebyshev, Chernoff; negative correlation; Markov chains: convergence, rapidly mixing; generating functions; Examples include: min cut, median, balls and bins, routing in hypercubes, 3SAT, card shuffling, random walks | |||||

Learning objective | After this course students will know fundamental techniques from probabilistic combinatorics for designing randomized algorithms and will be able to apply them to solve typical problems in these areas. | |||||

Content | Randomized Algorithms are algorithms that "flip coins" to take certain decisions. This concept extends the classical model of deterministic algorithms and has become very popular and useful within the last twenty years. In many cases, randomized algorithms are faster, simpler or just more elegant than deterministic ones. In the course, we will discuss basic principles and techniques and derive from them a number of randomized methods for problems in different areas. | |||||

Lecture notes | Yes. | |||||

Literature | - Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995) - Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005) | |||||

Selection: Further Realms | ||||||

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

401-3502-68L | Reading Course Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 2 credits | 4A | Professors | |

Abstract | ||||||

Learning objective | ||||||

401-3503-68L | Reading Course Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 3 credits | 6A | Professors | |

Abstract | ||||||

Learning objective | ||||||

401-3504-68L | Reading Course Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 4 credits | 9A | Professors | |

Abstract | ||||||

Learning objective | ||||||

401-0000-00L | Communication in Mathematics | W | 1 credit | 1V | W. Merry | |

Abstract | Don't hide your Next Great Theorem behind bad writing. This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. | |||||

Learning objective | Knowing how to present written mathematics in a structured and clear manner. | |||||

Content | Topics covered include: - How to write a thesis (more generally, a mathematics paper). - Elementary LaTeX skills and language conventions. - How to write a personal statement for Masters and PhD applications. | |||||

Lecture notes | Full lecture notes will be made available on my website: www.merry.io/communication-in-mathematics | |||||

Prerequisites / Notice | There are no formal mathematical prerequisites. | |||||

Application Area Only necessary and eligible for the Master degree in Applied Mathematics. One of the application areas specified must be selected for the category Application Area for the Master degree in Applied Mathematics. At least 8 credits are required in the chosen application area. | ||||||

Atmospherical Physics | ||||||

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

701-1221-00L | Dynamics of Large-Scale Atmospheric Flow | W | 4 credits | 2V + 1U | H. Wernli, L. Papritz | |

Abstract | This lecture course is about the fundamental aspects of the dynamics of extratropical weather systems (quasi-geostropic dynamics, potential vorticity, Rossby waves, baroclinic instability). The fundamental concepts are formally introduced, quantitatively applied and illustrated with examples from the real atmosphere. Exercises (quantitative and qualitative) form an essential part of the course. | |||||

Learning objective | Understanding the dynamics of large-scale atmospheric flow | |||||

Content | Dynamical Meteorology is concerned with the dynamical processes of the earth's atmosphere. The fundamental equations of motion in the atmosphere will be discussed along with the dynamics and interactions of synoptic system - i.e. the low and high pressure systems that determine our weather. The motion of such systems can be understood in terms of quasi-geostrophic theory. The lecture course provides a derivation of the mathematical basis along with some interpretations and applications of the concept. | |||||

Lecture notes | Dynamics of large-scale atmospheric flow | |||||

Literature | - Holton J.R., An introduction to Dynamic Meteorogy. Academic Press, fourth edition 2004, - Pichler H., Dynamik der Atmosphäre, Bibliographisches Institut, 456 pp. 1997 | |||||

Prerequisites / Notice | Physics I, II, Environmental Fluid Dynamics | |||||

Biology | ||||||

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

551-0015-00L | Biology I | W | 2 credits | 2V | R. Glockshuber, E. Hafen | |

Abstract | The lecture Biology I, together with the lecture Biology II in the following summer semester, is a basic, introductory course into Biology for Students of Materials Sciences and other students with biology as subsidiary subject. | |||||

Learning objective | The goal of this course is to give the students a basic understanding of the molecules that build a cell and make it function, and the basic principles of metabolism and molecular genetics. | |||||

Content | 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 1. Aufbau der Zelle Kapitel 5: Struktur und Funktion biologischer Makromoleküle Kapitel 6: Eine Tour durch die Zelle Kaptiel 7: Membranstruktur und-funktion Kapitel 8: Einführung in den Stoffwechsel Kapitel 9: Zelluläre Atmung und Speicherung chemischer Energie Kapitel 10: Photosynthese Kapitel 12: Der Zellzyklus Kapitel 17: Vom Gen zum Protein 2. Allgemeine Genetik Kapitel 13: Meiose und Reproduktionszyklen Kapitel 14: Mendel'sche Genetik Kapitel 15: Die chromosomale Basis der Vererbung Kapitel 16: Die molekulare Grundlage der Vererbung Kapitel 18: Genetik von Bakterien und Viren Kapitel 46: Tierische Reproduktion Grundlagen des Stoffwechsels und eines Überblicks über molekulare Genetik | |||||

Lecture notes | Der Vorlesungsstoff ist sehr nahe am Lehrbuch gehalten, Skripte werden ggf. durch die Dozenten zur Verfügung gestellt. | |||||

Literature | 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 | |||||

Prerequisites / Notice | Zur Vorlesung Biologie I gibt es während der Prüfungssessionen eine einstündige, schriftliche Prüfung. Die Vorlesung Biologie II wird separat geprüft. | |||||

636-0017-00L | Computational Biology | W | 6 credits | 3G + 2A | T. Stadler, C. Magnus, T. Vaughan | |

Abstract | The aim of the course is to provide up-to-date knowledge on how we can study biological processes using genetic sequencing data. Computational algorithms extracting biological information from genetic sequence data are discussed, and statistical tools to understand this information in detail are introduced. | |||||

Learning objective | Attendees will learn which information is contained in genetic sequencing data and how to extract information from this data using computational tools. The main concepts introduced are: * stochastic models in molecular evolution * phylogenetic & phylodynamic inference * maximum likelihood and Bayesian statistics Attendees will apply these concepts to a number of applications yielding biological insight into: * epidemiology * pathogen evolution * macroevolution of species | |||||

Content | The course consists of four parts. We first introduce modern genetic sequencing technology, and algorithms to obtain sequence alignments from the output of the sequencers. We then present methods for direct alignment analysis using approaches such as BLAST and GWAS. Second, we introduce mechanisms and concepts of molecular evolution, i.e. we discuss how genetic sequences change over time. Third, we employ evolutionary concepts to infer ancestral relationships between organisms based on their genetic sequences, i.e. we discuss methods to infer genealogies and phylogenies. Lastly, we introduce the field of phylodynamics, the aim of which is to understand and quantify population dynamic processes (such as transmission in epidemiology or speciation & extinction in macroevolution) based on a phylogeny. Throughout the class, the models and methods are illustrated on different datasets giving insight into the epidemiology and evolution of a range of infectious diseases (e.g. HIV, HCV, influenza, Ebola). Applications of the methods to the field of macroevolution provide insight into the evolution and ecology of different species clades. Students will be trained in the algorithms and their application both on paper and in silico as part of the exercises. | |||||

Lecture notes | Lecture slides will be available on moodle. | |||||

Literature | The course is not based on any of the textbooks below, but they are excellent choices as accompanying material: * Yang, Z. 2006. Computational Molecular Evolution. * Felsenstein, J. 2004. Inferring Phylogenies. * Semple, C. & Steel, M. 2003. Phylogenetics. * Drummond, A. & Bouckaert, R. 2015. Bayesian evolutionary analysis with BEAST. | |||||

Prerequisites / Notice | Basic knowledge in linear algebra, analysis, and statistics will be helpful. Programming in R will be required for the project work (compulsory continuous performance assessments). We provide an R tutorial and help sessions during the first two weeks of class to learn the required skills. However, in case you do not have any previous experience with R, we strongly recommend to get familiar with R prior to the semester start. For the D-BSSE students, we highly recommend the voluntary course „Introduction to Programming“, which takes place at D-BSSE from Wednesday, September 12 to Friday, September 14, i.e. BEFORE the official semester starting date http://www.cbb.ethz.ch/news-events.html For the Zurich-based students without R experience, we recommend the R course Link, or working through the script provided as part of this R course. | |||||

636-0007-00L | Computational Systems Biology | W | 6 credits | 3V + 2U | J. Stelling | |

Abstract | Study of fundamental concepts, models and computational methods for the analysis of complex biological networks. Topics: Systems approaches in biology, biology and reaction network fundamentals, modeling and simulation approaches (topological, probabilistic, stoichiometric, qualitative, linear / nonlinear ODEs, stochastic), and systems analysis (complexity reduction, stability, identification). | |||||

Learning objective | The aim of this course is to provide an introductory overview of mathematical and computational methods for the modeling, simulation and analysis of biological networks. | |||||

Content | Biology has witnessed an unprecedented increase in experimental data and, correspondingly, an increased need for computational methods to analyze this data. The explosion of sequenced genomes, and subsequently, of bioinformatics methods for the storage, analysis and comparison of genetic sequences provides a prominent example. Recently, however, an additional area of research, captured by the label "Systems Biology", focuses on how networks, which are more than the mere sum of their parts' properties, establish biological functions. This is essentially a task of reverse engineering. The aim of this course is to provide an introductory overview of corresponding computational methods for the modeling, simulation and analysis of biological networks. We will start with an introduction into the basic units, functions and design principles that are relevant for biology at the level of individual cells. Making extensive use of example systems, the course will then focus on methods and algorithms that allow for the investigation of biological networks with increasing detail. These include (i) graph theoretical approaches for revealing large-scale network organization, (ii) probabilistic (Bayesian) network representations, (iii) structural network analysis based on reaction stoichiometries, (iv) qualitative methods for dynamic modeling and simulation (Boolean and piece-wise linear approaches), (v) mechanistic modeling using ordinary differential equations (ODEs) and finally (vi) stochastic simulation methods. | |||||

Lecture notes | http://www.csb.ethz.ch/education/lectures.html | |||||

Literature | U. Alon, An introduction to systems biology. Chapman & Hall / CRC, 2006. Z. Szallasi et al. (eds.), System modeling in cellular biology. MIT Press, 2010. B. Ingalls, Mathematical modeling in systems biology: an introduction. MIT Press, 2013 | |||||

636-0009-00L | Evolutionary Dynamics | W | 6 credits | 2V + 1U + 2A | N. Beerenwinkel | |

Abstract | Evolutionary dynamics is concerned with the mathematical principles according to which life has evolved. This course offers an introduction to mathematical modeling of evolution, including deterministic and stochastic models. | |||||

Learning objective | The goal of this course is to understand and to appreciate mathematical models and computational methods that provide insight into the evolutionary process. | |||||

Content | Evolution is the one theory that encompasses all of biology. It provides a single, unifying concept to understand the living systems that we observe today. We will introduce several types of mathematical models of evolution to describe gene frequency changes over time in the context of different biological systems, focusing on asexual populations. Viruses and cancer cells provide the most prominent examples of such systems and they are at the same time of great biomedical interest. The course will cover some classical mathematical population genetics and population dynamics, and also introduce several new approaches. This is reflected in a diverse set of mathematical concepts which make their appearance throughout the course, all of which are introduced from scratch. Topics covered include the quasispecies equation, evolution of HIV, evolutionary game theory, birth-death processes, evolutionary stability, evolutionary graph theory, somatic evolution of cancer, stochastic tunneling, cell differentiation, hematopoietic tumor stem cells, genetic progression of cancer and the speed of adaptation, diffusion theory, fitness landscapes, neutral networks, branching processes, evolutionary escape, and epistasis. | |||||

Lecture notes | No. | |||||

Literature | - Evolutionary Dynamics. Martin A. Nowak. The Belknap Press of Harvard University Press, 2006. - Evolutionary Theory: Mathematical and Conceptual Foundations. Sean H. Rice. Sinauer Associates, Inc., 2004. | |||||

Prerequisites / Notice | Prerequisites: Basic mathematics (linear algebra, calculus, probability) | |||||

Control and Automation | ||||||

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

151-0563-01L | Dynamic Programming and Optimal Control | W | 4 credits | 2V + 1U | R. D'Andrea | |

Abstract | Introduction to Dynamic Programming and Optimal Control. | |||||

Learning objective | Covers the fundamental concepts of Dynamic Programming & Optimal Control. | |||||

Content | Dynamic Programming Algorithm; Deterministic Systems and Shortest Path Problems; Infinite Horizon Problems, Bellman Equation; Deterministic Continuous-Time Optimal Control. | |||||

Literature | Dynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. I, 3rd edition, 2005, 558 pages, hardcover. | |||||

Prerequisites / Notice | Requirements: Knowledge of advanced calculus, introductory probability theory, and matrix-vector algebra. | |||||

Economics | ||||||

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

401-3929-00L | Financial Risk Management in Social and Pension Insurance | W | 4 credits | 2V | P. Blum | |

Abstract | Investment returns are an important source of funding for social and pension insurance, and financial risk is an important threat to stability. We study short-term and long-term financial risk and its interplay with other risk factors, and we develop methods for the measurement and management of financial risk and return in an asset/liability context with the goal of assuring sustainable funding. | |||||

Learning objective | Understand the basic asset-liability framework: essential principles and properties of social and pension insurance; cash flow matching, duration matching, valuation portfolio and loose coupling; the notion of financial risk; long-term vs. short-term risk; coherent measures of risk. Understand the conditions for sustainable funding: derivation of required returns; interplay between return levels, contribution levels and other parameters; influence of guaranteed benefits. Understand the notion of risk-taking capability: capital process as a random walk; measures of long-term risk and relation to capital; short-term solvency vs. long-term stability; effect of embedded options and guarantees; interplay between required return and risk-taking capability. Be able to study empirical properties of financial assets: the Normal hypothesis and the deviations from it; statistical tools for investigating relevant risk and return properties of financial assets; time aggregation properties; be able to conduct analysis of real data for the most important asset classes. Understand and be able to carry out portfolio construction: the concept of diversification; limitations to diversification; correlation breakdown; incorporation of constraints; sensitivities and shortcomings of optimized portfolios. Understand and interpret the asset-liability interplay: the optimized portfolio in the asset-liability framework; short-term risk vs. long-term risk; the influence of constraints; feasible and non-feasible solutions; practical considerations. Understand and be able to address essential problems in asset / liability management, e.g. optimal risk / return positioning, optimal discount rate, target value for funding ratio or turnaround issues. Have an overall view: see the big picture of what asset returns can and cannot contribute to social security; be aware of the most relevant outcomes; know the role of the actuary in the financial risk management process. | |||||

Content | For pension insurance and other forms of social insurance, investment returns are an important source of funding. In order to earn these returns, substantial financial risks must be taken, and these risks represent an important threat to financial stability, in the long term and in the short term. Risk and return of financial assets cannot be separated from one another and, hence, asset management and risk management cannot be separated either. Managing financial risk in social and pension insurance is, therefore, the task of reconciling the contradictory dimensions of 1. Required return for a sustainable funding of the institution, 2. Risk-taking capability of the institution, 3. Returns available from financial assets in the market, 4. Risks incurred by investing in these assets. This task must be accomplished under a number of constraints. Financial risk management in social insurance also means reconciling the long time horizon of the promised insurance benefits with the short time horizon of financial markets and financial risk. It is not the goal of this lecture to provide the students with any cookbook recipes that can readily be applied without further reflection. The goal is rather to enable the students to develop their own understanding of the problems and possible solutions associated with the management of financial risks in social and pension insurance. To this end, a rigorous intellectual framework will be developed and a powerful set of mathematical tools from the fields of actuarial mathematics and quantitative risk management will be applied. When analyzing the properties of financial assets, an empirical viewpoint will be taken using statistical tools and considering real-world data. | |||||

Lecture notes | Extensive handouts will be provided. Moreover, practical examples and data sets in Excel and R will be made available. | |||||

Prerequisites / Notice | Solid base knowledge of probability and statistics is indispensable. Specialized concepts from financial and insurance mathematics as well as quantitative risk management will be introduced in the lecture as needed, but some prior knowledge in some of these areas would be an advantage. This course counts towards the diploma of "Aktuar SAV". The exams ONLY take place during the official ETH examination period. | |||||

363-0537-00L | Resource and Environmental Economics | W | 3 credits | 2G | L. Bretschger | |

Abstract | Relationship between economy and environment, market failure, external effects and public goods, contingent valuation, internalisation of externalities; economics of non-renewable resources, economics of renewable resources, cost-benefit analysis, sustainability, and international aspects of resource and environmental economics. | |||||

Learning objective | Understanding of the basic issues and methods in resource and environmental economics; ability to solve typical problems in the field using the appropriate tools, which are concise verbal explanations, diagrams or mathematical expressions. Topics are: Introduction to resource and environmental economics Importance of resource and environmental economics Main issues of resource and environmental economics Normative basis Utilitarianism Fairness according to Rawls Economic growth and environment Externalities in the environmental sphere Governmental internalisation of externalities Private internalisation of externalities: the Coase theorem Free rider problem and public goods Types of public policy Efficient level of pollution Tax vs. permits Command and Control Instruments Empirical data on non-renewable natural resources Optimal price development: the Hotelling-rule Effects of exploration and Backstop-technology Effects of different types of markets. Biological growth function Optimal depletion of renewable resources Social inefficiency as result of over-use of open-access resources Cost-benefit analysis and the environment Measuring environmental benefit Measuring costs Concept of sustainability Technological feasibility Conflicts sustainability / optimality Indicators of sustainability Problem of climate change Cost and benefit of climate change Climate change as international ecological externality International climate policy: Kyoto protocol Implementation of the Kyoto protocol in Switzerland | |||||

Content | Economy and natural environment, welfare concepts and market failure, external effects and public goods, measuring externalities and contingent valuation, internalising external effects and environmental policy, economics of non-renewable resources, renewable resources, cost-benefit-analysis, sustainability issues, international aspects of resource and environmental problems, selected examples and case studies. | |||||

Literature | Perman, R., Ma, Y., McGilvray, J, Common, M.: "Natural Resource & Environmental Economics", 3d edition, Longman, Essex 2003. | |||||

363-0503-00L | Principles of MicroeconomicsGESS (Science in Perspective): Suitable for Master students. Bachelor students should take the course ‚Einführung in die Mikroökonomie (363-1109-00L)‘. | W | 3 credits | 2G | M. Filippini | |

Abstract | The course introduces basic principles, problems and approaches of microeconomics. This provides them with reflective and contextual knowledge on how societies use scarce resources to produce goods and services and distribute them among themselves. | |||||

Learning objective | The learning objectives of the course are: (1) Students must be able to discuss basic principles, problems and approaches in microeconomics. (2) Students can analyse and explain simple economic principles in a market using supply and demand graphs. (3) Students can contrast different market structures and describe firm and consumer behaviour. (4) Students can identify market failures such as externalities related to market activities and illustrate how these affect the economy as a whole. (5) Students can also recognize behavioural failures within a market and discuss basic concepts related to behavioural economics. (6) Students can apply simple mathematical treatment of some basic concepts and can solve utility maximisation and cost minimisation problems. | |||||

Lecture notes | Lecture notes, exercises and reference material can be downloaded from Moodle. | |||||

Literature | N. Gregory Mankiw and Mark P. Taylor (2017), "Economics", 4th edition, South-Western Cengage Learning. The book can also be used for the course 'Principles of Macroeconomics' (Sturm) For students taking only the course 'Principles of Microeconomics' there is a shorter version of the same book: N. Gregory Mankiw and Mark P. Taylor (2017), "Microeconomics", 4th edition, South-Western Cengage Learning. Complementary: 1. R. Pindyck and D. Rubinfeld (2018), "Microeconomics", 9th edition, Pearson Education. 2. Varian, H.R. (2014), "Intermediate Microeconomics", 9th edition, Norton & Company | |||||

363-0565-00L | Principles of Macroeconomics | W | 3 credits | 2V | J.‑E. Sturm | |

Abstract | This course examines the behaviour of macroeconomic variables, such as gross domestic product, unemployment and inflation rates. It tries to answer questions like: How can we explain fluctuations of national economic activity? What can economic policy do against unemployment and inflation. What significance do international economic relations have for Switzerland? | |||||

Learning objective | This lecture will introduce the fundamentals of macroeconomic theory and explain their relevance to every-day economic problems. | |||||

Content | This course helps you understand the world in which you live. There are many questions about the macroeconomy that might spark your curiosity. Why are living standards so meagre in many African countries? Why do some countries have high rates of inflation while others have stable prices? Why have some European countries adopted a common currency? These are just a few of the questions that this course will help you answer. Furthermore, this course will give you a better understanding of the potential and limits of economic policy. As a voter, you help choose the policies that guide the allocation of society's resources. When deciding which policies to support, you may find yourself asking various questions about economics. What are the burdens associated with alternative forms of taxation? What are the effects of free trade with other countries? What is the best way to protect the environment? How does the government budget deficit affect the economy? These and similar questions are always on the minds of policy makers. | |||||

Lecture notes | The course webpage (to be found at https://moodle-app2.let.ethz.ch/course/view.php?id=4599) contains announcements, course information and lecture slides. | |||||

Literature | The set-up of the course will closely follow the book of N. Gregory Mankiw and Mark P. Taylor (2017), Economics, Cengage Learning, Fourth Edition. We advise you to also buy access to Aplia. This internet platform will support you in learning for this course. To save money, you should buy the book together with Aplia. This is sold as a bundle (ISBN: 978-1-473762008). Besides this textbook, the slides and lecture notes will cover the content of the lecture and the exam questions. | |||||

363-1021-00L | Monetary Policy | W | 3 credits | 2V | J.‑E. Sturm, A. Rathke | |

Abstract | The main aim of this course is to analyse the goals of monetary policy and to review the instruments available to central banks in order to pursue these goals. It will focus on the transmission mechanisms of monetary policy and the differences between monetary policy rules and discretionary policy. It will also make connections between theoretical economic concepts and current real world issues. | |||||

Learning objective | This lecture will introduce the fundamentals of monetary economics and explain the working and impact of monetary policy. | |||||

Literature | The course will be based on chapters of: Mishkin, Frederic S. (2015), The Economics of Money, Banking and Financial Markets 11th edition, Pearson. ISBN 10: 1-29-209418-4 ISBN 13: 978-1-292-09418-2 | |||||

Prerequisites / Notice | Basic knowledge in international economics and a good background in macroeconomics. The course website can be found at: https://moodle-app2.let.ethz.ch/course/view.php?id=2457 | |||||

Environmental Science | ||||||

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

701-0535-00L | Environmental Soil Physics/Vadose Zone Hydrology | W | 3 credits | 2G + 2U | D. Or | |

Abstract | The course provides theoretical and practical foundations for understanding and characterizing physical and transport properties of soils/ near-surface earth materials, and quantifying hydrological processes and fluxes of mass and energy at multiple scales. Emphasis is given to land-atmosphere interactions, the role of plants on hydrological cycles, and biophysical processes in soils. | |||||

Learning objective | Students are able to - characterize quantitative knowledge needed to measure and parameterize structural, flow and transport properties of partially-saturated porous media. - quantify driving forces and resulting fluxes of water, solute, and heat in soils. - apply modern measurement methods and analytical tools for hydrological data collection - conduct and interpret a limited number of experimental studies - explain links between physical processes in the vadose-zone and major societal and environmental challenges | |||||

Content | Weeks 1 to 3: Physical Properties of Soils and Other Porous Media – Units and dimensions, definitions and basic mass-volume relationships between the solid, liquid and gaseous phases; soil texture; particle size distributions; surface area; soil structure. Soil colloids and clay behavior Soil Water Content and its Measurement - Definitions; measurement methods - gravimetric, neutron scattering, gamma attenuation; and time domain reflectometry; soil water storage and water balance. Weeks 4 to 5: Soil Water Retention and Potential (Hydrostatics) - The energy state of soil water; total water potential and its components; properties of water (molecular, surface tension, and capillary rise); modern aspects of capillarity in porous media; units and calculations and measurement of equilibrium soil water potential components; soil water characteristic curves definitions and measurements; parametric models; hysteresis. Modern aspects of capillarity Demo-Lab: Laboratory methods for determination of soil water characteristic curve (SWC), sensor pairing Weeks 6 to 9: Water Flow in Soil - Hydrodynamics: Part 1 - Laminar flow in tubes (Poiseuille's Law); Darcy's Law, conditions and states of flow; saturated flow; hydraulic conductivity and its measurement. Lab #1: Measurement of saturated hydraulic conductivity in uniform and layered soil columns using the constant head method. Part 2 - Unsaturated steady state flow; unsaturated hydraulic conductivity models and applications; non-steady flow and Richard’s Eq.; approximate solutions to infiltration (Green-Ampt, Philip); field methods for estimating soil hydraulic properties. Midterm exam Lab #2: Measurement of vertical infiltration into dry soil column - Green-Ampt, and Philip's approximations; infiltration rates and wetting front propagation. Part 3 - Use of Hydrus model for simulation of unsaturated flow Week 10 to 11: Energy Balance and Land Atmosphere Interactions - Radiation and energy balance; evapotranspiration definitions and estimation; transpiration, plant development and transpirtation coefficients – small and large scale influences on hydrological cycle; surface evaporation. Week 12 to 13: Solute Transport in Soils – Transport mechanisms of solutes in porous media; breakthrough curves; convection-dispersion eq.; solutions for pulse and step solute application; parameter estimation; salt balance. Lab #3: Miscible displacement and breakthrough curves for a conservative tracer through a column; data analysis and transport parameter estimation. Additional topics: Temperature and Heat Flow in Porous Media - Soil thermal properties; steady state heat flow; nonsteady heat flow; estimation of thermal properties; engineering applications. Biological Processes in the Vaodse Zone – An overview of below-ground biological activity (plant roots, microbial, etc.); interplay between physical and biological processes. Focus on soil-atmosphere gaseous exchange; and challenges for bio- and phytoremediation. | |||||

Lecture notes | Classnotes on website: Vadose Zone Hydrology, by Or D., J.M. Wraith, and M. Tuller (available at the beginning of the semester) http://www.step.ethz.ch/education/vadose-zone-hydrology.html | |||||

Literature | Supplemental textbook (not mandatory) -Environmental Soil Physics, by: D. Hillel | |||||

Finance | ||||||

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

401-8905-00L | Financial Engineering (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MFOEC200 Mind the enrolment deadlines at UZH: https://www.uzh.ch/cmsssl/en/studies/application/mobilitaet.html | W | 6 credits | 4G | University lecturers | |

Abstract | This lecture is intended for students who would like to learn more on equity derivatives modelling and pricing. | |||||

Learning objective | Quantitative models for European option pricing (including stochastic volatility and jump models), volatility and variance derivatives, American and exotic options. | |||||

Content | After introducing fundamental concepts of mathematical finance including no-arbitrage, portfolio replication and risk-neutral measure, we will present the main models that can be used for pricing and hedging European options e.g. Black- Scholes model, stochastic and jump-diffusion models, and highlight their assumptions and limitations. We will cover several types of derivatives such as European and American options, Barrier options and Variance- Swaps. Basic knowledge in probability theory and stochastic calculus is required. Besides attending class, we strongly encourage students to stay informed on financial matters, especially by reading daily financial newspapers such as the Financial Times or the Wall Street Journal. | |||||

Lecture notes | Script. | |||||

Prerequisites / Notice | Basic knowledge of probability theory and stochastic calculus. Asset Pricing. | |||||

401-8913-00L | Advanced Corporate Finance I (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MOEC0455 Mind the enrolment deadlines at UZH: https://www.uzh.ch/cmsssl/en/studies/application/mobilitaet.html | W | 6 credits | 4G | University lecturers | |

Abstract | This course develops and refines tools for evaluating investments (capital budgeting), capital structure, and corporate securities. The course seeks to deepen students' understanding of the link between corporate finance theory and practice. | |||||

Learning objective | This course develops and refines tools for evaluating investments (capital budgeting), capital structure, and corporate securities. With respect to capital structure, we start with the famous Miller and Modigliani irrelevance proposition and then move on to study the effects of taxes, bankruptcy costs, information asymmetries between firms and the capital markets, and agency costs. In this context, we will also study how leverage affects some central financial ratios that are often used in practice to assess firms and their stock. Other topics include corporate cash holdings, the use and pricing of convertible bonds, and risk management. The latter two topics involve option pricing. With respect to capital budgeting, the course pays special attention to tax effects in valuation, including in the estimation of the cost of capital. We will also study payout policy (dividends and share repurchases). The course seeks to deepen students' understanding of the link between corporate finance theory and practice. Various cases will be assigned to help reach this objective. | |||||

Content | Topics covered 1. Capital structure: Perfect markets and irrelevance 2. Risk, leverage, taxes, and the cost of capital 3. Leverage and financial ratios 4. Payout policy: Dividends and share repurchases 5. Capital structure: Taxes and bankruptcy costs 6. Capital structure: Information asymmetries, agency costs, cash holdings 7. Valuation: DCF, adjusted present value and WACC 8. Valuation using options 9. The use and pricing of convertible bonds 10. Corporate risk management | |||||

Prerequisites / Notice | This course replaces "Advanced Corporate Finance I" (MOEC0288), which will be discontinued from HS16. | |||||

Image Processing and Computer Vision | ||||||

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

227-0447-00L | Image Analysis and Computer Vision | W | 6 credits | 3V + 1U | L. Van Gool, O. Göksel, E. Konukoglu | |

Abstract | Light and perception. Digital image formation. Image enhancement and feature extraction. Unitary transformations. Color and texture. Image segmentation. Motion extraction and tracking. 3D data extraction. Invariant features. Specific object recognition and object class recognition. Deep learning and Convolutional Neural Networks. | |||||

Learning objective | Overview of the most important concepts of image formation, perception and analysis, and Computer Vision. Gaining own experience through practical computer and programming exercises. | |||||

Content | This course aims at offering a self-contained account of computer vision and its underlying concepts, including the recent use of deep learning. The first part starts with an overview of existing and emerging applications that need computer vision. It shows that the realm of image processing is no longer restricted to the factory floor, but is entering several fields of our daily life. First the interaction of light with matter is considered. The most important hardware components such as cameras and illumination sources are also discussed. The course then turns to image discretization, necessary to process images by computer. The next part describes necessary pre-processing steps, that enhance image quality and/or detect specific features. Linear and non-linear filters are introduced for that purpose. The course will continue by analyzing procedures allowing to extract additional types of basic information from multiple images, with motion and 3D shape as two important examples. Finally, approaches for the recognition of specific objects as well as object classes will be discussed and analyzed. A major part at the end is devoted to deep learning and AI-based approaches to image analysis. Its main focus is on object recognition, but also other examples of image processing using deep neural nets are given. | |||||

Lecture notes | Course material Script, computer demonstrations, exercises and problem solutions | |||||

Prerequisites / Notice | Prerequisites: Basic concepts of mathematical analysis and linear algebra. The computer exercises are based on Python and Linux. The course language is English. | |||||

Information and Communication Technology | ||||||

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

227-0427-00L | Signal Analysis, Models, and Machine Learning | W | 6 credits | 4G | H.‑A. Loeliger | |

Abstract | Mathematical methods in signal processing and machine learning. I. Linear signal representation and approximation: Hilbert spaces, LMMSE estimation, regularization and sparsity. II. Learning linear and nonlinear functions and filters: neural networks, kernel methods. III. Structured statistical models: hidden Markov models, factor graphs, Kalman filter, Gaussian models with sparse events. | |||||

Learning objective | The course is an introduction to some basic topics in signal processing and machine learning. | |||||

Content | Part I - Linear Signal Representation and Approximation: Hilbert spaces, least squares and LMMSE estimation, projection and estimation by linear filtering, learning linear functions and filters, L2 regularization, L1 regularization and sparsity, singular-value decomposition and pseudo-inverse, principal-components analysis. Part II - Learning Nonlinear Functions: fundamentals of learning, neural networks, kernel methods. Part III - Structured Statistical Models and Message Passing Algorithms: hidden Markov models, factor graphs, Gaussian message passing, Kalman filter and recursive least squares, Monte Carlo methods, parameter estimation, expectation maximization, linear Gaussian models with sparse events. | |||||

Lecture notes | Lecture notes. | |||||

Prerequisites / Notice | Prerequisites: - local bachelors: course "Discrete-Time and Statistical Signal Processing" (5. Sem.) - others: solid basics in linear algebra and probability theory | |||||

227-0101-00L | Discrete-Time and Statistical Signal Processing | W | 6 credits | 4G | H.‑A. Loeliger | |

Abstract | The course introduces some fundamental topics of digital signal processing with a bias towards applications in communications: discrete-time linear filters, inverse filters and equalization, DFT, discrete-time stochastic processes, elements of detection theory and estimation theory, LMMSE estimation and LMMSE filtering, LMS algorithm, Viterbi algorithm. | |||||

Learning objective | The course introduces some fundamental topics of digital signal processing with a bias towards applications in communications. The two main themes are linearity and probability. In the first part of the course, we deepen our understanding of discrete-time linear filters. In the second part of the course, we review the basics of probability theory and discrete-time stochastic processes. We then discuss some basic concepts of detection theory and estimation theory, as well as some practical methods including LMMSE estimation and LMMSE filtering, the LMS algorithm, and the Viterbi algorithm. A recurrent theme throughout the course is the stable and robust "inversion" of a linear filter. | |||||

Content | 1. Discrete-time linear systems and filters: state-space realizations, z-transform and spectrum, decimation and interpolation, digital filter design, stable realizations and robust inversion. 2. The discrete Fourier transform and its use for digital filtering. 3. The statistical perspective: probability, random variables, discrete-time stochastic processes; detection and estimation: MAP, ML, Bayesian MMSE, LMMSE; Wiener filter, LMS adaptive filter, Viterbi algorithm. | |||||

Lecture notes | Lecture Notes | |||||

227-0417-00L | Information Theory I | W | 6 credits | 4G | A. Lapidoth | |

Abstract | This course covers the basic concepts of information theory and of communication theory. Topics covered include the entropy rate of a source, mutual information, typical sequences, the asymptotic equi-partition property, Huffman coding, channel capacity, the channel coding theorem, the source-channel separation theorem, and feedback capacity. | |||||

Learning objective | The fundamentals of Information Theory including Shannon's source coding and channel coding theorems | |||||

Content | The entropy rate of a source, Typical sequences, the asymptotic equi-partition property, the source coding theorem, Huffman coding, Arithmetic coding, channel capacity, the channel coding theorem, the source-channel separation theorem, feedback capacity | |||||

Literature | T.M. Cover and J. Thomas, Elements of Information Theory (second edition) | |||||

Material Modelling and Simulation | ||||||

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

327-1201-00L | Transport Phenomena I | W | 5 credits | 4G | H. C. Öttinger | |

Abstract | Phenomenological approach to "Transport Phenomena" based on balance equations supplemented by thermodynamic considerations to formulate the undetermined fluxes in the local species mass, momentum, and energy balance equations; fundamentals, applications, and simulations | |||||

Learning objective | The teaching goals of this course are on five different levels: (1) Deep understanding of fundamentals: local balance equations, constitutive equations for fluxes, entropy balance, interfaces, idea of dimensionless numbers, ... (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, finite differences, lattice Boltzmann, Brownian dynamics, ... | |||||

Content | Approach to Transport Phenomena Diffusion Equation Brownian Dynamics Refreshing Topics in Equilibrium Thermodynamics Balance Equations Forces and Fluxes Measuring Transport Coefficients Pressure-Driven Flows Driven Separations Complex Fluids | |||||

Lecture notes | The course is based on the book D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018) | |||||

Literature | 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. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, 2nd Ed. (Dover, 1984) 4. W. M. Deen, Analysis of Transport Phenomena (Oxford University Press, 1998) 5. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287 | |||||

Prerequisites / Notice | 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). Equilibrium thermodynamics (Gibbs' fundamental equation; thermodynamic potentials; Legendre transforms). Maxwell equations. Programming and simulation techniques (Matlab, Monte Carlo simulations). | |||||

Quantum Chemistry | ||||||

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

529-0003-01L | Advanced Quantum Chemistry | W | 6 credits | 3G | M. Reiher, S. Knecht | |

Abstract | Advanced, but fundamental topics central to the understanding of theory in chemistry and for solving actual chemical problems with a computer. Examples are: * Operators derived from principles of relativistic quantum mechanics * Relativistic effects + methods of relativistic quantum chemistry * Open-shell molecules + spin-density functional theory * New electron-correlation theories | |||||

Learning objective | The aim of the course is to provide an in-depth knowledge of theory and method development in theoretical chemistry. It will be shown that this is necessary in order to be able to solve actual chemical problems on a computer with quantum chemical methods. The relativistic re-derivation of all concepts known from (nonrelativistic) quantum mechanics and quantum-chemistry lectures will finally explain the form of all operators in the molecular Hamiltonian - usually postulated rather than deduced. From this, we derive operators needed for molecular spectroscopy (like those required by magnetic resonance spectroscopy). Implications of other assumptions in standard non-relativistic quantum chemistry shall be analyzed and understood, too. Examples are the Born-Oppenheimer approximation and the expansion of the electronic wave function in a set of pre-defined many-electron basis functions (Slater determinants). Overcoming these concepts, which are so natural to the theory of chemistry, will provide deeper insights into many-particle quantum mechanics. Also revisiting the workhorse of quantum chemistry, namely density functional theory, with an emphasis on open-shell electronic structures (radicals, transition-metal complexes) will contribute to this endeavor. It will be shown how these insights allow us to make more accurate predictions in chemistry in practice - at the frontier of research in theoretical chemistry. | |||||

Content | 1) Introductory lecture: basics of quantum mechanics and quantum chemistry 2) Einstein's special theory of relativity and the (classical) electromagnetic interaction of two charged particles 3) Klein-Gordon and Dirac equation; the Dirac hydrogen atom 4) Numerical methods based on the Dirac-Fock-Coulomb Hamiltonian, two-component and scalar relativistic Hamiltonians 5) Response theory and molecular properties, derivation of property operators, Breit-Pauli-Hamiltonian 6) Relativistic effects in chemistry and the emergence of spin 7) Spin in density functional theory 8) New electron-correlation theories: Tensor network and matrix product states, the density matrix renormalization group 9) Quantum chemistry without the Born-Oppenheimer approximation | |||||

Lecture notes | A set of detailed lecture notes will be provided, which will cover the whole course. | |||||

Literature | 1) M. Reiher, A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH, 2014, 2nd edition 2) F. Schwabl: Quantenmechanik für Fortgeschrittene (QM II), Springer-Verlag, 1997 [english version available: F. Schwabl, Advanced Quantum Mechanics] 3) R. McWeeny: Methods of Molecular Quantum Mechanics, Academic Press, 1992 4) C. R. Jacob, M. Reiher, Spin in Density-Functional Theory, Int. J. Quantum Chem. 112 (2012) 3661 http://onlinelibrary.wiley.com/doi/10.1002/qua.24309/abstract 5) K. H. Marti, M. Reiher, New Electron Correlation Theories for Transition Metal Chemistry, Phys. Chem. Chem. Phys. 13 (2011) 6750 http://pubs.rsc.org/en/Content/ArticleLanding/2011/CP/c0cp01883j 6) K.H. Marti, M. Reiher, The Density Matrix Renormalization Group Algorithm in Quantum Chemistry, Z. Phys. Chem. 224 (2010) 583 http://www.oldenbourg-link.com/doi/abs/10.1524/zpch.2010.6125 7) E. Mátyus, J. Hutter, U. Müller-Herold, M. Reiher, On the emergence of molecular structure, Phys. Rev. A 83 2011, 052512 http://pra.aps.org/abstract/PRA/v83/i5/e052512 Note also the standard textbooks: A) A. Szabo, N.S. Ostlund. Verlag, Dover Publications B) I. N. Levine, Quantum Chemistry, Pearson C) T. Helgaker, P. Jorgensen, J. Olsen: Molecular Electronic-Structure Theory, Wiley, 2000 D) R.G. Parr, W. Yang: Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1994 E) R.M. Dreizler, E.K.U. Gross: Density Functional Theory, Springer-Verlag, 1990 | |||||

Prerequisites / Notice | Strongly recommended (preparatory) courses are: quantum mechanics and quantum chemistry | |||||

Simulation of Semiconductor Devices | ||||||

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

227-0157-00L | Semiconductor Devices: Physical Bases and Simulation | W | 4 credits | 3G | A. Schenk | |

Abstract | The course addresses the physical principles of modern semiconductor devices and the foundations of their modeling and numerical simulation. Necessary basic knowledge on quantum-mechanics, semiconductor physics and device physics is provided. Computer simulations of the most important devices and of interesting physical effects supplement the lectures. | |||||

Learning objective | The course aims at the understanding of the principle physics of modern semiconductor devices, of the foundations in the physical modeling of transport and its numerical simulation. During the course also basic knowledge on quantum-mechanics, semiconductor physics and device physics is provided. | |||||

Content | The main topics are: transport models for semiconductor devices (quantum transport, Boltzmann equation, drift-diffusion model, hydrodynamic model), physical characterization of silicon (intrinsic properties, scattering processes), mobility of cold and hot carriers, recombination (Shockley-Read-Hall statistics, Auger recombination), impact ionization, metal-semiconductor contact, metal-insulator-semiconductor structure, and heterojunctions. The exercises are focussed on the theory and the basic understanding of the operation of special devices, as single-electron transistor, resonant tunneling diode, pn-diode, bipolar transistor, MOSFET, and laser. Numerical simulations of such devices are performed with an advanced simulation package (Sentaurus-Synopsys). This enables to understand the physical effects by means of computer experiments. | |||||

Lecture notes | The script (in book style) can be downloaded from: https://iis-students.ee.ethz.ch/lectures/ | |||||

Literature | The script (in book style) is sufficient. Further reading will be recommended in the lecture. | |||||

Prerequisites / Notice | Qualifications: Physics I+II, Semiconductor devices (4. semester). | |||||

Systems Design | ||||||

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

363-0541-00L | Systems Dynamics and Complexity | W | 3 credits | 3G | F. Schweitzer, G. Casiraghi, V. Nanumyan | |

Abstract | Finding solutions: what is complexity, problem solving cycle. Implementing solutions: project management, critical path method, quality control feedback loop. Controlling solutions: Vensim software, feedback cycles, control parameters, instabilities, chaos, oscillations and cycles, supply and demand, production functions, investment and consumption | |||||

Learning objective | A successful participant of the course is able to: - understand why most real problems are not simple, but require solution methods that go beyond algorithmic and mathematical approaches - apply the problem solving cycle as a systematic approach to identify problems and their solutions - calculate project schedules according to the critical path method - setup and run systems dynamics models by means of the Vensim software - identify feedback cycles and reasons for unintended systems behavior - analyse the stability of nonlinear dynamical systems and apply this to macroeconomic dynamics | |||||

Content | Why are problems not simple? Why do some systems behave in an unintended way? How can we model and control their dynamics? The course provides answers to these questions by using a broad range of methods encompassing systems oriented management, classical systems dynamics, nonlinear dynamics and macroeconomic modeling. The course is structured along three main tasks: 1. Finding solutions 2. Implementing solutions 3. Controlling solutions PART 1 introduces complexity as a system immanent property that cannot be simplified. It introduces the problem solving cycle, used in systems oriented management, as an approach to structure problems and to find solutions. PART 2 discusses selected problems of project management when implementing solutions. Methods for identifying the critical path of subtasks in a project and for calculating the allocation of resources are provided. The role of quality control as an additional feedback loop and the consequences of small changes are discussed. PART 3, by far the largest part of the course, provides more insight into the dynamics of existing systems. Examples come from biology (population dynamics), management (inventory modeling, technology adoption, production systems) and economics (supply and demand, investment and consumption). For systems dynamics models, the software program VENSIM is used to evaluate the dynamics. For economic models analytical approaches, also used in nonlinear dynamics and control theory, are applied. These together provide a systematic understanding of the role of feedback loops and instabilities in the dynamics of systems. Emphasis is on oscillating phenomena, such as business cycles and other life cycles. Weekly self-study tasks are used to apply the concepts introduced in the lectures and to come to grips with the software program VENSIM. | |||||

Lecture notes | The lecture slides are provided as handouts - including notes and literature sources - to registered students only. All material is to be found on the Moodle platform. More details during the first lecture | |||||

Prerequisites / Notice | Self-study tasks (discussion exercises, Vensim exercises) are provided as home work. Weekly exercise sessions (45 min) are used to discuss selected solutions. Regular participation in the exercises is an efficient way to understand the concepts relevant for the final exam. | |||||

Theoretical Physics In the Master's programme in Applied Mathematics 402-0205-00L Quantum Mechanics I is eligible as a course unit in the application area Theoretical Physics, but only if 402-0224-00L Theoretical Physics wasn't or isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

402-0809-00L | Introduction to Computational Physics | W | 8 credits | 2V + 2U | H. J. Herrmann | |

Abstract | This course offers an introduction to computer simulation methods for physics problems and their implementation on PCs and super computers: classical equations of motion, partial differential equations (wave equation, diffusion equation, Maxwell's equation), Monte Carlo simulations, percolation, phase transitions | |||||

Learning objective | ||||||

Content | Einführung in die rechnergestützte Simulation physikalischer Probleme. Anhand einfacher Modelle aus der klassischen Mechanik, Elektrodynamik und statistischen Mechanik sowie interdisziplinären Anwendungen werden die wichtigsten objektorientierten Programmiermethoden für numerische Simulationen (überwiegend in C++) erläutert. Daneben wird eine Einführung in die Programmierung von Vektorsupercomputern und parallelen Rechnern, sowie ein Überblick über vorhandene Softwarebibliotheken für numerische Simulationen geboten. | |||||

Prerequisites / Notice | Lecture and exercise lessons in english, exams in German or in English | |||||

402-2203-01L | Classical Mechanics | W | 7 credits | 4V + 2U | C. Anastasiou | |

Abstract | A conceptual introduction to theoretical physics: Newtonian mechanics, central force problem, oscillations, Lagrangian mechanics, symmetries and conservation laws, spinning top, relativistic space-time structure, particles in an electromagnetic field, Hamiltonian mechanics, canonical transformations, integrable systems, Hamilton-Jacobi equation. | |||||

Learning objective | ||||||

402-0861-00L | Statistical Physics | W | 10 credits | 4V + 2U | G. Blatter | |

Abstract | The lecture focuses on classical and quantum statistical physics. Various techniques, cumulant expansion, path integrals, and specific systems are discussed: Fermions, photons/phonons, Bosons, magnetism, van der Waals gas. Phase transitions are studied in mean field theory (Weiss, Landau). Including fluctuations leads to critical phenomena, scaling, and the renormalization group. | |||||

Learning objective | This lecture gives an introduction into the the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education. | |||||

Content | Thermodynamics, three laws of thermodynamics, thermodynamic potentials, phenomenology of phase transitions. Classical statistical physics: micro-canonical-, canonical-, and grandcanonical ensembles, applications to simple systems. Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons, statistical interaction. Techniques: variational approach, cumulant expansion, path integral formulation. Degenerate fermions: Fermi gas, electrons in magnetic field. Bosons: photons and phonons, Bose-Einstein condensation. Magnetism: Ising-, XY-, Heisenberg models, Weiss mean-field theory. Van der Waals gas-liquid transition. Landau theory of phase transitions, first- and second order, tricritical. Fluctuations: field theory approach, Gauss theory, self-consistent field, Ginzburg criterion. Critical phenomena: scaling theory, universality. Renormalization group: general theory and applications to spin models (real space RG), phi^4 theory (k-space RG), Kosterlitz-Thouless theory. | |||||

Lecture notes | Lecture notes available in English. | |||||

Literature | No specific book is used for the course. Relevant literature will be given in the course. | |||||

402-0843-00L | Quantum Field Theory I | W | 10 credits | 4V + 2U | A. Gehrmann-De Ridder | |

Abstract | This course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity. Topics include: - Relativistic quantum mechanics - Quantisation of bosonic and fermionic fields - Interactions in perturbation theory - Scattering processes and decays - Elementary processes in QED - Radiative corrections | |||||

Learning objective | The goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics. | |||||

Literature | as provided in the entity Lernmaterialien | |||||

402-0830-00L | General Relativity | W | 10 credits | 4V + 2U | R. Renner | |

Abstract | Manifold, Riemannian metric, connection, curvature; Special Relativity; Lorentzian metric; Equivalence principle; Tidal force and spacetime curvature; Energy-momentum tensor, field equations, Newtonian limit; Post-Newtonian approximation; Schwarzschild solution; Mercury's perihelion precession, light deflection. | |||||

Learning objective | Basic understanding of general relativity, its mathematical foundations, and some of the interesting phenomena it predicts. | |||||

Literature | Suggested textbooks: C. Misner, K, Thorne and J. Wheeler: Gravitation S. Carroll - Spacetime and Geometry: An Introduction to General Relativity R. Wald - General Relativity S. Weinberg - Gravitation and Cosmology N. Straumann - General Relativity with applications to Astrophysics | |||||

» Electives Theoretical Physics | ||||||

Transportation Science | ||||||

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

101-0417-00L | Transport Planning Methods | W | 6 credits | 4G | K. W. Axhausen | |

Abstract | The course provides the necessary knowledge to develop models supporting and also evaluating the solution of given planning problems. The course is composed of a lecture part, providing the theoretical knowledge, and an applied part in which students develop their own models in order to evaluate a transport project/ policy by means of cost-benefit analysis. | |||||

Learning objective | - Knowledge and understanding of statistical methods and algorithms commonly used in transport planning - Comprehend the reasoning and capabilities of transport models - Ability to independently develop a transport model able to solve / answer planning problem - Getting familiar with cost-benefit analysis as a decision-making supporting tool | |||||

Content | The course provides the necessary knowledge to develop models supporting the solution of given planning problems and also introduces cost-benefit analysis as a decision-making tool. Examples of such planning problems are the estimation of traffic volumes, prediction of estimated utilization of new public transport lines, and evaluation of effects (e.g. change in emissions of a city) triggered by building new infrastructure and changes to operational regulations. To cope with that, the problem is divided into sub-problems, which are solved using various statistical models (e.g. regression, discrete choice analysis) and algorithms (e.g. iterative proportional fitting, shortest path algorithms, method of successive averages). The course is composed of a lecture part, providing the theoretical knowledge, and an applied part in which students develop their own models in order to evaluate a transport project/ policy by means of cost-benefit analysis. Interim lab session take place regularly to guide and support students with the applied part of the course. | |||||

Lecture notes | Moodle platform (enrollment needed) | |||||

Literature | Willumsen, P. and J. de D. Ortuzar (2003) Modelling Transport, Wiley, Chichester. Cascetta, E. (2001) Transportation Systems Engineering: Theory and Methods, Kluwer Academic Publishers, Dordrecht. Sheffi, Y. (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, Prentice Hall, Englewood Cliffs. Schnabel, W. and D. Lohse (1997) Verkehrsplanung, 2. edn., vol. 2 of Grundlagen der Strassenverkehrstechnik und der Verkehrsplanung, Verlag für Bauwesen, Berlin. McCarthy, P.S. (2001) Transportation Economics: A case study approach, Blackwell, Oxford. | |||||

Seminars and Semester Papers | ||||||

Seminars Early enrolments for seminars in myStudies are encouraged, so that we will recognise need for additional seminars in a timely manner. Some seminars have waiting lists. Nevertheless, register for at most two mathematics seminars. | ||||||

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

401-4140-68L | Étale Cohomology | W | 4 credits | 2S | R. Pink, M. Mornev | |

Abstract | Approximate programme: Left and right derived functors on abelian categories. Flat morphisms revisited. Étale morphisms. The (big and small) étale site of a scheme. Étale site and Galois cohomology. Faithfully flat descent. First cohomology of the multiplicative group. Cohomology of curves. Brauer groups and Galois cohomology. Vanishing theorems in Galois cohomology. | |||||

Learning objective | ||||||

401-4220-68L | Symmetric Spaces of Non-Compact Type Number of participants limited to 10. | W | 4 credits | 2S | A. Iozzi | |

Abstract | ||||||

Learning objective | ||||||

Content | 1) Root systems of symmetric spaces and the Weyl group 2) Action of the Weyl group 3) The geodesic boundary 4) SL(n,R)/SO(n,R) 5) Parabolic subgroups 6) Iwasawa decomposition 7) The Tits metric | |||||

Prerequisites / Notice | If you are interested in the seminar, please send an e-mail to yannick.krifka@math.ethz.ch with your mathematical background before Tuesday, August 28th. Priority will be given to students as follows: 1) Students knowledgeable about Lie groups and symmetric spaces; 2) Students knowledgeable about symmetric spaces. A limited number of spots might be allocated to students who do not satisfy either of the above requirements, depending on availability and background. | |||||

401-3350-68L | Introduction to Optimal Transport Number of participants limited to 11. | W | 4 credits | 2S | A. Figalli, further speakers | |

Abstract | Introductory seminar about the theory of optimal transport. Starting from Monge's and Kantorovich's statements of the optimal transport problem, we will investigate the theory of duality necessary to prove the fundamental Brenier's theorem. After some applications, we will study the properties of the Wasserstein space and we will conclude introducing the dynamical point of view on the problem. | |||||

Learning objective | ||||||

Content | Given two distributions of mass, it is natural to ask ourselves what is the "best way" to transport one into the other. What are mathematically acceptable definitions of "distributions of mass" and "to transport one into the other"? Measures are perfectly suited to play the role of the distributions of mass, whereas a map that pushes-forward one measure into the other is the equivalent of transporting the distributions. By "best way" we mean that we want to minimize the map in some norm. The original problem of Monge is to understand whether there is an optimal map and to study its properties. In order to attack the problem we will need to relax the formulation (Kantorovich's statement) and to apply a little bit of duality theory. The main theorem we will prove in this direction is Brenier's theorem that answers positively to the existence problem of optimal maps (under certain conditions). The Helmotz's decomposition and the isoperimetric inequality will then follow rather easily as applications of the theory. Finally, we will see how the optimal transport problem gives a natural way to define a distance on the space of probabilities (Wasserstein distance) and we will study some of its properties. | |||||

Literature | "Optimal Transport, Old and New", C. Villani [http://cedricvillani.org/wp-content/uploads/2012/08/preprint-1.pdf] "Optimal Transport for Applied Mathematicians", F. Santambrogio [https://www.math.u-psud.fr/~filippo/OTAM-cvgmt.pdf] | |||||

Prerequisites / Notice | The students are expected to have mastered the content of the first two years taught at ETH, especially Measure Theory. The seminar is mainly intended for Bachelor students. In order to obtain the 4 credit points, each student is expected to give two 1h-talks and regularly attend the seminar. Moreover some exercises will be assigned. Further information can be found at https://metaphor.ethz.ch/x/2018/hs/401-3350-68L/ | |||||

401-3620-68L | Student Seminar in Statistics: Statistical Learning with Sparsity Number of participants limited to 24. Mainly for students from the Mathematics Bachelor and Master Programmes who, in addition to the introductory course unit 401-2604-00L Probability and Statistics, have heard at least one core or elective course in statistics. Also offered in the Master Programmes Statistics resp. Data Science. | W | 4 credits | 2S | M. Mächler, M. H. Maathuis, N. Meinshausen, S. van de Geer | |

Abstract | We study selected chapters from the 2015 book "Statistical Learning with Sparsity" by Trevor Hastie, Rob Tibshirani and Martin Wainwright. (details see below) | |||||

Learning objective | During this seminar, we will study roughly one chapter per week from the book. You will obtain a good overview of the field of sparse & high-dimensional modeling of modern statistics. Moreover, you will practice your self-studying and presentation skills. | |||||

Content | (From the book's preface:) "... summarize the actively developing field of statistical learning with sparsity. A sparse statistical model is one having only a small number of nonzero parameters or weights. It represents a classic case of “less is more”: a sparse model can be much easier to estimate and interpret than a dense model. In this age of big data, the number of features measured on a person or object can be large, and might be larger than the number of observations. The sparsity assumption allows us to tackle such problems and extract useful and reproducible patterns from big datasets." For presentation of the material, occasionally you'd consider additional published research, possibly e.g., for "High-Dimensional Inference" | |||||

Lecture notes | Website: with groups, FAQ, topics, slides, and Rscripts : https://stat.ethz.ch/lectures/as18/seminar.php#course_materials | |||||

Literature | Trevor Hastie, Robert Tibshirani, Martin Wainwright (2015) Statistical Learning with Sparsity: The Lasso and Generalization Monographs on Statistics and Applied Probability 143 Chapman Hall/CRC ISBN 9781498712170 Access : - https://www.taylorfrancis.com/books/9781498712170 (full access via ETH (library) network, if inside ETH (VPN)) - Author's website (includes errata, updated pdf, data): https://web.stanford.edu/~hastie/StatLearnSparsity/ | |||||

Prerequisites / Notice | We require at least one course in statistics in addition to the 4th semester course Introduction to Probability and Statistics, as well as some experience with the statistical software R. Topics will be assigned during the first meeting. | |||||

401-3910-68L | Topics in Mathematical Finance and Machine Learning Number of participants limited to 20. | W | 4 credits | 2S | J. Teichmann | |

Abstract | ||||||

Learning objective | ||||||

401-3650-68L | Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Number of participants limited to 6. | W | 4 credits | 2S | C. Schwab | |

Abstract | This seminar will review recent (2016-) _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. | |||||

Learning objective | ||||||

Content | 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 application areas (Chess, Go, autonomous driving, language translation, image classification, etc.). In many cases, these successes have been achieved by implementations, based on heuristics, with massive compute power and training data. This seminar will review recent (2016-) _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. Also here, there is mounting mathematical evidence that DNNs equalize or outperform the best known mathematical results. 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. The format will be oral student presentations in December 2018 based on a recent research paper selected in two meetings at the start of the semester. | |||||

Literature | Partial reading list: arXiv:1809.07669 DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing Authors: Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen, Christoph Schwab arXiv:1806.08459 Topological properties of the set of functions generated by neural networks of fixed size Authors: Philipp Petersen, Mones Raslan, Felix Voigtlaender arXiv:1804.10306 Universal approximations of invariant maps by neural networks Author: Dmitry Yarotsky arXiv:1802.03620 Optimal approximation of continuous functions by very deep ReLU networks Author: Dmitry Yarotsky arXiv:1709.05289 Optimal approximation of piecewise smooth functions using deep ReLU neural networks Authors: Philipp Petersen, Felix Voigtlaender arXiv:1706.03301 Neural networks and rational functions Author: Matus Telgarsky arXiv:1705.05502 The power of deeper networks for expressing natural functions Authors: David Rolnick, Max Tegmark arXiv:1705.01365 Quantified advantage of discontinuous weight selection in approximations with deep neural networks Author: Dmitry Yarotsky arXiv:1610.01145 Error bounds for approximations with deep ReLU networks Author: Dmitry Yarotsky arXiv:1608.03287 Deep vs. shallow networks : An approximation theory perspective Authors: Hrushikesh Mhaskar, Tomaso Poggio arXiv:1602.04485 Benefits of depth in neural networks Author: Matus Telgarsky | |||||

Prerequisites / Notice | Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. Disclaimer: The seminar will _not_ address recent developments in DNN software, such as training heuristics, or programming techniques for various specific applications. | |||||

401-3640-13L | Seminar in Applied Mathematics: Shape Calculus Number of participants limited to 10 | W | 4 credits | 2S | R. Hiptmair | |

Abstract | Shape calculus studies the dependence of solutions of partial differential equations on deformations of the domain and/or interfaces. It is the foundation of gradient methods for shape optimization. The seminar will rely on several sections of monographs and research papers covering analytical and numerical aspects of shape calculus. | |||||

Learning objective | * Understanding of concepts like shape derivative, shape gradient, shape Hessian, and adjoint problem. * Ability to derive analytical formulas for shape gradients * Knowledge about numerical methods for the computation of shape gradients. | |||||

Content | Topics: 1. The velocity method and Eulerian shape gradients: Main reference [SZ92, Sect. 2.8–2.11, 2.1, 2.18], covers the “velocity method”, the Hadamard structure theorem and formulas for shape gradients of particular functionals. Several other sections of [SZ92,Ch. 2] provide foundations and auxiliary results and should be browsed, too. 2. Material derivatives and shape derivatives, based on [SZ92, Sect. 2.25–2.32]. 3. Shape calculus with exterior calculus, following [HL13] (without Sections 5 & 6). Based on classical vector analysis the formulas are also derived in [SZ92, Sects 2,19,2.20] and [DZ10, Ch. 9, Sect. 5]. Important background and supplementary information about the shape Hessian can be found in [DZ91, BZ97] and [DZ10, Ch. 9, Sect. 6]. 4. Shape derivatives of solutions of PDEs using exterior calculus [HL17], see also [HL13,Sects. 5 & 6]. From the perspective of classical calculus the topic is partly covered in [SZ92, Sects. 3.1-3.2]. 5. Shape gradients under PDE constraints according to [Pag16, Sect. 2.1] including a presentation of the adjoint method for differentiating constrained functionals [HPUU09, Sect. 1.6]. Related information can be found in [DZ10, Ch. 10, Sect. 2.5] and [SZ92, Sect. 3.3]. 6. Approximation of shape gradients following [HPS14]. Comparison of discrete shape gradients based on volume and boundary formulas, see also [DZ10, Ch. 10, Sect. 2.5]. 7. Optimal shape design based on boundary integral equations following [Epp00b], with some additional information provided in [Epp00a]. 8. Convergence in elliptic shape optimization as discussed in [EHS07]. Relies on results reported in [Epp00b] and [DP00]. Discusses Ritz-Galerkin discretization of optimality conditions for normal displacement parameterization. 9. Shape optimization by pursuing diffeomorphisms according to [HP15], see also [Pag16,Ch. 3] for more details, and [PWF17] for extensions. 10. Distributed shape derivative via averaged adjoint method following [LS16]. | |||||

Literature | References: [BZ97] Dorin Bucur and Jean-Paul Zolsio. Anatomy of the shape hessian via lie brackets. Annali di Matematica Pura ed Applicata, 173:127–143, 1997. 10.1007/BF01783465. [DP00] Marc Dambrine and Michel Pierre. About stability of equilibrium shapes. M2AN Math. Model. Numer. Anal., 34(4):811–834, 2000. [DZ91] Michel C. Delfour and Jean-Paul Zolésio. Velocity method and Lagrangian formulation for the computation of the shape Hessian. SIAM J. Control Optim., 29(6):1414–1442, 1991. [DZ10] M.C. Delfour and J.-P. Zolésio. Shapes and Geometries, volume 22 of Advances in Design and Control. SIAM, Philadelphia, 2nd edition, 2010. [EHS07] Karsten Eppler, Helmut Harbrecht, and Reinhold Schneider. On convergence in elliptic shape optimization. SIAM J. Control Optim., 46(1):61–83 2007. [Epp00a] Karsten Eppler. Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim., 20(1):63–78, 2000. German-Polish Conference on Optimization—Methods and Applications (Żagań, 1999). [Epp00b] Karsten Eppler. Optimal shape design for elliptic equations via BIE-methods. Int. J. Appl. Math. Comput. Sci., 10(3):487–516, 2000. [HL13] Ralf Hiptmair and Jingzhi Li. Shape derivatives in differential forms I: an intrinsic perspective. Ann. Mat. Pura Appl. (4), 192(6):1077–1098, 2013. [HL17] R. Hiptmair and J.-Z. Li. Shape derivatives in differential forms II: Application to scattering problems. Report 2017-24, SAM, ETH Zürich, 2017. To appear in Inverse Problems. [HP15] Ralf Hiptmair and Alberto Paganini. Shape optimization by pursuing diffeomorphisms. Comput. Methods Appl. Math., 15(3):291–305, 2015. [HPS14] R. Hiptmair, A. Paganini, and S. Sargheini. Comparison of approximate shape gradients. BIT Numerical Mathematics, 55:459–485, 2014. [HPUU09] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints, volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York, 2009. [LS16] Antoine Laurain and Kevin Sturm. Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal., 50(4):1241–1267,2016. [Pag16] A. Paganini. Numerical shape optimization with finite elements. Eth dissertation 23212, ETH Zurich, 2016. [PWF17] A. Paganini, F. Wechsung, and P.E. Farell. Higher-order moving mesh methods for pde-constrained shape optimization. Preprint arXiv:1706.03117 [math.NA], arXiv, 2017. [SZ92] J. Sokolowski and J.-P. Zolesio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer, Berlin, 1992. | |||||

Prerequisites / Notice | Knowledge of analysis and functional analysis; knowledge of PDEs is an advantage and so is some familiarity with numerical methods for PDEs | |||||

401-4640-68L | Uncertainty Quantification in Electromagnetism Number of participants limited to 10 | W | 4 credits | 2S | C. Jerez Hanckes | |

Abstract | In this seminar we will discuss, through the reading of recent papers, state-of-the-art techniques in uncertainty quantification as well as some typical numerical methods to model electromagnetic problems with uncertainty. | |||||

Learning objective | We will understand state-of-the-art techniques in uncertainty quantification as well as some typical numerical methods to model electromagnetic problems with uncertainty. | |||||

Content | 1. Basics of Electromastatics and Electromagnetic 2. Basics of Finite Element and Boundary Element Methods 3. Introduction of different UQ techniques 4. Application of UQ in Maxwell equations | |||||

Literature | - K. Beddek, Y. Le Menach, S. Clenet, and O. Moreau. 3-d stochastic spectral finite- element method in static electromagnetism using vector potential formulation. Mag- netics, IEEE Transactions on, 47(5):1250–1253, May 2011 - J. Castrillón-Candás, F. Nobile, and R. Tempone. Analytic regularity and collocation approximation for elliptic PDEs with Random domain deformations. Computers and Mathematics with Applications, 71(6):1173–1197, 2016. - C.Chauvière,J.S.Hesthaven,andL.Lurati.Computational modeling of uncertainty in time-domain electromagnetics. SIAM J. Sci. Comput., 28(2):751–775, 2006. - Cohen, Albert; Devore, Ronald; Schwab, Christoph Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE's - Dick, Josef; Gantner, Robert N.; Le Gia, Quoc T.; Schwab, Christoph Multilevel higher-order quasi-Monte Carlo Bayesian estimation. Math. Models Methods Appl. Sci. 27 (2017), no. 5, 953–995. - Gantner, Robert N.; Schwab, Christoph Computational higher order quasi-Monte Carlo integration. Monte Carlo and quasi-Monte Carlo methods, 271–288, Springer Proc. Math. Stat., 163, - Harbrecht, Helmut; Schneider, Reinhold; Schwab, Christoph Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109 (2008), no. 3, - Hiptmair, R., Scarabosio, L., Schillings, C. et al. Adv Comput Math (2018). - Jerez-Hanckes, Carlos; Schwab, Christoph; Zech, Jakob Electromagnetic wave scattering by random surfaces: shape holomorphy. Math. Models Methods Appl. Sci. 27 (2017), no. 12,2229–2259 - Jerez-Hanckes, Carlos; Schwab, Christoph Electromagnetic wave scattering by random surfaces: uncertainty quantification via sparse tensor boundary elements, IMA J. Numer. Anal. 37 (2017), no. 3, 1175–1210 - von Petersdorff, Tobias; Schwab, Christoph Sparse finite element methods for operator equations with stochastic data. Appl. Math. 51 (2006), no. 2, 145–180. | |||||

Prerequisites / Notice | - Assistance is mandatory - Students will choose 1 or 2 articles for presenting - Individual meetings with lecturer will be scheduled on ad hoc basis | |||||

Semester Papers There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. | ||||||

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

401-3750-01L | Semester Paper Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see www.math.ethz.ch/intranet/students/study-administration/theses.html | W | 8 credits | 11A | Supervisors | |

Abstract | Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing. | |||||

Learning objective | ||||||

Prerequisites / Notice | There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. | |||||

401-3750-02L | Semester Paper Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see www.math.ethz.ch/intranet/students/study-administration/theses.html | W | 8 credits | 11A | Supervisors | |

Abstract | Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing. | |||||

Learning objective | ||||||

Prerequisites / Notice | There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again. | |||||

401-3750-03L | Semester Paper Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see www.math.ethz.ch/intranet/students/study-administration/theses.html | W | 8 credits | 11A | Supervisors | |

Abstract | Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing. | |||||

Learning objective | ||||||

Prerequisites / Notice | ||||||

GESS Science in Perspective | ||||||

» Recommended Science in Perspective (Type B) for D-MATH. | ||||||

» see Science in Perspective: Language Courses ETH/UZH | ||||||

» see Science in Perspective: Type A: Enhancement of Reflection Capability | ||||||

Master's Thesis | ||||||

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

401-2000-00L | Scientific Works in MathematicsTarget audience: Third year Bachelor students; Master students who cannot document to have received an adequate training in working scientifically. | O | 0 credits | E. Kowalski | ||

Abstract | 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.) | |||||

Learning objective | Learn the basic standards of scientific works in mathematics. | |||||

Content | - Types of mathematical works - Publication standards in pure and applied mathematics - Data handling - Ethical issues - Citation guidelines | |||||

Lecture notes | Moodle of the Mathematics Library: https://moodle-app2.let.ethz.ch/course/view.php?id=519 | |||||

Prerequisites / Notice | Directive Link | |||||

401-2000-01L | Training Course "Search for Documents in Mathematics" [under revision]Details and registration for the optional MathBib training course: https://www.math.ethz.ch/mathbib-schulungen | Z | 0 credits | Speakers | ||

Abstract | Optional course "Recherchieren in der Mathematik" (held in German) by the Mathematics Library. | |||||

Learning objective | ||||||

401-4990-00L | Master's Thesis Only students who fulfil the following criteria are allowed to begin with their Master's thesis: a. successful completion of the Bachelor's programme; b. fulfilling of any additional requirements necessary to gain admission to the Master's programme. Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required. For more information, see www.math.ethz.ch/intranet/students/study-administration/theses.html | O | 30 credits | 57D | Supervisors | |

Abstract | The master's thesis concludes the study programme. Writing up the master's thesis allows students to independently produce a major piece of work on a mathematical topic. It generally involves consulting the literature, solving any ensuing problems, and putting together the results in writing. | |||||

Learning objective | ||||||

Additional Courses | ||||||

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

401-5000-00L | Zurich Colloquium in Mathematics | E- | 0 credits | A. Iozzi, S. Mishra, R. Pandharipande, University lecturers | ||

Abstract | 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. | |||||

Learning objective | ||||||

401-5990-00L | Zurich Graduate Colloquium | E- | 0 credits | 1K | A. Iozzi, University lecturers | |

Abstract | 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. | |||||

Learning objective | ||||||

401-5110-00L | Number Theory Seminar | E- | 0 credits | 1K | Ö. Imamoglu, P. S. Jossen, E. Kowalski, P. D. Nelson, R. Pink, G. Wüstholz | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5350-00L | Analysis Seminar | E- | 0 credits | 1K | M. Struwe, A. Carlotto, F. Da Lio, A. Figalli, N. Hungerbühler, T. Ilmanen, T. Rivière, University lecturers | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5370-00L | Ergodic Theory and Dynamical Systems | E- | 0 credits | 1K | M. Einsiedler, University lecturers, further lecturers | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5530-00L | Geometry Seminar | E- | 0 credits | 1K | M. Burger, M. Einsiedler, A. Iozzi, U. Lang, A. Sisto, University lecturers | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5580-00L | Symplectic Geometry Seminar | E- | 0 credits | 2K | P. Biran, A. Cannas da Silva | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5330-00L | Talks in Mathematical Physics | E- | 0 credits | 1K | A. Cattaneo, G. Felder, M. Gaberdiel, G. M. Graf, T. H. Willwacher, University lecturers | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5650-00L | Zurich Colloquium in Applied and Computational Mathematics | E- | 0 credits | 2K | R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, A. Jentzen, C. Jerez Hanckes, S. Mishra, S. Sauter, C. Schwab | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5600-00L | Seminar on Stochastic Processes | E- | 0 credits | 1K | J. Bertoin, A. Nikeghbali, B. D. Schlein, A.‑S. Sznitman, V. Tassion, W. Werner | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5620-00L | Research Seminar on Statistics | E- | 0 credits | 2K | L. Held, T. Hothorn, D. Kozbur, M. H. Maathuis, N. Meinshausen, S. van de Geer, M. Wolf | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

401-5640-00L | ZüKoSt: Seminar on Applied Statistics | E- | 0 credits | 1K | M. Kalisch, R. Furrer, L. Held, T. Hothorn, M. H. Maathuis, M. Mächler, L. Meier, N. Meinshausen, M. Robinson, C. Strobl, S. van de Geer | |

Abstract | About 5 talks on applied statistics. | |||||

Learning objective | See how statistical methods are applied in practice. | |||||

Content | There will be about 5 talks on how statistical methods are applied in practice. | |||||

Prerequisites / Notice | This is no lecture. There is no exam and no credit points will be awarded. The current program can be found on the web: http://stat.ethz.ch/events/zukost Course language is English or German and may depend on the speaker. | |||||

401-5910-00L | Talks in Financial and Insurance Mathematics | E- | 0 credits | 1K | P. Cheridito, M. Schweizer, M. Soner, J. Teichmann, M. V. Wüthrich | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

Content | Regular research talks on various topics in mathematical finance and actuarial mathematics | |||||

401-5900-00L | Optimization Seminar | E- | 0 credits | 1K | R. Weismantel, R. Zenklusen | |

Abstract | Lectures on current topics in optimization | |||||

Learning objective | Expose graduate students to ongoing research acitivites (including applications) in the domain of otimization. | |||||

Content | This seminar is a forum for researchers interested in optimization theory and its applications. Speakers are expected to stimulate discussions on theoretical and applied aspects of optimization and related subjects. The focus is on efficient 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. | |||||

401-5960-00L | Colloquium on Mathematics, Computer Science, and Education Subject didactics for mathematics and computer science teachers. | E- | 0 credits | N. Hungerbühler, M. Akveld, J. Hromkovic, H. Klemenz | ||

Abstract | Didactics colloquium | |||||

Learning objective | ||||||

402-0101-00L | The Zurich Physics Colloquium | E- | 0 credits | 1K | R. Renner, G. Aeppli, C. Anastasiou, G. Blatter, S. Cantalupo, C. Degen, G. Dissertori, K. Ensslin, T. Esslinger, J. Faist, M. Gaberdiel, T. K. Gehrmann, G. M. Graf, R. Grange, J. Home, S. Huber, A. Imamoglu, P. Jetzer, S. Johnson, U. Keller, K. S. Kirch, S. Lilly, L. M. Mayer, J. Mesot, B. Moore, D. Pescia, A. Refregier, A. Rubbia, T. C. Schulthess, M. Sigrist, A. Vaterlaus, R. Wallny, A. Wallraff, W. Wegscheider, A. Zheludev, O. Zilberberg | |

Abstract | Research colloquium | |||||

Learning objective | ||||||

402-0800-00L | The Zurich Theoretical Physics Colloquium | E- | 0 credits | 1K | O. Zilberberg, C. Anastasiou, G. Blatter, M. Gaberdiel, T. K. Gehrmann, G. M. Graf, S. Huber, P. Jetzer, L. M. Mayer, B. Moore, R. Renner, T. C. Schulthess, M. Sigrist, University lecturers | |

Abstract | Research colloquium | |||||

Learning objective | The Zurich Theoretical Physics Colloquium is jointly organized by the University of Zurich and ETH Zurich. Its mission is to bring both students and faculty with diverse interests in theoretical physics together. Leading experts explain the basic questions in their field of research and communicate the fascination for their work. | |||||

251-0100-00L | Computer Science Colloquium | E- | 0 credits | 2K | Lecturers | |

Abstract | Invited talks, covering the entire scope of computer science. External Listeners are welcome at no charge. A detailed schedule is published at the beginning of each semester. | |||||

Learning objective | Top international computer scientists take the floor at the distinguished computer science colloquium. Our guest speakers present impacting topics across various areas of the discipline. The colloquium series is held every semester and also includes inaugural and farewell lectures of the department's professors. The colloquium is a noteworthy event for all graduate students. Outside attendance is equally welcome. | |||||

Content | 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 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. | E- | 2 credits | 2S | E. Welzl, B. Gärtner, M. Hoffmann, J. Lengler, A. Steger, B. Sudakov | |

Abstract | Presentation of recent publications in theoretical computer science, including results by diploma, masters and doctoral candidates. | |||||

Learning objective | The goal is to introduce students to current research, and to enable them to read, understand, and present scientific papers. | |||||

Course Units for Additional Admission Requirements The courses below are only available for MSc students with additional admission requirements. | ||||||

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

406-2004-AAL | Algebra IIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 5 credits | 11R | M. Burger | |

Abstract | Galois theory and Representations of finite groups, algebras. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

Learning objective | Introduction to fundamentals of Galois theory, and representation theory of finite groups and algebras | |||||

Content | Fundamentals of Galois theory Representation theory of finite groups and algebras | |||||

Lecture notes | For a summary of the content and exercises with solutions of my lecture course in FS2016 see: https://www2.math.ethz.ch/education/bachelor/lectures/fs2016/math/algebra2/ | |||||

Literature | S. Lang, Algebra, Springer Verlag B.L. van der Waerden: Algebra I und II, Springer Verlag I.R. Shafarevich, Basic notions of algebra, Springer verlag G. Mislin: Algebra I, vdf Hochschulverlag U. Stammbach: Algebra, in der Polybuchhandlung erhältlich I. Stewart: Galois Theory, Chapman Hall (2008) G. Wüstholz, Algebra, vieweg-Verlag, 2004 J-P. Serre, Linear representations of finite groups, Springer Verlag | |||||

Prerequisites / Notice | Algebra I | |||||

406-2005-AAL | Algebra I and IIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 12 credits | 26R | M. Burger, E. Kowalski | |

Abstract | 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. | |||||

Learning objective | ||||||

Content | 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 | |||||

Lecture notes | For a summary of the content and exercises with solutions of my lecture courses in HS2015 and FS2016 see: Link https://www2.math.ethz.ch/education/bachelor/lectures/fs2016/math/algebra2/ | |||||

Literature | S. Lang, Algebra, Springer Verlag B.L. van der Waerden: Algebra I und II, Springer Verlag I.R. Shafarevich, Basic notions of algebra, Springer verlag G. Mislin: Algebra I, vdf Hochschulverlag U. Stammbach: Algebra, in der Polybuchhandlung erhältlich I. Stewart: Galois Theory, Chapman Hall (2008) G. Wüstholz, Algebra, vieweg-Verlag, 2004 J-P. Serre, Linear representations of finite groups, Springer Verlag | |||||

406-2303-AAL | Complex AnalysisEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | M. Struwe | |

Abstract | 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. | |||||

Learning objective | ||||||

Literature | 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 | |||||

406-2284-AAL | Measure and IntegrationAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | U. Lang | |

Abstract | Introduction to the abstract measure theory and integration, including the following topics: Lebesgue measure and Lebesgue integral, Lp-spaces, convergence theorems, differentiation of measures, product measures (Fubini's theorem), abstract measures, Radon-Nikodym theorem, probabilistic language. | |||||

Learning objective | Basic acquaintance with the theory of measure and integration, in particular, Lebesgue's measure and integral. | |||||

Literature | 1. Lecture notes by Professor Michael Struwe (http://www.math.ethz.ch/~struwe/Skripten/AnalysisIII-SS2007-18-4-08.pdf) 2. L. Evans and R.F. Gariepy "Measure theory and fine properties of functions" 3. Walter Rudin "Real and complex analysis" 4. R. Bartle The elements of Integration and Lebesgue Measure 5. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis. http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf | |||||

406-2554-AAL | TopologyAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | A. Sisto | |

Abstract | Topological spaces, continuous maps, connectedness, compactness, separation axioms, metric spaces, quotient spaces, homotopy, fundamental group and covering spaces, van Kampen Theorem, surfaces and manifolds. | |||||

Learning objective | ||||||

Literature | Klaus Jänich: Topologie (Springer-Verlag) Link James Munkres: Topology (Prentice Hall) William Massey: Algebraic Topology: an Introduction (Springer-Verlag) Alan Hatcher: Algebraic Topology (Cambridge University Press) http://www.math.cornell.edu/~hatcher/AT/ATpage.html | |||||

Prerequisites / Notice | The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

406-2604-AAL | Probability and StatisticsAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 7 credits | 15R | J. Teichmann | |

Abstract | Introduction to probability and statistics with many examples, based on chapters from the books "Probability and Random Processes" by G. Grimmett and D. Stirzaker and "Mathematical Statistics and Data Analysis" by J. Rice. | |||||

Learning objective | The goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. In addition to a mathematically rigorous treatment, also an intuitive understanding and familiarity with the ideas behind the definitions are emphasized. Measure theory is not used systematically, but it should become clear why and where measure theory is needed. | |||||

Content | Probability: Chapters 1-5 (Probabilities and events, Discrete and continuous random variables, Generating functions) and Sections 7.1-7.5 (Convergence of random variables) from the book "Probability and Random Processes". Most of this material is also covered in Chap. 1-5 of "Mathematical Statistics and Data Analysis", on a slightly easier level. Statistics: Sections 8.1 - 8.5 (Estimation of parameters), 9.1 - 9.4 (Testing Hypotheses), 11.1 - 11.3 (Comparing two samples) from "Mathematical Statistics and Data Analysis". | |||||

Literature | Geoffrey Grimmett and David Stirzaker, Probability and Random Processes. 3rd Edition. Oxford University Press, 2001. John A. Rice, Mathematical Statistics and Data Analysis, 3rd edition. Duxbury Press, 2006. | |||||

406-3461-AAL | Functional Analysis IAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 10 credits | 21R | M. Einsiedler | |

Abstract | Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces; Fourier transform and applications. | |||||

Learning objective | Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps. | |||||

Literature | We will be using the book Functional Analysis, Spectral Theory, and Applications by Manfred Einsiedler and Thomas Ward and available by SpringerLink. Other useful, and recommended references include the following: Lecture Notes on "Funktionalanalysis I" by Michael Struwe Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. | |||||

406-3621-AAL | Fundamentals of Mathematical StatisticsAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 10 credits | 21R | S. van de Geer | |

Abstract | The course covers the basics of inferential statistics. | |||||

Learning objective |