# Search result: Catalogue data in Autumn Semester 2021

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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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401-3225-00L | Introduction to Lie Groups | W | 8 credits | 4G | A. Iozzi | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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: Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3001-61L | Algebraic Topology I | W | 8 credits | 4G | W. Merry | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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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: Link See also: Link 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

Abstract | This course provides an introduction to commutative algebra. It serves in particular as a foundation for modern algebraic geometry. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The topics presented in the course will include: * Basics facts about rings, ideals and modules * Constructions of rings: quotients, polynomial rings, localization * Noetherian rings and modules * The tensor product of modules over commutative rings and its applications * Krull dimension * Integral extensions and the Cohen-Seidenberg theorems * Finitely generated algebrais over fields, including the Noether Normalization Theorem and the Nullstellensatz * Primary decomposition * Discrete valuation rings and some applications | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Primary Reference: "(Mostly) Commutative Algebra", by A. Chambert-Loir; Springer 2021, available on the author's web page. Secondary References: 1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969) 2. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995) 3. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989) 4. "Commutative Algebra" by N. Bourbaki | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Prerequisites: Algebra I/II (or a similar introduction to the basic concepts of ring theory, including field theory). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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 (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH as an incoming student. UZH Module Code: MAT802 Mind the enrolment deadlines at UZH: Link 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 | 9 credits | 6G | S. Sauter | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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 | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | 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). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | 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 credits | 4V + 1U | S. van de Geer | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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401-3622-00L | Statistical Modelling | W | 8 credits | 4G | C. Heinze-Deml | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | 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 credits | 4V + 2U | D. Possamaï | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Advanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | The course is based on different parts from different books as well as on original research literature. Lecture notes will not be available. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | (will be updated later) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | 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 | Linear & Combinatorial Optimization | W | 11 credits | 4V + 2U | R. Zenklusen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Mathematical treatment of optimization techniques for linear and combinatorial optimization problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The goal of this course is to get a thorough understanding of various classical mathematical optimization techniques for linear and combinatorial optimization problems, 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 such structural insights. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Key topics include: - Linear programming and polyhedra; - Flows and cuts; - Combinatorial optimization problems and polyhedral techniques; - Equivalence between optimization and separation. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | - 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Solid background in linear algebra. Former course title: Mathematical Optimization. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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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 For the category assignment take contact with the Study Administration Office (Link) after having received the credits. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3461-00L | Functional Analysis IAt most one of the three course units (Bachelor Core Courses) 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. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (Link) after having received the credits. | E- | 10 credits | 4V + 1U | J. Teichmann | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Recommended references include the following: Michael Struwe: "Funktionalanalysis I" (Skript available at Link) Haim Brezis: "Functional analysis, Sobolev spaces and partial differential equations". Springer, 2011. 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. Manfred Einsiedler and Thomas Ward: "Functional Analysis, Spectral Theory, and Applications", Graduate Text in Mathematics 276. Springer, 2017. Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with topology and measure theory, in part. Lebesgue integration and L^p spaces). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3531-00L | Differential Geometry I At most one of the three course units (Bachelor Core Courses) 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. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (Link) after having received the credits. | E- | 10 credits | 4V + 1U | J. Serra | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Provide insightful knowledge about the classical theory of curves and surfaces (which is the precursor of modern differential geometry). Invite students to use and sharpen their geometric intuition. Introduce the language, basic tools, and some fundamental results in modern differential geometry. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Partial lecture notes are available from Prof. Lang's website Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - John M. Lee: Introduction to Smooth Manifolds - S. Montiel, A. Ros: Curves and Surfaces - S. Kobayashi: Differential Geometry of Curves and Surfaces - Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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 (Link) after having received the credits. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3601-00L | Probability Theory At most one of the three course units (Bachelor Core Courses) 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. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (Link) after having received the credits. | E- | 10 credits | 4V + 1U | W. Werner | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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, series of independent random variables, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson processes, Markov chains (classification and convergence results). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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 processes, Markov chains (classification and convergence results). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | will be available in electronic form. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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 | M. Gaberdiel | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | General structure of quantum theory: Hilbert spaces, states and observables, equations of motion, Heisenberg uncertainty relation, symmetries, angular momentum addition, EPR paradox, Schrödinger and Heisenberg picture. Applications: simple potentials in wave mechanics, scattering and resonance, harmonic oscillator, hydrogen atom, and perturbation theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Introduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, angular momentum, perturbation theory) and generic examples and applications (bound states, tunneling, hydrogen atom, harmonic oscillator). Ability to solve simple problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | The beginnings of quantum theory with Planck, Einstein and Bohr; Wave mechanics; Simple examples; The formalism of quantum mechanics (states and observables, Hilbert spaces and operators, the measurement process); Heisenberg uncertainty relation; Harmonic oscillator; Symmetries (in particular rotations); Hydrogen atom; Angular momentum addition; Quantum mechanics and classical physics (EPR paradoxon and Bell's inequality); Perturbation theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Auf Moodle, in deutscher Sprache | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | G. Baym, 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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Electives 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. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Electives: Pure Mathematics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3059-00L | Combinatorics II | W | 4 credits | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3033-00L | Gödel's Theorems | W | 8 credits | 3V + 1U | L. Halbeisen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Die Vorlesung besteht aus drei Teilen: Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe. Teil II behandelt den Gödel'schen Vollständigkeitssatz Teil III behandelt die Gödel'schen Unvollständigkeitssätze | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Das Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Syntax und Semantik der Prädikatenlogik Gödel'scher Vollständigkeitssatz Gödel'sche Unvollständigkeitssätze | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | L. Halbeisen and R. Krapf: Gödel's Theorems and Zermelo's Axioms: a firm foundation of mathematics, Birkhäuser-Verlag, Basel (2020) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Geometry | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3533-70L | Topics in Riemannian Geometry | W | 6 credits | 3V | U. Lang | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Selected topics from Riemannian geometry in the large: triangle and volume comparison theorems, Milnor's results on growth of the fundamental group, Gromov-Hausdorff convergence, Cheeger's diffeomorphism finiteness theorem, the Besson-Courtois-Gallot barycenter method and the proofs of the minimal entropy theorem and the Mostow rigidity theorem for rank one locally symmetric spaces. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture notes will be provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4207-71L | Coxeter Groups from a Geometric Viewpoint | W | 4 credits | 2V | M. Cordes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Introduction to Coxeter groups and the spaces on which they act. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Understand the basic properties of Coxeter groups. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Brown, Kenneth S. "Buildings" Davis, Michael "The geometry and topology of Coxeter groups" | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Students must have taken a first course in algebraic topology or be familiar with fundamental groups and covering spaces. They should also be familiar with groups and group actions. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3057-00L | Finite Geometries IIDoes not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | - Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Analysis | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-4421-71L | Harmonic Analysis | W | 4 credits | 2V | A. Figalli | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The goal of this class is to give an introduction to harmonic analysis, covering a series of classical important results such as: 1) interpolation theorems 2) convergence properties of Fourier series 3) Calderón-Zygmund operators 4) Littlewood-Paley decomposition 5) Hardy and BMO spaces | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | I plan to write some notes of the class. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | There is no official textbook. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4475-71L | Microlocal Analysis | W | 6 credits | 3G | P. Hintz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Microlocal analysis is the analysis of partial differential equations in phase space. The first half of the course introduces basic notions such as pseudodifferential operators, wave front sets of distributions, and elliptic parametrices. The second half develops modern tools for the study of nonelliptic equations, with applications to wave equations arising in general relativity. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Students will be able to analyze linear partial differential operators (with smooth coefficients) and their solutions in phase space, i.e. in the cotangent bundle. For various classes of operators including, but not limited to, elliptic and hyperbolic operators, they will be able to prove existence and uniqueness (possibly up to finite-dimensional obstructions) of solutions, and study the precise regularity properties of solutions. The first goal is to construct and apply parametrices (approximate inverses) or approximate solutions of PDEs using suitable calculi of pseudodifferential operators (ps.d.o.s). This requires defining ps.d.o.s and the associated symbol calculus on Euclidean space, proving the coordinate invariance of ps.d.o.s, and defining a ps.d.o. calculus on manifolds (including mapping properties on Sobolev spaces). The second goal is to analyze distributions and operations on them (such as: products, restrictions to submanifolds) using information about their wave front sets or other microlocal regularity information. Students will in particular be able to compute the wave front set of distributions. The third goal is to infer microlocal properties (in the sense of wave front sets) of solutions of general linear PDEs, with a focus on elliptic, hyperbolic and certain degenerate hyperbolic PDE. For hyperbolic operators, this includes proving the Duistermaat-Hörmander theorem on the propagation of singularities. For certain degenerate hyperbolic operators, students will apply positive commutator methods to prove results on the propagation of microlocal regularity at critical or invariant sets for the Hamiltonian vector field of the principal symbol of the partial differential operator under study. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Tempered distributions, Sobolev spaces, Schwartz kernel theorem. Symbols, asymptotic summation. Pseudodifferential operators on Euclidean space: composition, principal symbols and the symbol calculus, elliptic parametrix construction, boundedness on Sobolev spaces. Pseudodifferential operators on manifolds, elliptic operators on compact manifolds and Fredholm theory, basic symplectic geometry. Microlocalization: wave front set, characteristic set; pairings, products, restrictions of distributions. Hyperbolic evolution equations: existence and uniqueness of solutions, Egorov's theorem. Propagation of singularities: the Duistermaat-Hörmander theorem, microlocal estimates at radial sets. Applications to general relativity: asymptotic behavior of waves on de Sitter space. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture notes will be made available on the course website. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Lars Hörmander, "The Analysis of Linear Partial Differential Operators", Volumes I and III. Alain Grigis and Johannes Sjöstrand, "Microlocal Analysis for differential operators: an introduction". | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Students are expected to have a good understanding of functional analysis. Familiarity with distribution theory, the Fourier transform, and analysis on manifolds is useful but not strictly necessary; the relevant notions will be recalled in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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Selection: Further Realms | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3502-71L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 2 credits | 4A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective |

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