Course on Singular Analysis

(online, Winter Semester 2020/21, Prof. Dr. Daniel Grieser)

Lecture Notes

Literature

These two books by R. Melrose contain many details of what we are doing:
Differential Analysis on Manifolds with Corners (very systematic)
The Atiyah-Patodi-Singer Index Theorem (has many details on b-calculus)

Notes of mine of a more introductory nature:
Basics of the b-calculus
Notes on polyhomogeneous functions (somewhat preliminary, will be included in lecture notes)
Lecture Notes on Pseudodifferential Operators (in German)

Exercises

First sheet
Second sheet: Blow-ups and resolutions
Third sheet: Mellin transform, regularized integrals and push-forward theorem

Syllabus

Click on the date to see blackboards

10-21  What is singular analysis?
10-22  Manifolds with corners, smooth maps
10-28  p-submanifolds, smooth maps
10-29  b-vector fields; Laplace equation on a cone
11-05  Blow-up of a point
11-11  Blow-up of p-submanifolds; lifts and resolutions
11-12  Lifting vector fields; commuting blow-ups; b-tangent bundle
11-18  b-normal spaces; b-differential; polyhomogeneous functions on R_+
11-19  Polyhomogeneous functions
11-26  Mellin transform; regularized integral
12-02  Densities; push-forward; Ehresmann theorem
12-03  Singular Asymptotics Lemma; b-fibrations
12-09  Push-forward theorem; pull-back theorem; application; Schwartz kernels and half-densities
12-10  b-calculus (smoothing part): mapping and composition theorems
12-17  Conormal distributions
01-06   Conormal distributions: Push-forward, Pull-back
01-07   Classical pseudodifferential calculus
01-13    Applications of PsiDO calculus; conical singularities
01-14    Small b-calculus: Schwartz kernels of b-differential operators
01-20    Small b-calculus: definition, symbol, composition, parametric, conormal elliptic regularity
01-21    L^2 boundedness, b-Sobolev spaces, shortcomings of small calculus
01-27    Full b-calculus: indicial operator and indicial family
01-28    Full b-calculus: mapping and composition; inverse of indicial family
02-03    Full b-calculus: inverse of indicial operator; parametrix construction; consequences
02-04    Examples, summary, outlook on other singular settings

 

 

(Stand: 09.06.2021)