Spectral Geometry
Code  Completion  Credits  Range 

D01SG  ZK  2P 
 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 selfadjoint 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.Quasiconical domains. Location of the essential spectrum. Criticality versus subcriticality.5.Quasibounded domains. Compactness of the Sobolev embedding and counterexamples.6.Bounded domains. The symmetric rearrangement and the FaberKrahn inequality. Properties of nodal sets. Vibrational systems.7.Quasiconical domains. Geometrically induced discrete eigenvalues and Hardytype 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 timetable has been prepared for this course
 The course is a part of the following study plans: