# Course on Singular Analysis

## 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)

Grades 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, regularised 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; regularised 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

(Changed: 28 Feb 2024)  |
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