Mathematisches Seminar: Pseudodifferential operators (WiSe 2018/19)

Time and place: Wednesday 08:30 – 10:00   in B 251.


Ankündigung/announcement.

First meeting: 08:30, October 17th (2018) (Intro, motivation, topics).

Vorträge können auch auf Deutsch gehalten werden!

If interested, please sign up via email ( sorensen-a-t-math.lmu.de ) until October 15th 2018.




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), GÃ¥rding'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 (not needed!):
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 & pseudo-differential operators, Pitman, Boston, 1983.
L. Hörmander, The analysis of linear partial differential operators III, Pseudo-Differential Operators, corr. reprint, Springer, Berlin, 2007.
M. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer, Berlin, 2001.
A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators, Cambridge University Press, 1994.

A longer list can be found here.

(For more on Distribution Theory, see
[F-J] F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions (2nd Edition), Cambridge University Press, 1999. (Available in several copies in the library) - Errata 1 Errata 2.)

Office hours: Wed 10:15-11:00 (Room B 408) or by appointment via email.

Programme (Talks start at 08:30):

Datum	Speaker	Title	Remarks
17.10.2018	Thomas Sørensen		Intro, motivation, topics.
24.10.2018	Thomas Sørensen	More motivation on ΨDO's.
07.11.2018	Christian Koke	Fourier-transf. & distrib. in R^n I.	[R] p.5-10.
09.11.2018 (B 046) (10.15-12.00)	Christian Koke Thomas Sørensen	Fourier-transf. & distrib. in R^n II.	[R] p.10-15.     Notes
14.11.2018	Jakob Stern	Sobolev spaces.	[R] p.17-23.
21.11.2018	Borislav Polovnikov	Payley-Wiener-Schwartz Theorem.	[R] p.15-17(Thm1.13)+ex.1.7,1.8.
28.11.2018	Kevin Kögler	Def. and approx. of symbols.	[R] p.29-32+ex.2.1,2.2,2.8(a)+(b).
05.12.2018	Jago Silberbauer	Oscillatory integrals.	[R] p.32-37+ex.2.3,2.4.
12.12.2018	Andreas Tsevas	Operations on symbols.	[R] p.37-41+ex.2.5,2.8(c)+(d).
19.12.2018	Jago Silberbauer	Ellipticity.	[R] Thm2.10+ex.2.6+2.9.
09.01.2019	Gao Zhen	ΨDO's: Action on S and S'.	[R] p.47-52.
16.01.2019	Kevin Kögler	Action in Sobolev spaces.	[R] p.52-56+ex.3.7(b)(2nd half), maybe ex.3.3,3.4,3.5.
23.01.2019	Jonas Stelzig	GÃ¥rding's Inequality.	[R] Thm3.9+ex.3.7(a)+(b)(1st half).
30.01.2019		No seminar.	Conference.
06.02.2019	Thomas Sørensen	Overview & Outlook.

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


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Letzte Änderung: 06 Februar 2019 (wird nicht mehr geändert).

Thomas Østergaard Sørensen