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Course Information and Syllabus
110.635: Microlocal Analysis
Please consult this course homepage for updates on the course schedule, texts, exercises, handouts...
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Course title
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Microlocal Analysis |
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Course number
Meeting times/Location |
110.635
TuTh 12-1: 15 Krieger 302 |
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Professor:
Office Hours
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Steve
Zelditch
M
2 - 3PM in Krieger 406
and by appointment.
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Course description
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Microlocal Analysis is the phase space analysis of solutions of PDE's. Microlocal = local in phase space, i.e. the space of (positions, momenta). MA is the rigorous theory of quantization and the semi-classical limit. Roughly, it is the systematic study of WKB approximations. We will emphasize its applications to the wave equation and eigenfunctions on a Riemannian manifold and to the dbar equation on a complex manifold.
Mathematically, phase space is a symplectic manifold. Its quantization is a Hilbert space. Functions on phase space are quantized as operators on the Hilbert space and symplectic maps are quantized as unitary operators. The purpose of the course is to define this quantization on cotangent bundles, where it becomes the theory of pseudo-differential and Fourier integral operators, and on Kahler manifolds, where it becomes the theory of Toeplitz operators.
Although we will cover basic definitions and constructions, we hope to emphasize microlocal analysis in the complex domain. In the cotangent setting, we hope to discuss Grauert tubes and analytic continuation of the wave kernel and eigenfunctions. In the Kahler setting, we hope to cover Bergman kernels and Toeplitz FIO's. In fact, a real analytic Riemannian metric on a manifold M induces a Kahler structure on T^*M, so the two settings overlap in a fruitful way.
The first series of main results and topics are: Stationary Phase, WKB, Lagrangian manifolds, Egorov's theorem on conjugations of pseudo-differential operators by Fourier integral operators.
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Pre-requisites: Some familiarity with manifolds, vector fields, flows of vector fields, differential forms and Riemannian metrics is needed.
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| EXERCISES |
I will post and grade exercises |
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Textbooks
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I will use many sources as needed. These are texts that cover the basic results. I will also copy articles. |
1.
Grigis, Alain;
Sjöstrand, Johannes Microlocal analysis for
differential operators. An introduction.
London Mathematical Society Lecture Note Series, 196. Cambridge
University Press, Cambridge, 1994. 2. Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit. London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. 3. Martinez, André An introduction to semiclassical and microlocal analysis. Universitext. Springer-Verlag, New York, 2002 4. Duistermaat, J. J. Fourier integral operators. Progress in Mathematics, 130. Birkhäuser Boston, Inc., Boston, MA, 1996 5. Hörmander, Lars The analysis of linear partial differential operators. I - IV. Pseudo-differential operators. Reprint of the 1994 edition. Classics in Mathematics. Springer, Berlin, 2007 6.
Bates, Sean;
Weinstein, Alan |
| Supplementary articles | |