Department Mathematik



Mathematisches Seminar: Pseudodifferential operators (PsiDO) (SoSe 2023)

Time and place: Tuesday 10:15 – 12:00   in B 252.   LSF

First meeting: 10:15 Uhr, April 18th (2023) (Intro, motivation, topics).

If interested, please sign up via email ( ) until April 15th 2023.

Synopsis (Kurzbeschreibung):
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 Partial Differential Equations, Harmonic Analysis, Mathematical Physics, 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.

Audience (Hörerkreis):
Master students of Mathematics (PO 2011: WP 16, 17, 41.1, 42.1, 43.1, 43.2, 43.3, 44.1, 44.2, 45.1, 46.2, 47.3, 48.3
PO 2021: P 1 (2 talks), WP 12, 15, 38, 39), TMP-Master, and motivated bachelor students.


Prerequisites (Vorkenntnisse):
Analysis I-III. Basic knowledge of Functional Analysis and/or Partial Differential Equations is helpful, but not required.

Language (Sprache):

[R] X. Saint Raymond, Elementary introduction to the theory of pseudodifferential operators, CRC Press, Boca Raton, 1991.
(Also available in several copies in the library).

Supplementary literature (not needed!):
H. Abels, Pseudodifferential and Singular Integral Operators, De Gruyter Textbook, 2012.
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.)

Programme (Talks start at 10:15 sharp):

Datum Speaker           Title             Remarks        
18.04.2023     Thomas Sørensen Intro, motivation, topics.
25.04.2023 Thomas Sørensen More motivation on ΨDO's.
02.05.2023 Fourier-transf. & distrib. in R^n I. [R] p.5-10.
09.05.2023 Fourier-transf. & distrib. in R^n II. [R] p.10-15.
16.05.2023 Sobolev spaces. [R] p.17-23.
23.05.2023 Def. and approx. of symbols. [R] p.29-32+ex.2.1,2.2,2.8(a)+(b).
30.05.2023 No seminar Dienstag nach Pfingsten.
06.06.2023 Oscillatory integrals. [R] p.32-37+ex.2.3,2.4.
13.06.2023 Operations on symbols. [R] p.37-41+ex.2.5,2.8(c)+(d).
20.06.2023 No seminar Conference
27.06.2023 Ellipticity. [R] Thm2.10+ex.2.6+2.9.
04.07.2023 ΨDO's: Action on S and S'. [R] p.47-52.
11.07.2023 Action in Sobolev spaces. [R] p.52-56+ex.3.7(b)(2nd half), maybe ex.3.3,3.4,3.5.
18.07.2023 Gårding's Inequality. [R] Thm3.9+ex.3.7(a)+(b)(1st half).

Wie halte ich einen Seminarvortrag? by Prof. Dr. Manfred Lehn, Johannes Gutenberg-Universität Mainz.
(Google translation.)


Letzte Änderung: 03 August 2023 (No more updates).

Thomas Østergaard Sørensen