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 ( sorensen-a-t-math.lmu.de ) 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.

Credits:
3 ECTS.

Prerequisites (Vorkenntnisse):

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

Language (Sprache):
English.

Literature:
[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.)

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Letzte Änderung: 03 August 2023 (

No more updates

).

Thomas Østergaard Sørensen