# Search result: Catalogue data in Spring Semester 2019

Mathematics Bachelor | ||||||

Core Courses | ||||||

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

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3532-08L | Differential Geometry II | W | 10 credits | 4V + 1U | W. Merry | |

Abstract | This is a continuation course of Differential Geometry I. Topics covered include: - Connections and curvature, - Riemannian geometry, - Gauge theory and Chern-Weil theory. | |||||

Objective | ||||||

Lecture notes | I will produce full lecture notes, available from my website at Link | |||||

Literature | There are many excellent textbooks on differential geometry. A friendly and readable book that contains everything covered in Differential Geometry I is: John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag. For Differential Geometry II, the textbooks: - S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley, - I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP, are both excellent. The monograph - A. L. Besse "Einstein Manifolds", (1987), Springer, gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.) | |||||

Prerequisites / Notice | Familiarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). lecture notes for Differential Geometry I can be found on my webpage. | |||||

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

Abstract | Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity, spectral theory, and unitary representations. | |||||

Objective | Acquiring the language and methods for boundary value problems, Sobolev spaces, Banach algebras, Spectral theory of bounded and unbounded selfadjoint operators, and Unitary representations. | |||||

Lecture notes | Functional Analysis, Spectral Theory and Applications. Manfred Einsiedler and Thomas Ward, GTM Springer 2017 | |||||

Literature | Functional Analysis, Spectral Theory and Applications. Manfred Einsiedler and Thomas Ward, GTM Springer 2017 | |||||

Prerequisites / Notice | Functional Analysis I plus a 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). | |||||

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

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, Link * Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link "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, Link 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. Biran | |

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. | |||||

Objective | ||||||

Literature | 1) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 2) E. Spanier, "Algebraic topology", Springer-Verlag 3) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link | |||||

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. | |||||

» Core Courses: Pure Mathematics (Mathematics Master) | ||||||

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. | |||||

Prerequisites / Notice | Students are expected to have a mathematical background and should be able to write rigorous proofs. | |||||

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

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" | |||||

» Core Courses: Applied Mathematics and Further Appl.-Oriented Fields (Mathematics Master) | ||||||

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

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 | - J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016). - I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991). - D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005). - L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000). - D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006). | |||||

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 | 8 credits | 3V + 1U | M. H. Maathuis | |

Abstract | We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R. | |||||

Objective | The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R. | |||||

Content | See the class website | |||||

Prerequisites / Notice | At least one semester of (basic) probability and statistics. Programming experience is helpful but not required. | |||||

401-3602-00L | Applied Stochastic Processes | W | 8 credits | 3V + 1U | V. Tassion | |

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: Link R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: Link M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: Link 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, high-dimensional linear models, nonlinear models and generalized linear models, robust methods, model choice and nonparametric models. Several numerical examples will illustrate the theory. | |||||

Objective | Introduction into theory and practice of a broad and popular area of statistics, from a modern viewpoint. | |||||

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, Hoch-dimensionale Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste 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). | |||||

Core Courses: Further Application-Oriented Fields 402-0204-00L Electrodynamics is eligible in the Bachelor's programme in Mathematics 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 (Link) after having received the credits. | ||||||

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

402-0204-00L | Electrodynamics | W | 7 credits | 4V + 2U | G. M. Graf | |

Abstract | Derivation and discussion of Maxwell's equations, from the static limit to the full dynamical case. Wave equation, waveguides, cavities. Generation of electromagnetic radiation, scattering and diffraction of light. Structure of Maxwell's equations, relativity theory and covariance, Lagrangian formulation. Dynamics of relativistic particles in the presence of fields and radiation properties. | |||||

Objective | Develop a physical understanding for static and dynamic phenomena related to (moving) charged objects and understand the structure of the classical field theory of electrodynamics (transverse versus longitudinal physics, invariances (Lorentz-, gauge-)). Appreciate the interrelation between electric, magnetic, and optical phenomena and the influence of media. Understand a set of classic electrodynamical phenomena and develop the ability to solve simple problems independently. Apply previously learned mathematical concepts (vector analysis, complete systems of functions, Green's functions, co- and contravariant coordinates, etc.). Prepare for quantum mechanics (eigenvalue problems, wave guides and cavities). | |||||

Content | Classical field theory of electrodynamics: Derivation and discussion of Maxwell equations, starting from the static limit (electrostatics, magnetostatics, boundary value problems) in the vacuum and in media and subsequent generalization to the full dynamical case (Faraday's law, Ampere/Maxwell law; potentials and gauge invariance). Wave equation and solutions in full space, half-space (Snell's law), waveguides, cavities, generation of electromagnetic radiation, scattering and diffraction of light (optics). Application to various specific examples. Discussion of the structure of Maxwell's equations, Lorentz invariance, relativity theory and covariance, Lagrangian formulation. Dynamics of relativistic particles in the presence of fields and their radiation properties (synchrotron). | |||||

Literature | J.D. Jackson, Classical Electrodynamics W.K.H Panovsky and M. Phillis, Classical electricity and magnetism L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of continuus media A. Sommerfeld, Electrodynamics / Optics (Lectures on Theoretical Physics) M. Born and E. Wolf, Principles of optics R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures of Physics, Vol II W. Nolting, Elektrodynamik (Grundkurs Theoretische Physik 3) |

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