Suchergebnis: Katalogdaten im Herbstsemester 2019
Mathematik Master | ||||||
Kernfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen. | ||||||
Kernfächer aus Bereichen der reinen Mathematik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
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401-3225-00L | Introduction to Lie Groups | W | 8 KP | 4G | P. D. Nelson | |
Kurzbeschreibung | 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. | |||||
Lernziel | 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. | |||||
Literatur | 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) | |||||
Voraussetzungen / Besonderes | 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: Link | |||||
401-3001-61L | Algebraic Topology I | W | 8 KP | 4G | A. Sisto | |
Kurzbeschreibung | This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms. | |||||
Lernziel | ||||||
Literatur | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link 2) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||
Voraussetzungen / Besonderes | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not strictly necessary. Some (elementary) group theory and algebra will also be needed. | |||||
401-3114-69L | Introduction to Algebraic Number Theory | W | 8 KP | 3V + 1U | Ö. Imamoglu | |
Kurzbeschreibung | This is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory. | |||||
Lernziel | ||||||
Inhalt | algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory. | |||||
401-3132-00L | Commutative Algebra | W | 10 KP | 4V + 1U | E. Kowalski | |
Kurzbeschreibung | This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. | |||||
Lernziel | 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 | |||||
Literatur | 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) | |||||
Voraussetzungen / Besonderes | Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory). | |||||
Kernfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3651-00L | Numerical Analysis 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 KP | 4V + 1U | C. Schwab | |
Kurzbeschreibung | This course gives a comprehensive introduction into the numerical treatment of linear and nonlinear 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. | |||||
Lernziel | 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 | |||||
Inhalt | The course will address the mathematical analysis of numerical solution methods for linear and nonlinear elliptic and parabolic partial differential equations. Functional analytic and algebraic (De Rham complex) tools will be provided. Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed. Particular attention will be placed on developing mathematical foundations (Regularity, Approximation theory) for a-priori convergence rate analysis. A-posteriori error analysis and mathematical proofs of adaptivity and optimality will be covered. Implementations for model problems in MATLAB and python will illustrate the theory. 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 | |||||
Literatur | Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp. A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 Additional Literature: D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp. D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications, Springer, 2012 [DOI: 10.1007/978-3-642-22980-0] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006). | |||||
Voraussetzungen / Besonderes | Practical exercises based on MATLAB Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations | |||||
401-3621-00L | Fundamentals of Mathematical Statistics | W | 10 KP | 4V + 1U | S. van de Geer | |
Kurzbeschreibung | The course covers the basics of inferential statistics. | |||||
Lernziel | ||||||
401-3622-00L | Statistical Modelling | W | 8 KP | 4G | C. Heinze-Deml | |
Kurzbeschreibung | In der Regression wird die Abhängigkeit einer zufälligen Response-Variablen von anderen Variablen untersucht. Wir betrachten die Theorie der linearen Regression mit einer oder mehreren Ko-Variablen, hoch-dimensionale lineare Modelle, nicht-lineare Modelle und verallgemeinerte lineare Modelle, Robuste Methoden, Modellwahl und nicht-parametrische Modelle. | |||||
Lernziel | Einführung in Theorie und Praxis eines umfassenden und vielbenutzten Teilgebiets der Statistik, unter Berücksichtigung neuerer Entwicklungen. | |||||
Inhalt | 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. | |||||
Skript | Vorlesungsskript | |||||
Voraussetzungen / Besonderes | This is the course unit with former course title "Regression". Credits cannot be recognised for both courses 401-3622-00L Statistical Modelling 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). | |||||
401-4889-00L | Mathematical Finance | W | 11 KP | 4V + 2U | J. Teichmann | |
Kurzbeschreibung | Advanced course on mathematical finance: - semimartingales and general stochastic integration - absence of arbitrage and martingale measures - fundamental theorem of asset pricing - option pricing and hedging - hedging duality - optimal investment problems - additional topics | |||||
Lernziel | Advanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes) | |||||
Inhalt | This is an advanced course on 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 mostly in continuous-time models. Topics include - semimartingales and general stochastic integration - absence of arbitrage and martingale measures - fundamental theorem of asset pricing - option pricing and hedging - hedging duality - optimal investment problems - and probably others | |||||
Skript | The course is based on different parts from different books as well as on original research literature. Lecture notes will not be available. | |||||
Literatur | (will be updated later) | |||||
Voraussetzungen / Besonderes | Prerequisites are the standard courses - Probability Theory (for which lecture notes are available) - Brownian Motion and Stochastic Calculus (for which lecture notes are available) Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts. This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II. For an overview of courses offered in the area of mathematical finance, see Link. | |||||
401-3901-00L | Mathematical Optimization | W | 11 KP | 4V + 2U | R. Zenklusen | |
Kurzbeschreibung | Mathematical treatment of diverse optimization techniques. | |||||
Lernziel | The goal of this course is to get a thorough understanding of various classical mathematical optimization techniques with an emphasis on polyhedral approaches. In particular, we want students to develop a good understanding of some important problem classes in the field, of structural mathematical results linked to these problems, and of solution approaches based on this structural understanding. | |||||
Inhalt | Key topics include: - Linear programming and polyhedra; - Flows and cuts; - Combinatorial optimization problems and techniques; - Equivalence between optimization and separation; - Brief introduction to Integer Programming. | |||||
Literatur | - Bernhard Korte, Jens Vygen: Combinatorial Optimization. 6th edition, Springer, 2018. - Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. This work has 3 volumes. - Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993. - Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986. | |||||
Voraussetzungen / Besonderes | Solid background in linear algebra. | |||||
Bachelor-Kernfächer aus Bereichen der reinen Mathematik Nebst weiteren Einschränkungen gilt: Die Anrechnung von 401-3531-00L Differentialgeometrie I / Differential Geometry I im Master-Studiengang ist nur dann zulässig, wenn 401-3532-00L Differentialgeometrie II / Differential Geometry II nicht für den Bachelor-Studiengang angerechnet wurde. Ebenso für: 401-3461-00L Funktionalanalysis I / Functional Analysis I - 401-3462-00L Funktionalanalysis II / Functional Analysis II 401-3001-61L Algebraische Topologie I / Algebraic Topology I - 401-3002-12L Algebraische Topologie II / Algebraic Topology II 401-3132-00L Kommutative Algebra / Commutative Algebra - 401-3146-12L Algebraische Geometrie / Algebraic Geometry Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link). | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3461-00L | Functional Analysis I Höchstens eines der drei Bachelor-Kernfächer 401-3461-00L Funktionalanalysis I / Functional Analysis I 401-3531-00L Differentialgeometrie I / Differential Geometry I 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist im Master-Studiengang Mathematik anrechenbar. | E- | 10 KP | 4V + 1U | M. Struwe | |
Kurzbeschreibung | Baire category; Banach and Hilbert spaces, bounded linear operators; basic 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. | |||||
Lernziel | Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps. | |||||
Literatur | We will be using the Lecture Notes on "Funktionalanalysis I" by Michael Struwe. Other useful, and recommended references include the following books: Haim Brezis: "Functional analysis, Sobolev spaces and partial differential equations". Springer, 2011. Manfred Einsiedler and Thomas Ward: "Functional Analysis, Spectral Theory, and Applications", Graduate Text in Mathematics 276. Springer, 2017. Peter D. Lax: "Functional analysis". Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Elias M. Stein and Rami Shakarchi: "Functional analysis" (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. Dirk Werner, "Funktionalanalysis". Springer-Lehrbuch, 8. Auflage. Springer, 2018 | |||||
Voraussetzungen / Besonderes | 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). | |||||
401-3531-00L | Differential Geometry I Höchstens eines der drei Bachelor-Kernfächer 401-3461-00L Funktionalanalysis I / Functional Analysis I 401-3531-00L Differentialgeometrie I / Differential Geometry I 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist im Master-Studiengang Mathematik anrechenbar. | E- | 10 KP | 4V + 1U | U. Lang | |
Kurzbeschreibung | Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||
Lernziel | ||||||
Skript | Partial lecture notes are available from Link | |||||
Literatur | Differential geometry in R^n: - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten - Christian Bär: Elementare Differentialgeometrie 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 KP | 4V + 1U | W. Merry | |
Kurzbeschreibung | This course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics. | |||||
Lernziel | Mastery of the basic methods and principal themes of some aspects of dynamical systems. | |||||
Inhalt | 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]) | |||||
Literatur | The most relevant textbook for this course is Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002. I will also produce full lecture notes, available from my website Link | |||||
Voraussetzungen / Besonderes | 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. | |||||
Bachelor-Kernfächer aus Bereichen der angewandten Mathematik .. Nebst weiteren Einschränkungen gilt: Die Anrechnung von 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory im Master-Studiengang ist nur dann zulässig, wenn weder 401-3642-00L Brownian Motion and Stochastic Calculus noch 401-3602-00L Applied Stochastic Processes für den Bachelor-Studiengang angerechnet wurde. Ausserdem ist 402-0205-00L Quantenmechanik I als angewandtes Kernfach anrechenbar, aber nur unter der Bedingung, dass 402-0224-00L Theoretische Physik (letztmals im FS 2016 angeboten) nicht angerechnet wird oder wurde (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link). | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3601-00L | Probability Theory Höchstens eines der drei Bachelor-Kernfächer 401-3461-00L Funktionalanalysis I / Functional Analysis I 401-3531-00L Differentialgeometrie I / Differential Geometry I 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist im Master-Studiengang Mathematik anrechenbar. | E- | 10 KP | 4V + 1U | A.‑S. Sznitman | |
Kurzbeschreibung | Basics of probability theory and the theory of stochastic processes in discrete time | |||||
Lernziel | 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. | |||||
Inhalt | 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. | |||||
Skript | available, will be sold in the course | |||||
Literatur | 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 | Quantenmechanik I | W | 10 KP | 3V + 2U | G. Blatter | |
Kurzbeschreibung | Einfuehrung in die Quantentheorie: Wellenmechanik, Schroedingergleichung, Drehimpuls, Zentralkraftprobleme, Potentialstreuung, Spin. Allgemeine Struktur der Quantentheorie: Hilbertraeume, Zustaende und Observable, Bewegungsgleichung, Dichtematrizen, Symmetrien, Heisenberg- und Wechselwirkungs Bild. Naeherungsmethoden: Stoerungstheorie, Variations-Verfahren, quasi-Klassik. | |||||
Lernziel | Einführung in die Einteilchen Quantenmechanik. Beherrschung grundlegender Ideen (Quantisierung, Operatorformalismus, Symmetrien, Drehimpuls, Störungstheorie) und generischer Beispiele und Anwendungen (gebundene Zustände, Tunneleffekt, Wasserstoffatom, harmonischer Oszillator). Fähigkeit zur Lösung einfacher Probleme. | |||||
Inhalt | Feynmansche Pfadintegrale fuehren uns von der klassischen- zur Quantenmechanik, ihre infinitesimale Zeitentwicklung fuehrt auf den Operator Formalismus (Schroedinger Gleichung, Dirac Formalismus). Die Einteilchen-Quantenmechanik wird entwickelt anhand von ein-dimensionalen Problemen (gebundene Zustaende, Streuprobleme, Tunneleffekt, Resonanzen, periodische und ungeordnete Potential). Der Einfuehrung von Drehungen und dem Drehimpuls folgen die Diskussion von Zentralpotentialen, Streuprobleme in drei Dimensionen, Spin, und Drehimpuls/Spin Addition. Verschiedene Bilder (Schroedinger, Heisenberg, Dirac) werden in der Diskussion approximativer Loesungenmethoden (Variationsrechnung, Stoerungstheorie, Quasiklassik/WKB) benutzt. | |||||
Skript | Auf Moodle, in deutscher Sprache | |||||
Literatur | G. Baim, Lectures on Quantum Mechanics E. Merzbacher, Quantum Mechanics L.I. Schiff, Quantum Mechanics R. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals J.J. Sakurai: Modern Quantum Mechanics A. Messiah: Quantum Mechanics I S. Weinberg: Lectures on Quantum Mechanics |
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