# Search result: Catalogue data in Spring Semester 2017

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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401-3146-12L | Algebraic Geometry | W | 10 credits | 4V + 1U | R. Pink | |

Abstract | This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes). | |||||

Objective | Learning Algebraic Geometry. | |||||

Literature | Primary reference: * Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer. Secondary reference: * Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications. * Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer. * Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013). Other good textbooks and online texts are: * David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer. * Ravi Vakil, Foundations of Algebraic Geometry, http://math.stanford.edu/~vakil/216blog/ * Jean Gallier and Stephen S. Shatz, Algebraic Geometry http://www.cis.upenn.edu/~jean/algeom/steve01.html "Classical" Algebraic Geometry over an algebraically closed field: * Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer. * J.S. Milne, Algebraic Geometry, http://www.jmilne.org/math/CourseNotes/AG.pdf Further readings: * Günter Harder: Algebraic Geometry 1 & 2 * I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag. * Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA * Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag. | |||||

Prerequisites / Notice | Requirement: Some knowledge of Commutative Algebra. | |||||

401-3002-12L | Algebraic Topology II | W | 8 credits | 4G | P. S. Jossen | |

Abstract | This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as: products, duality, cohomology operations, characteristic classes, spectral sequences etc. | |||||

Objective | ||||||

Literature | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: http://www.math.cornell.edu/~hatcher/AT/ATpage.html See also: http://www.math.cornell.edu/~hatcher/#anchor1772800 2) E. Spanier, "Algebraic topology", Springer-Verlag 3) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 4) R. Bott & L. Tu, "Differential forms in algebraic topology", Graduate Texts in Mathematics, 82. Springer-Verlag, 1982. 5) J. Milnor & J. Stasheff, "Characteristic classes", Annals of Mathematics Studies, No. 76. Princeton University Press, 1974. | |||||

Prerequisites / Notice | General topology, linear algebra. Basic knowledge of singular homolgoy and cohomology of topological spaces (e.g. as taught in "Algebraic topology I"). Some knowledge of differential geometry and differential topology is useful but not absolutely necessary. | |||||

401-3226-01L | Representation Theory of Lie Groups | W | 8 credits | 4G | E. Kowalski | |

Abstract | This course will contain two parts: * Introduction to unitary representations of Lie groups * Introduction to the study of discrete subgroups of Lie groups and some applications. | |||||

Objective | The goal is to acquire familiarity with the basic formalism and results concerning unitary representations of Lie groups, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups. | |||||

Content | * Unitary representations of compact Lie groups: Peter-Weyl theory, weights, Weyl character formula * Introduction to unitary representations of non-compact Lie groups: the examples of SL(2,R), SL(2,C) * Example: Property (T) for SL(n,R) * Discrete subgroups of Lie groups: examples and some applications | |||||

Literature | Bekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press. | |||||

Prerequisites / Notice | Differential geometry, Functional analysis, Introduction to Lie Groups (or equivalent). Notice that this course has a large overlap with 401-3226-01L Unitary Representations of Lie Groups and Discrete Subgroups of Lie Groups taught in FS 2016. Therefore it is not possible to acquire credits for both courses. | |||||

401-3372-00L | Dynamical Systems II | W | 10 credits | 4V + 1U | W. Merry | |

Abstract | This course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic dynamics. | |||||

Objective | Mastery of the basic methods and principal themes of some aspects of hyperbolic dynamical systems. | |||||

Content | Topics covered include: - Circle homeomorphisms and rotation numbers. - Hyperbolic linear dynamical systems, hyperbolic fixed points, the Hartman-Grobman Theorem. - Hyperbolic sets, Anosov diffeomorphisms. - The (Un)stable Manifold Theorem. - Shadowing Lemmas and stability. - The Lambda Lemma. - Transverse homoclinic points, horseshoes, and chaos. | |||||

Lecture notes | I will provide full lecture notes, available here: http://www.merry.io/dynamical-systems/ | |||||

Literature | The most useful textbook is - Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002. Another (more advanced) useful book is - Introduction to the Modern Theory of Dynamical Systems, Katok and Hasselblatt, CUP, 1995. | |||||

Prerequisites / Notice | It will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here: http://www.merry.io/dynamical-systems/ However we will only really use material covered in the first 12 lectures of Dynamical Systems I, so if you did not attend Dynamical Systems I, it is sufficient to read through the notes from the first 12 lectures. In addition, it would be useful to have some familiarity with basic differential geometry. | |||||

401-3532-08L | Differential Geometry II | W | 10 credits | 4V + 1U | U. Lang | |

Abstract | Introduction to Riemannian Geometry in combination with some elements of modern metric geometry. Contents: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, curvature and topology, spaces of riemannian manifolds. | |||||

Objective | The aim of this course is to give an introduction to Riemannian Geometry in combination with some elements of modern metric geometry. | |||||

Content | Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form of submanifolds, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of riemannian manifolds. | |||||

Literature | Riemannian Geometry: - M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992 - S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004 - B. O'Neill, Semi-Riemannian Geometry, With Applications to Relativity, Academic Press 1983 Metric Geometry: - M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999 - D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Amer. Math. Soc. 2001 | |||||

Prerequisites / Notice | Prerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, tangent and tensor bundles, and differential forms. | |||||

401-3462-00L | Functional Analysis II | W | 10 credits | 4V + 1U | M. Struwe | |

Abstract | Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity theory, Schauder estimates | |||||

Objective | The lecture course will focus on weak solutions of elliptic boundary value problems in Sobolev spaces and discuss their regularity properties, possibly followed by a proof of the Calderon-Zygmund inequality and some basic results on parabolic regularity, with applications to geometry, if time allows. | |||||

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

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

401-3052-10L | Graph Theory | W | 10 credits | 4V + 1U | B. Sudakov | |

Abstract | Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem | |||||

Objective | The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems. | |||||

Lecture notes | Lecture will be only at the blackboard. | |||||

Literature | West, D.: "Introduction to Graph Theory" Diestel, R.: "Graph Theory" Further literature links will be provided in the lecture. | |||||

401-3652-00L | Numerical Methods for Hyperbolic Partial Differential Equations | W | 10 credits | 4V + 1U | U. S. Fjordholm | |

Abstract | This course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB. | |||||

Objective | The goal of this course is familiarity with the fundamental ideas and mathematical consideration underlying modern numerical methods for conservation laws and wave equations. | |||||

Content | * Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering. * Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes. * Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes. * Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods. * Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes. * Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory. | |||||

Lecture notes | Lecture slides will be made available to participants. However, additional material might be covered in the course. | |||||

Literature | H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online. R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online. E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991. | |||||

Prerequisites / Notice | Having attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite. Programming exercises in MATLAB Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations" | |||||

401-3642-00L | Brownian Motion and Stochastic Calculus | W | 10 credits | 4V + 1U | M. Larsson | |

Abstract | This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations. | |||||

Objective | This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations. | |||||

Lecture notes | Lecture notes will be distributed in class. | |||||

Literature | - I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991). - D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005). - L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000). - D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006). | |||||

Prerequisites / Notice | Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in - J. Jacod, P. Protter, Probability Essentials, Springer (2004). - R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010). | |||||

401-3632-00L | Computational Statistics | W | 10 credits | 3V + 2U | M. Mächler, P. L. Bühlmann | |

Abstract | "Computational Statistics" deals with modern methods of data analysis (aka "data science") for prediction and inference. An overview of existing methodology is provided and also by the exercises, the student is taught to choose among possible models and about their algorithms and to validate them using graphical methods and simulation based approaches. | |||||

Objective | Getting to know modern methods of data analysis for prediction and inference. Learn to choose among possible models and about their algorithms. Validate them using graphical methods and simulation based approaches. | |||||

Content | Course Synopsis: multiple regression, nonparametric methods for regression and classification (kernel estimates, smoothing splines, regression and classification trees, additive models, projection pursuit, neural nets, ridging and the lasso, boosting). Problems of interpretation, reliable prediction and the curse of dimensionality are dealt with using resampling, bootstrap and cross validation. Details are available via https://stat.ethz.ch/lectures/ . Exercises will be based on the open-source statistics software R (http://www.R-project.org/). Emphasis will be put on applied problems. Active participation in the exercises is strongly recommended. More details are available via the webpage https://stat.ethz.ch/lectures/ (-> "Computational Statistics"). | |||||

Lecture notes | lecture notes are available online; see http://stat.ethz.ch/lectures/ (-> "Computational Statistics"). | |||||

Literature | (see the link above, and the lecture notes) | |||||

Prerequisites / Notice | Basic "applied" mathematical calculus (incl. simple two-dimensional) and linear algebra (including Eigenvalue decomposition) similar to two semester "Analysis" in an ETH (math or) engineer's bachelor. At least one semester of (basic) probability and statistics, as e.g., taught in an ETH engineer's or math bachelor. Programming experience in either a compiler-based computer language (such as C++) or a high-level language such as python, R, julia, or matlab. The language used in the exercises and the final exam will be R (https://www.r-project.org) exclusively. If you don't know it already, some extra effort will be required for the exercises. | |||||

401-3602-00L | Applied Stochastic Processes | W | 8 credits | 3V + 1U | A.‑S. Sznitman | |

Abstract | Poisson processes; renewal processes; Markov chains in discrete and in continuous time; some applications. | |||||

Objective | Stochastic processes are a way to describe and study the behaviour of systems that evolve in some random way. In this course, the evolution will be with respect to a scalar parameter interpreted as time, so that we discuss the temporal evolution of the system. We present several classes of stochastic processes, analyse their properties and behaviour and show by some examples how they can be used. The main emphasis is on theory; in that sense, "applied" should be understood to mean "applicable". | |||||

Literature | R. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: http://epubs.siam.org/doi/book/10.1137/1.9780898718997 R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: http://link.springer.com/book/10.1007/978-1-4614-3615-7/page/1 M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: http://link.springer.com/book/10.1007/978-0-387-48976-6/page/1 S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005) | |||||

Prerequisites / Notice | Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L). | |||||

401-3622-00L | RegressionDoes not take place this semester. | W | 8 credits | 4G | not available | |

Abstract | In regression, the dependency of a random response variable on other variables is examined. We consider the theory of linear regression with one or more covariates, nonlinear models and generalized linear models, robust methods, model choice and nonparametric models. Several numerical examples will illustrate the theory. | |||||

Objective | Einführung in Theorie und Praxis eines umfassenden und vielbenutzten Teilgebiets der angewandten Statistik, unter Berücksichtigung neuerer Entwicklungen. | |||||

Content | In der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression, Numerik, Ridge Regression. Durchrechnung und Diskussion von Anwendungsbeispielen. | |||||

Lecture notes | Lecture notes | |||||

Prerequisites / Notice | Credits cannot be recognised for both courses 401-3622-00L Regression and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden). |

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