Search result: Catalogue data in Autumn Semester 2017

High-Energy Physics (Joint Master with EP Paris) Information
Electives
Optional Subjects in Mathematics
NumberTitleTypeECTSHoursLecturers
401-3531-00LDifferential Geometry I Information
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.
W10 credits4V + 1UD. A. Salamon
AbstractSubmanifolds of R^n, tangent bundle,
embeddings and immersions, vector fields, Lie bracket, Frobenius' Theorem.
Geodesics, exponential map, completeness, Hopf-Rinow.
Levi-Civita connection, parallel transport,
motions without twisting, sliding, and wobbling.
Isometries, Riemann curvature, Theorema Egregium.
Cartan-Ambrose-Hicks, symmetric spaces, constant curvature,
Hadamard's theorem.
ObjectiveIntroduction to Differential Geometry.
Submanifolds of Euclidean space, tangent bundle,
embeddings and immersions, vector fields and flows,
Lie bracket, foliations, the Theorem of Frobenius.
Geodesics, exponential map, injectivity radius, completeness
Hopf-Rinow Theorem, existence of minimal geodesics.
Levi-Civita connection, parallel transport, Frame bundle,
motions without twisting, sliding, and wobbling.
Isometries, the Riemann curvature tensor, Theorema Egregium.
Cartan-Ambrose-Hicks, symmetric spaces, constant curvature,
nonpositive sectional curvature, Hadamard's theorem.
LiteratureJoel Robbin and Dietmar Salamon "Introduction to Differential Geometry",
Link
401-3461-00LFunctional Analysis I Information
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.
W10 credits4V + 1UA. Carlotto
AbstractBaire 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; Fourier transform and applications.
ObjectiveAcquire 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.
Lecture notesLecture Notes on "Funktionalanalysis I" by Michael Struwe
LiteratureA primary reference for the course is the textbook by H. Brezis:

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

Other useful, and recommended references are the following:

Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.

Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.

Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
Prerequisites / NoticeSolid 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).
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