# Search result: Catalogue data in Autumn Semester 2016

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 | |
---|---|---|---|---|---|---|

401-3225-00L | Introduction to Lie Groups | W | 8 credits | 4G | P. D. Nelson | |

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. | |||||

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: http://www.math.ethz.ch/education/bachelor/lectures/hs2014/math/introlg | |||||

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 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. | W | 10 credits | 4V + 1U | C. Schwab | |

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. | |||||

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 | n.a. | |||||

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

401-3621-00L | Fundamentals of Mathematical Statistics | W | 10 credits | 4V + 1U | F. Balabdaoui | |

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

Objective | ||||||

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

Abstract | Advanced introduction to mathematical finance: - absence of arbitrage and martingale measures - option pricing and hedging - optimal investment problems - additional topics | |||||

Objective | Advanced level introduction to mathematical finance, presupposing knowledge in probability theory and stochastic processes | |||||

Content | This is an advanced level introduction to mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this in both discrete- and continuous-time models. Topics include absence of arbitrage and martingale measures, option pricing and hedging, optimal investment problems, and probably others. Prerequisites are probability theory and stochastic processes (for which lecture notes are available). | |||||

Lecture notes | None available | |||||

Literature | Details will be announced in the course. | |||||

Prerequisites / Notice | Prerequisites are probability theory and stochastic processes (for which lecture notes are available). | |||||

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

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

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. | |||||

(also Bachelor) Core Courses: Pure Mathematics Further restrictions apply, but in particular: 401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme. Analogously for: 401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II 401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II 401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry 401-3371-00L Dynamical Systems I - 401-3372-00L Dynamical Systems II For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

401-3461-00L | Functional Analysis IThis course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from neither 401-3461-00L Functional Analysis I nor 401-3462-00L Functional Analysis II for their Bachelor's degree still can have recognised this course for the Master's degree. Furthermore, at most one of the three course units 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | M. Struwe | |

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

Objective | ||||||

Lecture notes | Lecture Notes on "Funktionalanalysis I" by Michael Struwe | |||||

401-3531-00L | Differential Geometry IThis course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from neither 401-3531-00L Differential Geometry I nor 401-3532-00L Differential Geometry II for their Bachelor's degree still can have recognised this course for the Master's degree. Furthermore, at most one of the three course units 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | U. Lang | |

Abstract | Curves in R^n, inner geometry of hypersurfaces in R^n, curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, tangent bundle, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||

Objective | Introduction to elementary differential geometry and differential topology. | |||||

Content | - Differential geometry in R^n: theory of curves, submanifolds and immersions, inner geometry of hypersurfaces, Gauss map and curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet, Poincaré Index Theorem. - The hyperbolic space. - Differential topology: differentiable manifolds, tangent bundle, immersions and embeddings in R^n, Sard's Theorem, transversality, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||

Literature | Differential Geometry in R^n: - Manfredo P. do Carmo: Differential geometry of curves and surfaces - Wolfgang Kühnel: Differentialgeometrie. Curves-surfaces-manifolds - Christian Bär: Elementary differential geometry Differential Topology: - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds - Victor Guillemin & Alan Pollack: Differential Topology - Morris W. Hirsch: Differential Topology | |||||

401-3371-00L | Dynamical Systems I | W | 10 credits | 4V + 1U | W. Merry | |

Abstract | This course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics. | |||||

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

Content | Topics covered include: 1. Topological dynamics (transitivity, attractors, chaos, structural stability) 2. Ergodic theory (Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures) 3. Low-dimensional dynamics (Poincare rotation number, dynamical systems on [0,1]) | |||||

Literature | The most relevant textbook for this course is Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002. I will also produce full lecture notes. | |||||

Prerequisites / Notice | The material of the basic courses of the first two years of the program at ETH is assumed. In particular, you should be familiar with metric spaces and elementary measure theory. | |||||

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

Abstract | This is an introductory course in algebraic topology. The course will cover the following main topics: introduction to homotopy theory, homology and cohomology of spaces. | |||||

Objective | ||||||

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/%7ehatcher/AT/ATpage.html See also: http://www.math.cornell.edu/%7eehatcher/#anchor1772800 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||

Prerequisites / Notice | General topology, linear algebra. Some knowledge of differential geometry and differential topology is useful but not absolutely necessary. | |||||

401-3132-00L | Commutative Algebra | W | 10 credits | 4V + 1U | R. Pink | |

Abstract | This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. The material in this course will be assumed in the lecture course "Algebraic Geometry" in the spring semester 2017. | |||||

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). | |||||

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

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

401-3601-00L | Probability TheoryThis course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from none of the three course units 401-3601-00L Probability Theory, 401-3642-00L Brownian Motion and Stochastic Calculus resp. 401-3602-00L Applied Stochastic Processes for their Bachelor's degree still can have recognised this course for the Master's degree. Furthermore, at most one of the three course units 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | A.‑S. Sznitman | |

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

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 | T. K. Gehrmann | |

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 and the description of observables as operators on a Hilbert space, and the formulation of symmetries will be discussed. Basic phenomena will be analysed and illustrated by generic examples. | |||||

Objective | Introduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, perturbation theory) and generic examples and applications (bound states, tunneling, scattering states, in one- and three-dimensional settings). Ability to solve simple problems. | |||||

Content | Keywords: Schrödinger equation, basic formalism of quantum mechanics (states, operators, commutators, measuring process), symmetries (translations, rotations), quantum mechanics in one dimension, spherically symmetric problems in three dimensions, scattering theory, perturbation theory, variational techniques, spin, addition of angular momenta, relation between QM and classical physics. | |||||

Literature | F. Schwabl: Quantum mechanics J.J. Sakurai: Modern Quantum Mechanics C. Cohen-Tannoudji: Quantum mechanics I |

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