Spectral Theory: Applications to Laplacian in Euclidean Space (MAGIC077)
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The module is intended to give students a brief introduction to the spectral theory of operators in Banach and Hilbert spaces and then concentrate on practical applications to spectral geometry of the Dirichlet and Neumann Laplacians in the Euclidean spaces. By the end of the module students are expected to be able to identify and demonstrate understanding of main definitions in spectral theory; formulate and solve problems involving bounded and unbounded operators; apply those to boundary value problems for some simple second order ordinary and partial differential operators. Students will understand the links between abstract analysis, applications in physics, and numerical techniques
Spring 2013 (Monday, January 21 to Friday, March 29)
No prerequisites information is available yet.
The topics will include (some of) : Review of the theory of operators and spectra in Banach spaces and Hilbert spaces; bounded, compact and Fredholm operators; domain and numerical range; symmetric and self-adjoint operators; semi-linear forms, quadratic forms, and weak spectral problems; variational principles; brief excurse into Sobolev spaces; Dirichlet, Neumann, and mixed Laplacians; eigenvalues; separation of variables; isoperimetric eigenvalues for the Laplacians; Weyl's spectral asymptotics; Friedlander's Theorem; introduction to projection and finite-element methods in applications to spectral problems.
No bibliography has been specified for this course.
No assessment information is available yet.
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