Search result: Catalogue data in Spring Semester 2022

High-Energy Physics (Joint Master with IP Paris) Information
Electives
Optional Subjects in Mathematics
NumberTitleTypeECTSHoursLecturers
401-3532-08LDifferential Geometry II Information W10 credits4V + 1UJ. Serra
AbstractThis is a continuation course of Differential Geometry I. Topics covered include:
Introduction to Riemannian geometry: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, and isoperimetric inequalities.
Learning objectiveProviding an introductory invitation to Riemannian geometry.
Literature- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,
Prerequisites / NoticeDifferential Geometry I (or basics of differentiable manifolds)
401-3462-00LFunctional Analysis IIW10 credits4V + 1UM. Burger
AbstractThe course will focus essentially on the theory of abelian Banach algebras and its applications to harmonic analysis on locally compact abelian groups, and spectral theorems. Time permitting we will talk about a fundamental property of highly non abelian groups, namely property (T); one of the spectacular applications thereof is the explicit construction of expander graphs.
Learning objectiveAcquire fluency with abelian Banach algebras in order to apply their theory to harmonic analysis on locally compact groups and to spectral theorems.
ContentBanach algebras and the spectral radius formula,

Guelfand's theory of abelian Banach algebras,

Locally compact groups, Haar measure, properties of the convolution product,

Locally compact abelian groups, the dual group, basic properties of the Fourier transform,

Positive definite functions and Bochner's theorem,

The Fourier inversion formula, Plancherel's theorem,

Pontryagin duality and consequences,

Regular abelian Banach algebras, minimal ideals and Wiener's theorem for general locally compact abelian groups. Applications to Wiener-Ikehara and the prime number theorem,

Guelfand's theory of abelian C*-algebras and applications to the spectral theorem for normal operators,

Property (T).
LiteratureM.Einsiedler, T. Ward: Functional Analysis, Spectral Theory, and Applications, GTM Springer, 2017

I. Gelfand, D. Raikov, G. Shilov: Commutative Normed Rings, Chelsea 1964

E. Kaniuth: A Course in Commutative Banach Algebras, GTM Springer, 2009

W. Rudin: Fourier Analysis on Groups, Dover, 1967

M. Takesaki: Theory of Operator Algebras, Springer, 1979
Prerequisites / NoticePoint set topology,

Basic measure theory,

Basics of functional analysis specifically: Banach-Steinhaus, Banach-Alaoglu, and Hahn-Banach.
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