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Syllabus of Math 689 (Special Topics in Applied Mathematics), Section 602

Introduction to Microlocal Analysis

Fall 2009

Instructor Peter Kuchment

Office Rm. Blocker 614A

Telephone (979)862-3257

E-mail: kuchment@math.tamu.edu

Home Page: http://www.math.tamu.edu/~kuchment


Introduction

Microlocal analysis is an important technique in the contemporary partial diferential equations, harmonic analysis, and complex analysis, knowledge of the basics of which should be in the toolbox of any analyst working in these areas and their applications. Among the applied topics where microlocal analysis plays a major role one can mention for instance medical imaging, geophysics, seismology, industrial non-distructive testing, and mathematical physics. To a large extent, development of microlocal analysis was related to classical and quantum mechanics and optics.

The course will start with a brief review of distribution theory and Fourier transform and will include wavefront sets of functions and distributions, oscillatory integrals, Pseudo-Differential Operators (YDOs), and a touch of Fourier Integral Operators (FIOs).

Wavefront sets of functions give a rather precise answer to the questions: what does it mean for a function to be non-smooth at a point? Are there different ways for a function to be non-smooth at this point? This is the language in which one can understand much better properties of functions, especially of solutions of partial differential equations (in particular, propagation of singularities of solutions).

Pseudo-differential operators that originate from work by Kohn and Nirenberg present a wide range extension of differential operators. Using this class of operators, one can answer many standard PDE questions in a much simpler manner than one could before this tool was developed. This is currently a leading technique in linear PDEs.

Prerequisites

Some familiarity with basics of Fourier transform and partial differential equations is expected.

Grading

No exams will be conducted. Grading will be based upon attendance and class participation.

Tentative schedule of the course

Recommended additional reading

There are quite a few very good books devoted to the topics of the class, so the list below is not intended to be comprehensive.

Very friendly introductions to distribution theory and related issues of Fourier transform are provided in [1,9]. The classical book [3] is still very useful.

Nice discussions of wave front sets and their applications one can find in [1,9] and on-line notes [2].

A very intuitive introduction to pseudo-differential and Fourier integral operators can be found in the notes [7], see also [9] for a brief view of pseudo-differential operators. Books [8,10] contain good expositions of the pseudo-differential operator theory, see also the first chapter of [4].

For students interested in Fourier integral operators (which will be just touched upon in the class), the classical paper [12] is still a great source. The book [5] is also recommended, but maybe not for the first reading.

Much more advanced sources are for instance the classical treatise [6], which covers in detail all the topics of the class, as well as [11].

Applications in geophysics are surveyed in [14]. Medical imaging applications can be found, for instance, in [15-17]. Microlocal techniques in inverse problems in general are discussed in [18].

  1. F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press 1999. ISBN: 0521649714
  2. L. Friedlander, notes to the class Introduction to Micro-local Analysis.
  3. I. Gelfand and G. Shilov, Generalized Functions, Academic Press 1968. ISBN: 0122795024
  4. Peter B Gilkey, Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem, CRC Press; 2nd edition 1994. ISBN: 0849378745
    available on-line here
  5. Alain Grigis and Johannes Sjöstrand, Microlocal Analysis for Differential Operators : An Introduction (London Mathematical Society Lecture Note Series), Cambridge University Press 1994. ISBN: 0521449863
  6. Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer-Verlag; 2nd Rep edition 2003. ISBN: 3540006621. The Analysis of Linear Partial Differential Operators II, Springer-Verlag 1983. ISBN: 0387121390. The Analysis of Linear Partial Differential Operators III, Springer-Verlag 1985. ISBN: 0387138285. The Analysis of Linear Partial Differential Operators IV, Springer-Verlag; Corrected edition 1994. ISBN: 0387138293
  7. Louis Nirenberg, Lectures on Linear Partial Differential Equations (CBMS Regional Conference Series in Mathematics No. 17), Amer. Math. Soc.; Reprint edition 1983. ISBN: 0821816675
  8. M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag 2001. ISBN: 354041195X
  9. Robert S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World. Sci. 2003. ISBN: 9812384308
  10. Michael E. Taylor, Pseudodifferential Operators, Princeton Univ. Press 1981. ISBN: 0691082820
  11. Francois Treves, Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. ISBN: 0306404044
  12. L. Hörmander, Fourier integral operators, I, Acta Math. 127 (1971), 79--183
  13. H. Bremermann, Distributions, complex variables, and Fourier transforms, Addison-Wesley Pub. Co. 1965
  14. M. de Hoop, Microlocal Analysis of Seismic Inverse Scattering, in "Inside Out: Inverse Problems and Applications," 219--296, Math. Sci. Res. Inst. Publ., v. 47, Cambridge Univ. Press, Cambridge, 2003.
  15. E. T. Quinto,Singularities of the X-ray transform and limited data tomography in R^2 and R^3, SIAM J. Math. Anal. 24 (1993), 1215--1225.
  16. E. T. Quinto, An introduction to X-ray tomography and Radon transforms, in The Radon transform, inverse problems, and tomography, pp. 1--23, Proc. Sympos. Appl. Math., v. 63, Amer. Math. Soc., Providence, RI, 2006.
  17. P. Kuchment, K. Lancaster, and L. Mogilevskaya, On the local tomography, Inverse Problems, 11 (1995), 571--589.
  18. G. Uhlmann, Microlocal analysis and inverse problems, slides (2 parts) http://www.msri.org/publications/ln/msri/2001/jiw2001/uhlmann/1/index.html https://secure.msri.org/communications/ln/msri/2001/jiw2001/uhlmann/2/banner/01.html

Students with Disabilities:

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Copyright policy:

All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Scholastic dishonesty:

Copying work done by others, either in class or out of class, looking on other students papers during exams or quizzes, having possession of unapproved information in your calculator/computer/phone, etc., and/or having someone else do your work for you are all acts of scholastic dishonesty. These acts, and other acts that can be classified as scholastic dishonesty, will be prosecuted to the full extent allowed by University policy. In this class, collaboration on graded assignments, either in class or out of class, is forbidden unless permission to do so is granted by the instructor. For more information on university policy regarding scholastic dishonesty, see University Student Rules at http://studentrules.tamu.edu/.
"An Aggie does not lie, cheat, steal, or tolerate those who do." Visit http://www.tamu.edu/aggiehonor and follow the rules of the Aggie Honor Code.

GOOD LUCK IN YOUR STUDIES!!!

This syllabus is subject to change at the instructor's discretion

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Last revised August 31st, 2009