Suchergebnis: Katalogdaten im Frühjahrssemester 2018
Mathematik Bachelor | ||||||
Basisjahr | ||||||
» Ergänzende Fächer | ||||||
» GESS Wissenschaft im Kontext | ||||||
» Obligatorische Fächer des Basisjahres | ||||||
Obligatorische Fächer des Basisjahres | ||||||
Basisprüfungsblock 1 Wird im Herbstsemester angeboten. | ||||||
Basisprüfungsblock 2 | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
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401-1262-07L | Analysis II | O | 10 KP | 6V + 3U | M. Einsiedler | |
Kurzbeschreibung | Einführung in die Differential- und Integralrechnung in mehreren reellen Veränderlichen, Vektoranalysis: Differential, partielle Ableitungen, Satz über implizite Funktionen, Umkehrsatz, Extrema mit Nebenbedingungen; Riemannsches Integral, Vektorfelder und Differentialformen, Wegintegrale, Oberflächenintegrale, Integralsätze von Gauss und Stokes. | |||||
Lernziel | ||||||
Inhalt | Mehrdimensionale Differential- und Integralrechnung; Kurven und Flächen im R^n; Extremalaufgaben; Mehrfache Integrale; Vektoranalysis. | |||||
Literatur | H. Amann, J. Escher: Analysis II Link J. Appell: Analysis in Beispielen und Gegenbeispielen Link R. Courant: Vorlesungen über Differential- und Integralrechnung Link O. Forster: Analysis 2 Link H. Heuser: Lehrbuch der Analysis Link K. Königsberger: Analysis 2 Link W. Walter: Analysis 2 Link V. Zorich: Mathematical Analysis II (englisch) Link | |||||
401-1152-02L | Lineare Algebra II | O | 7 KP | 4V + 2U | M. Akveld | |
Kurzbeschreibung | Eigenwerte und Eigenvektoren, Jordan-Normalform, Bilinearformen, Euklidische und Unitäre Vektorräume, ausgewählte Anwendungen. | |||||
Lernziel | Verständnis der wichtigsten Grundlagen der Linearen Algebra. | |||||
401-1652-10L | Numerische Mathematik I | O | 6 KP | 3V + 2U | C. Schwab | |
Kurzbeschreibung | Dieser Kurs gibt eine Einführung in numerische Methoden für Studierende der Mathematik im 2. Semester. Abgedeckt werden Methoden der linearen Algebra (lineare Gleichungssysteme, Matrixeigenwertprobleme) sowie der Analysis (Nullstellensuche von Funktionen sowie numerische Interpolation, Integration und Approximation) in Theorie und Implementierung. | |||||
Lernziel | Kenntnis der grundlegenden numerischen Verfahren sowie `numerische Kompetenz': Anwendung der numerischen Verfahren zur Problemloesung, Mathematische Beweistechniken fuer den Nachweis von Stabilitaet, Konsistenz u. Konvergenz der Verfahren sowie deren MATLAB Implementierung. | |||||
Inhalt | Rundungsfehler, lineare Gleichungssysteme, nichtlineare Gleichungen (Skalar und Systeme), Interpolation, Extrapolation, lineare und nichtlineare Ausgleichsrechnung, elementare Optimierungsverfahren, numerische Integration. | |||||
Skript | Skript zur Vorlesung sowie Leseliste sind auf der Webseite der Vorlesung verfügbar. | |||||
Literatur | Skript wird eingeschriebenen Studierenden des ETH BSc Mathematik zur Verfuegung gestellt. _Zusaetzlich_ wird empfohlen: Quarteroni, Sacco und Saleri, Numerische Mathematik 1 + 2, Springer Verlag 2002. | |||||
Voraussetzungen / Besonderes | Zulassungsbedingungen: Linear Algebra I , Analysis I in ETH BSc MATH u. parallele Belegung von Linear Algebra II, Analysis II in ETH BSc MATH Woechentliche Hausuebungsserien sind integraler Bestandteil des Kurses; die Hausuebungen involvieren MATLAB Programmieraufgaben, u. werden bewertet. | |||||
402-1782-00L | Physik II Flankierend zur Vorlesung "Physik II" wird das folgende Fach aus GESS Wissenschaft im Kontext angeboten: 851-0147-01L Philosophische Betrachtungen zur Physik II | O | 7 KP | 4V + 2U | K. S. Kirch | |
Kurzbeschreibung | Einführung in die Wellenlehre, Elektrizität und Magnetismus. Diese Vorlesung stellt die Weiterführung von Physik I dar, in der die Grundlagen der Mechanik gegeben wurden. | |||||
Lernziel | Grundkenntnisse zur Mechanik sowie Elektrizität und Magnetismus sowie die Fähigkeit, physikalische Problemstellungen zu diesen Themen eigenhändig zu lösen. | |||||
Obligatorische Fächer | ||||||
Prüfungsblock II | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-2284-00L | Mass und Integral | O | 6 KP | 3V + 2U | U. Lang | |
Kurzbeschreibung | Abstrakte Mass- und Integrationstheorie, inklusive: Satz von Caratheodory, Lebesgue-Mass, Konvergenzsätze, L^p-Räume, Satz von Radon-Nikodym, Produktmasse und Satz von Fubini, Masse auf topologischen Räumen | |||||
Lernziel | Grundlagen der abstrakten Mass- und Integrationstheorie | |||||
Inhalt | Abstrakte Mass- und Integrationstheorie, inklusive: Satz von Caratheodory, Lebesgue-Mass, Konvergenzsätze, L^p-Räume, Satz von Radon-Nikodym, Produktmasse und Satz von Fubini, Masse auf topologischen Räumen | |||||
Skript | Link | |||||
Literatur | - D. A. Salamon, Measure and Integration, EMS 2016 - W. Rudin, Real and Complex Analysis, McGraw-Hill 1987 - P. R. Halmos, Measure Theory, Springer 1950 | |||||
401-2004-00L | Algebra II | O | 5 KP | 2V + 2U | M. Burger | |
Kurzbeschreibung | Die Hauptthemen der Vorlesung sind Körpererweiterungen und Galoistheorie. | |||||
Lernziel | Einführung in die Grundlagen der Körpererweiterungen, der Galoistheorie, sowie verwandter Gebiete. | |||||
Inhalt | Das Hauptthema wird die Galoistheorie sein. Ausgansgpunkt ist das Problem der Loesung algebraischen Gleichungen mit Radikalen. Galoistheorie loest dieses Problem in dem es einen Zusammenhang herstellt zwischen Koerpererweiterungen und endlichen Gruppen. Insbesondere werden wir den Satz von Abels-Ruffini, dass es Gleichungen fuenften Grades gibt die nicht mittels Radikalen loesbar sind beweisen, sowie das Theorem von Galois das die Polynome charakterisiert deren Wurzeln mittels Radikalen dargestellt werden koennen. | |||||
Literatur | Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society Galois Theory is the topic treated in Chapter A5. | |||||
401-2554-00L | Topology | O | 6 KP | 3V + 2U | A. Sisto | |
Kurzbeschreibung | Topics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces. | |||||
Lernziel | An introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures. | |||||
Literatur | James Munkres: Topology | |||||
401-2654-00L | Numerical Analysis II | O | 6 KP | 3V + 2U | H. Ammari | |
Kurzbeschreibung | The central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation. | |||||
Lernziel | The course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in MATLAB and test them in numerical experiments. | |||||
Inhalt | Chapter 1. Some basics 1.1. What is a differential equation? 1.2. Some methods of resolution 1.3. Important examples of ODEs Chapter 2. Existence, uniqueness, and regularity in the Lipschitz case 2.1. Banach fixed point theorem 2.2. Gronwall’s lemma 2.3. Cauchy-Lipschitz theorem 2.4. Stability 2.5. Regularity Chapter 3. Linear systems 3.1. Exponential of a matrix 3.2. Linear systems with constant coefficients 3.3. Linear system with non-constant real coefficients 3.4. Second order linear equations 3.5. Linearization and stability for autonomous systems Chapter 4. Numerical solution of ordinary differential equations 4.1. Introduction 4.2. The general explicit one-step method 4.3. Example of linear systems 4.4. Runge-Kutta methods 4.5. Multi-step methods 4.6. Stiff equations and systems 4.7. Perturbation theories for differential equations Chapter 5. Geometrical numerical integration methods for differential equation 5.1. Introduction 5.2. Structure preserving methods for Hamiltonian systems 5.3. Runge-Kutta methods 5.4. Long-time behaviour of numerical solutions Chapter 6. Finite difference methods 6.1. Introduction 6.2. Numerical algorithms for the heat equation 6.3. Numerical algorithms for the wave equation 6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimension Chapter 7. Stochastic differential equations 7.1. Introduction 7.2. Langevin equation 7.3. Ornstein-Uhlenbeck equation 7.4. Existence and uniqueness of solutions in dimension one 7.5. Numerical solution of stochastic differential equations | |||||
Skript | Lecture notes including supplements will be provided electronically. Please find the lecture homepage here: Link All assignments and some previous exam problems will be available for download on lecture homepage. | |||||
Literatur | Note: Extra reading is not considered important for understanding the course subjects. Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994. Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996. Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002. L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009. Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993. Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972. Walter: Ordinary differential equations, Springer-Verlag, New York, 1998. | |||||
Voraussetzungen / Besonderes | Homework problems involve MATLAB implementation of numerical algorithms. | |||||
401-2604-00L | Wahrscheinlichkeit und Statistik | O | 7 KP | 4V + 2U | J. Teichmann | |
Kurzbeschreibung | - Laplace-Modelle, Irrfahrten, bedingte Wahrscheinlichkeiten, Unabhängigkeit. - Axiome von Kolmogorov, Zufallsvariablen, Momente, mehrdimensionale Verteilungen, Gesetze der grossen Zahlen und zentraler Grenzwertsatz. - Punktschätzungen, Tests und Vertrauensinvervalle. | |||||
Lernziel | Ziel der Vorlesung ist die Vermittlung der Grundkonzepte von Wahrscheinlichkeitstheorie und mathematischer Statistik. Neben der mathematisch präzisen Behandlung wird auch Wert auf Intuition und Anschauung gelegt. Die Vorlesung setzt die Masstheorie nicht systematisch ein, verweist aber auf die Zusammenhänge. | |||||
Inhalt | - Diskrete Wahrscheinlichkeitsräume: Laplace-Modelle, Binomial- und Poissonverteilung, bedingte Wahrscheinlichkeiten, Unabhängigkeit, Irrfahrten, erzeugende Funktionen, eventuell Markovketten. - Allgemeine Wahrscheinlichkeitsräume: Axiome von Kolmogorov, Zufallsvariablen und ihre Verteilungen, Erwartungswert und andere Kennzahlen, Entropie, charakteristische Funktionen, mehrdimensionale Verteilung inkl. Normalverteilung, Summen von Zufallsvariablen. - Grenzwertsätze: Schwaches und starkes Gesetz der grossen Zahlen, zentraler Grenzwertsatz. - Statistik: Fragestellungen der Statistik (Schätzen, Vertrauensintervalle, Testen), Verknüpfung Statistik und Wahrscheinlichkeit, Neyman-Pearson Lemma, Wilcoxon-, t- und Chiquadrat-Test, Beurteilung von Schätzern, kleinste Quadrate. | |||||
Kernfächer | ||||||
Kernfächer aus Bereichen der reinen Mathematik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3532-08L | Differential Geometry II | W | 10 KP | 4V + 1U | D. A. Salamon | |
Kurzbeschreibung | Introduction to Differential Topology, including degree theory and intersection theory; Differential forms, including deRham cohomology and Poincare duality; Vector bundles, including Thom isomorphism and Euler number. | |||||
Lernziel | The aim of this course is to give an introduction to Differential Topology including the degree of a mapping and intersection theory, differential forms including deRham cohomology and Poincare duality, and vector bundles including the Thom isomorphism theorem. | |||||
Inhalt | Introduction to Differential Topology, including the mod-2 degree, orientation and the Brouwer degree, Poincare-Hopf Theorem, the Pontryagin construction, Hopf Degree Theorem., intersection theory, Lefschetz numbers; Differential forms, Stokes, Cartan's formula, deRham cohomology, Mayer-Vietoris, Poincare duality, Euler characteristic, Degree Theorem, Gauss-Bonnet, Moser isotopy, Cech-DeRham complex and finite-dimensionality; Vector bundles, Thom isomorphism, Euler number. | |||||
Literatur | - J. Milnor, Topology from the Differential Viewpoint. Univ Virginia Press, 1969. - V. Guillemin, A. Pollack, Differential Topology. Prentice-Hall, 1974. - R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. - J. Robbin, D. Salamon, Introduction to Differential Topology, in preparation. Link | |||||
Voraussetzungen / Besonderes | Prerequisite is a working knowledge of the introductory material in Differential Geometry I, including smooth manifolds, tangent bundles, vector fields and flows. see Link | |||||
401-3462-00L | Functional Analysis II | W | 10 KP | 4V + 1U | A. Carlotto | |
Kurzbeschreibung | Fundamentals of the theory of distributions, Sobolev spaces, weak solutions of elliptic boundary value problems (solvability results both via linear methods and via direct variational methods), elliptic regularity theory, Schauder estimates, selected applications coming from physics and differential geometry. | |||||
Lernziel | Acquiring the language and methods of the theory of distributions in order to study differential operators and their fundamental solutions; mastering the notion of weak solutions of elliptic problems both for scalar and vector-valued maps, proving existence of weak solutions in various contexts and under various classes of assumptions; learning the basic tools and ideas of elliptic regularity theory and gaining the ability to apply these methods in important instances of contemporary mathematics. | |||||
Skript | Lecture notes "Funktionalanalysis II" by Michael Struwe. | |||||
Literatur | Useful references for the course are the following textbooks: Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011. Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003. | |||||
Voraussetzungen / Besonderes | 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). | |||||
401-3114-00L | Algebraic Number Theory | W | 10 KP | 4V + 1U | R. Pink | |
Kurzbeschreibung | Algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, cyclotomic fields, ramification theory, valuations, p-adic numbers, local fields, Galois theory of valuations, (+ other material from Neukirch's book for which time remains) | |||||
Lernziel | ||||||
Literatur | Jürgen Neukirch: Algebraic number theory. Springer-Verlag, 1999. | |||||
Voraussetzungen / Besonderes | Algebra II with Galois theory is a must; some commutative algebra of modules and Dedekind rings is desired. Lecture homepage: Link | |||||
401-3146-12L | Algebraic Geometry | W | 10 KP | 4V + 1U | E. Kowalski | |
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, 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. | |||||
Voraussetzungen / Besonderes | Requirement: Some knowledge of Commutative Algebra. | |||||
401-3002-12L | Algebraic Topology II | W | 8 KP | 4G | W. Merry | |
Kurzbeschreibung | This is a continuation course to Algebraic Topology I. Topics covered include: - Universal coefficients, - The Eilenberg-Zilber Theorem and the Künneth Formula), - The cohomology ring, - Fibre bundles, the Leray-Hirsch Theorem, and the Gysin sequence, - Topological manifolds and Poincaré duality, - Higher homotopy groups and fibrations. | |||||
Lernziel | ||||||
Skript | I will produce full lecture notes, available on my website at Link | |||||
Literatur | "Algebraic Topology" (CUP, 2002) by Hatcher is excellent and covers all the material from both Algebraic Topology I and Algebraic Topology II. You can also download it (legally!) for free from Hatcher's webpage: Link Another classic book is Spanier's "Algebraic Topology" (Springer, 1963). This book is very dense and somewhat old-fashioned, but again covers everything you could possibly want to know on the subject. | |||||
Voraussetzungen / Besonderes | Familiarity with all the material from Algebraic Topology I will be assumed (the fundamental group, singular homology, cell complexes, the Eilenberg-Steenrod axioms, the basics of homological algebra and category theory). Full lecture notes for Algebraic Topology I can be found on my webpage. | |||||
» 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. | |||||
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: Link | W | 10 KP | 4V + 1U | R. Abgrall | |
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" |
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