Suchergebnis: Katalogdaten im Frühjahrssemester 2020

Nummer	Titel	Typ	ECTS	Umfang	Dozierende
Mathematik Bachelor
Kernfächer
Kernfächer aus Bereichen der reinen Mathematik
401-3532-08L	Differential Geometry II	W	10 KP	4V + 1U	U. Lang
Kurzbeschreibung	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, relations between curvature and topology, spaces of Riemannian manifolds.
Lernziel	Learn the basics of Riemannian geometry and some elements of modern metric geometry.
Literatur	- 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
Voraussetzungen / Besonderes	Prerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, and differential forms.
401-3462-00L	Functional Analysis II	W	10 KP	4V + 1U	M. Struwe
Kurzbeschreibung	Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity
Lernziel	Acquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates
Skript	Funktionalanalysis II, Michael Struwe
Literatur	Funktionalanalysis II, Michael Struwe Functional Analysis, Spectral Theory and Applications. Manfred Einsiedler and Thomas Ward, GTM Springer 2017
Voraussetzungen / Besonderes	Functional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces.
401-3146-12L	Algebraic Geometry	W	10 KP	4V + 1U	D. Johnson
Kurzbeschreibung	This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
Lernziel	Learning Algebraic Geometry.
Literatur	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.
Voraussetzungen / Besonderes	Requirement: Some knowledge of Commutative Algebra.
401-3002-12L	Algebraic Topology II	W	8 KP	4G	A. Sisto
Kurzbeschreibung	This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality.
Lernziel
Literatur	1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. The book can be downloaded for free at: http://www.math.cornell.edu/~hatcher/AT/ATpage.html 2) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / Besonderes	General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I"). Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
401-3372-00L	Dynamical Systems II	W	10 KP	4V + 1U	W. Merry
Kurzbeschreibung	This course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic and complex dynamics.
Lernziel	Mastery of the basic methods and principal themes of some aspects of hyperbolic and complex dynamical systems.
Inhalt	Topics covered include: - 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. - Complex dynamics of rational maps on the Riemann sphere - Julia sets and Fatou sets. - Fractals and the Mandelbrot set.
Skript	I will provide full lecture notes, available here: https://www.merry.io/courses/dynamical-systems/
Literatur	The most useful textbook is - Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.
Voraussetzungen / Besonderes	It will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here: https://www.merry.io/courses/dynamical-systems/ However we will only really use material covered in the first 10 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 10 lectures. In addition, it would be useful to have some familiarity with basic differential geometry and complex analysis.
» Kernfächer aus Bereichen der reinen Mathematik (Mathematik Master)

Kernfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3052-10L	Graph Theory	W	10 KP	4V + 1U	B. Sudakov
Kurzbeschreibung	Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem
Lernziel	The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.
Skript	Lecture will be only at the blackboard.
Literatur	West, D.: "Introduction to Graph Theory" Diestel, R.: "Graph Theory" Further literature links will be provided in the lecture.
Voraussetzungen / Besonderes	Students are expected to have a mathematical background and should be able to write rigorous proofs.
401-3642-00L	Brownian Motion and Stochastic Calculus	W	10 KP	4V + 1U	W. Werner
Kurzbeschreibung	This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
Lernziel	This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
Skript	Lecture notes will be distributed in class.
Literatur	- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016). - I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991). - D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005). - L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000). - D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006).
Voraussetzungen / Besonderes	Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in - J. Jacod, P. Protter, Probability Essentials, Springer (2004). - R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).
401-3632-00L	Computational Statistics	W	8 KP	3V + 1U	M. H. Maathuis
Kurzbeschreibung	We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R.
Lernziel	The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R.
Voraussetzungen / Besonderes	At least one semester of (basic) probability and statistics. Programming experience is helpful but not required.
401-3602-00L	Applied Stochastic Processes Findet dieses Semester nicht statt.	W	8 KP	3V + 1U	keine Angaben
Kurzbeschreibung	Poisson-Prozesse; Erneuerungsprozesse; Markovketten in diskreter und in stetiger Zeit; einige Beispiele und Anwendungen.
Lernziel	Stochastische Prozesse dienen zur Beschreibung der Entwicklung von Systemen, die sich in einer zufälligen Weise entwickeln. In dieser Vorlesung bezieht sich die Entwicklung auf einen skalaren Parameter, der als Zeit interpretiert wird, so dass wir die zeitliche Entwicklung des Systems studieren. Die Vorlesung präsentiert mehrere Klassen von stochastischen Prozessen, untersucht ihre Eigenschaften und ihr Verhalten und zeigt anhand von einigen Beispielen, wie diese Prozesse eingesetzt werden können. Die Hauptbetonung liegt auf der Theorie; "applied" ist also im Sinne von "applicable" zu verstehen.
Literatur	R. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: http://epubs.siam.org/doi/book/10.1137/1.9780898718997 R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: http://link.springer.com/book/10.1007/978-1-4614-3615-7/page/1 M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: http://link.springer.com/book/10.1007/978-0-387-48976-6/page/1 S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005)
Voraussetzungen / Besonderes	Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).
401-3652-00L	Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich) Der Kurs muss direkt an der UZH belegt werden. UZH Modulkürzel: MAT827 Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/mobilitaet.html	W	10 KP	4V + 2U	Uni-Dozierende
Kurzbeschreibung	This course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB.
Lernziel	The goal of this course is familiarity with the fundamental ideas and mathematical consideration underlying modern numerical methods for conservation laws and wave equations.
Inhalt	* Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering. * Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes. * Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes. * Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods. * Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes. * Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory.
Skript	Lecture slides will be made available to participants. However, additional material might be covered in the course.
Literatur	H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online. R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online. E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.
Voraussetzungen / Besonderes	Having attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite. Programming exercises in MATLAB Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations"
» Kernfächer aus Bereichen der angewandten Mathematik ... (Mathematik Master)

Kernfächer aus weiteren anwendungsorientierten Gebieten 402-0204-00L Elektrodynamik ist als angewandtes Kernfach im Bachelor-Studiengang Mathematik anrechenbar, aber nur unter der Bedingung, dass 402-0224-00L Theoretische Physik (letztmals im FS 2016 angeboten) nicht angerechnet wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (www.math.ethz.ch/studiensekretariat).
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
402-0204-00L	Elektrodynamik	W	7 KP	4V + 2U	R. Renner
Kurzbeschreibung	Herleitung und Diskussion der Maxwellgleichungen, vom statischen Fall zur Elektrodynamik. Wellengleichung, Wellenleiter, Kavitäten. Erzeugung elektromagnetischer Strahlung, Streuung und Beugung von Licht. Struktur der Maxwellgleichungen, Lorentz-Invarianz, Relativitätstheorie und Kovarianz, Lagrange Formulierung. Dynamik relativistischer Teilchen im Feld und deren Strahlung.
Lernziel	Physikalisches Verständnis statischer und dynamischer Phänomene (bewegter) geladener Objekte, und der Struktur der klassischen Feldtheorie der Elektrodynamik (transversale versus longitudinale Physik, Invarianzen (Lorentz-, Eich-)). Erkennen des Zusammenhangs von elektrischen, magnetischen und optischen Phänomenen und Einfluss von Medien. Verständnis klassischer Phänomene der Elektrodynamik und Fähigkeit zur selbständigen Lösung einfacher Probleme. Anwendung mathematischer Fertigkeiten (Vektoranalysis, vollständige Funktionensysteme, Green'sche Funktionen, ko- und kontravariante Koordinaten, etc.). Vorbereitung auf die Quantenmechanik (Eigenwertprobleme, Lichtleiter und Kavitäten).
Inhalt	Klassische Feldtheorie der Elektrodynamik: Herleitung und Diskussion der Maxwellgleichungen, ausgehend vom statischen Fall (Elektrostatik, Magnetostatik, Randwertprobleme) im Vakuum und in Medien und Verallgemeinerung zur Elektrodynamik (Faraday Gesetz, Ampere/Maxwell; Potentiale, Eichinvarianz). Wellengleichung und Lösungen im vollen Raum, Halbraum (Snellius Gesetz), Wellenleiter, Kavitäten. Erzeugung elektromagnetischer Strahlung, Streuung und Beugung von Licht (Optik). Erarbeitung von Beispielen. Diskussion zur Struktur der Maxwellgleichungen, Lorentz-Invarianz, Relativitätstheorie und Kovarianz, Lagrange Formulierung. Dynamik relativistischer Teilchen im Feld und deren Strahlung (Synchrotron).
Literatur	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, Elektrodynamik, Optik (Vorlesungen über theoretische Physik) 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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