Office Rm. Blocker 614A
Telephone (979)862-3257
E-mail: kuchment@math.tamu.edu
Home Page: http://www.math.tamu.edu/~kuchment
Section: 602
Time: TR 2:20-3:35
Room: BLOC 624
Textbook: None. Some notes will be distributed. Several books that could be used for additional reading are recommended below.
Office hours: TR 1:20-2:15 pm Additional office hours can be arranged by appointment.
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.
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].
The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Services for Students with Disabilities (Cain Hall, Room B118, or call 845-1637). For additional information visit http://disability.tamu.edu.
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.
Copying work done by others, either in class or out of class, looking
on other student’s 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.