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.)

Supplementary literature:
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):

Datum Speaker           Title             Remarks        
20.10.2011     Thomas Sørensen     Intro, motivation, topics.
27.10.2011 Thomas Sørensen More motivation on ΨDO's.
03.11.2011 NO TALK
10.11.2011 NO TALK
17.11.2011 Anders Aaen Fourier-transf. & distrib. in R^n. [R] p.5-17(!!)
24.11.2011 Anders Aaen Sobolev spaces. [R] p.17-23.
01.12.2011 Christian Schmidbauer Def. and approx. of symbols. [R] p.29-32+ex.2.1,2.2,2.8(a)+(b).
08.12.2011 Sebastian Gottwald Oscillatory integrals. [R] p.32-37+ex.2.3,2.4.
15.12.2011 Jetlir Duraj Operations on symbols. [R] p.37-41+ex.2.5,2.8(c)+(d).
22.12.2011 Sebastian Gottwald Payley-Wiener-Schwartz Theorem. [R] p.17(Thm1.13)+ex.1.7,1.8.
12.01.2012 Matthias Bollwein ΨDO's: Action on S and S'. [R] p.47-52.
19.01.2012 Matthias Bollwein Action in Sobolev spaces. [R] p.52-56+ex.3.7(b)(2nd half), maybe ex.3.3,3.4,3.5.
26.01.2012 Jetlir Duraj Ellipticity. [R] Thm2.10+ex.2.6+??.
02.02.2012 Christian Schmidbauer Gårding's Inequality. [R] Thm3.9+ex.3.7(a)+(b)(1st half).

Wie halte ich einen Seminarvortrag? von Prof. Dr. Manfred Lehn, Johannes Gutenberg-Universitätt Mainz.

Curriculum Vitae