Mathematisches Seminar: Pseudodifferential operators (WS 2011/12)
Time, room: Thursday 8-10 in B 252.
Vorträge können auch auf Deutsch gehalten werden!
Synopsis: The theory of pseudodifferential operators arose in the 1960's as a tool in the study of elliptic partial differential equations (the Laplace equation, Poisson equation, Dirichlet and Neumann boundary value problems etc.). Such operators are a generalisation of Partial Differential Operators (PDO's), and they have since then become a strong and useful tool in many other areas of analysis, such as Harmonic Analysis, Spectral Theory, and Index Theory for elliptic operators on manifolds (they are an important ingredient in many proofs of the Atiyah-Singer Index Theorem).
This seminar will give an elementary introduction to the theory of pseudodifferential operators and their properties. It will include an introduction to the Fourier transform, (tempered) distributions, and Sobolev spaces, which are by themselves very useful tools.
Topics to be discussed: Schwartz functions (S) and tempered distributions (S'), The Fourier transform on S and S', Sobolev spaces, Pseudodifferential symbols, Oscillatory integrals, Pseudodifferential operators (ΨDO's), The action of ΨDO's on S, S', and Sobolev spaces, Global regularity of elliptic PDO's (and ΨDO's), Gårding's inequality, Applications.
Audience: 3rd year Bachelor students and Master students of Mathematics and Physics, TMP-Master.
Prerequisites: Analysis I-III. Basic knowledge of Functional Analysis and/or Partial Differential Equations is helpful, but not required.
Language: The speakers can choose between English and German.
Literature: [R] X. Saint Raymond, Elementary introduction to the theory of pseudodifferential operators, CRC Press, Boca Raton, 1991. (Available in several copies in the library.)
S. G. Krantz, Partial Differential Equations and Complex Analysis, CRC Press, Boca Raton, 1992.
M. M. Wong, An Introduction to pseudo-differential Operators, 2nd ed., World Scientific, Singapore, 1999.
B. E. Petersen, Introduction to the Fourier transform and pseudo-differential operators, Pitman, Boston, 1983.
L. Hörmander, The analysis of linear partial differential operators III, pseudo-differential operators, corr. reprint, Springer, Berlin, 1994.
M. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer, Berlin, 2001.
A. Grigis and J. Sjoestrand, Microlocal Analysis for Differential Operators, Cambridge University Press, 1994.
Office hours: Thursday 10:15-11:00 (Room B 408) or by appointment via email.
Programme (Talks start at 08:30):
Wie halte ich einen Seminarvortrag? von Prof. Dr. Manfred Lehn, Johannes Gutenberg-Universitätt Mainz.