Spectral Geometry

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

This is an advanced version of the master course having been given by the lecturer in thesummer semester since 2014 (SPEC: Geometrical aspects of spectral theory). The goal of the lecture is to acquaint the students with spectral methods in the theory of partial differential operators coming from physics and geometry, with a special emphasis put on geometrically induced spectral properties.

Requirements:

Functional analysis welcome but not required.

Syllabus of lectures:

1.Motivations. Spectral problems in classical and modern physics. Geometrical aspects.2.Definition of the Laplacian as a self-adjoint operator in a Hilbert space. Dirichlet, Neumann and Robin boundary conditions. Sobolev spaces and elliptic regularity.3.Glazman's classification of Euclidean domains. Basic spectral properties.4.Quasi-conical domains. Location of the essential spectrum. Criticality versus subcriticality.5.Quasi-bounded domains. Compactness of the Sobolev embedding and counter-examples.6.Bounded domains. The symmetric rearrangement and the Faber-Krahn inequality. Properties of nodal sets. Vibrational systems.7.Quasi-conical domains. Geometrically induced discrete eigenvalues and Hardy-type inequalities in tubes. Quantum waveguides.

Syllabus of tutorials:

Study Objective:

Study materials:

Key references: [1] B. Davies: Spectral theory and differential operators, Cambridge University Press, 1995.[2] H, Urakawa, Spectral Geometry of the Laplacian: Spectral Analysis and Differential Geometry of the Laplacian, World Scientific, 2017.Recommended references:[3] D. E. Edmunds and W. D. Evans: Spectral theory and differential operators, Oxford University Press, 1987.[4] Grigor'yan: Heat kernel and analysis on manifolds, AMS, 2009.[5] A. Henrot: Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser, Basel,2006.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: