Search result: Catalogue data in Autumn Semester 2024
High-Energy Physics (Joint Master with IP Paris)  | |||||||||||||||||||||||||||
 Electives | |||||||||||||||||||||||||||
| Optional Subjects in Mathematics | |||||||||||||||||||||||||||
| Number | Title | Type | ECTS | Hours | Lecturers | ||||||||||||||||||||||
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| 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 (www.math.ethz.ch/studiensekretariat) after having received the credits. | W | 9 credits | 4V + 1U | U. Lang | ||||||||||||||||||||||
| 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. | ||||||||||||||||||||||||||
| Learning objective | Learn the basic concepts and results in differential geometry and differential topology. Learn to describe, compute, and solve problems in the language of differential geometry. | ||||||||||||||||||||||||||
| Content | Curves, (hyper-)surfaces in R^n, first and second fundamental forms, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet, minimal surfaces. 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. | ||||||||||||||||||||||||||
| Lecture notes | Partial lecture notes are available from https://people.math.ethz.ch/~lang/ | ||||||||||||||||||||||||||
| Literature | Differential geometry in R^n: - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - S. Montiel, A. Ros: 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 - John M. Lee: Introduction to Smooth Manifolds | ||||||||||||||||||||||||||
| Competencies |
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| 401-3461-00L | Functional Analysis 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 (www.math.ethz.ch/studiensekretariat) after having received the credits. | W | 9 credits | 4V + 1U | M. Burger | ||||||||||||||||||||||
| 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 graph theorem; spectral theory of self-adjoint operators in Hilbert spaces. Basics of Sobolev spaces. | ||||||||||||||||||||||||||
| Learning 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 https://people.math.ethz.ch/~struwe/Skripten/FA-I-2019.pdf) 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 importantly: fluency with point set topology and measure theory, in part. Lebesgue integration and L^p spaces. | ||||||||||||||||||||||||||
| Competencies |
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