Global Analysis on Manifolds with Singularities

Selected Topics in Analysis (V5B4)

Discussion of manifolds with singular or degenerate geometries forms a central aspect of modern Global Analysis. The simplest example of such a singularity is a cone, with a rich and beautiful theory and still many interesting unanswered questions. We introduce the basic frameworks for dealing with phenomena that arise in those singular geometric situations. We discuss elements of Melrose's approach, where the singularity is literally "blown up" and apply these ideas to the analysis of the heat kernel and heat trace asymptotics.

We plan to cover a subset of the following topics

Asymptotic Expansions, Pushforward Theorem
Singular Asymptotics Lemma
Heat operator on a Cone, Scaling Property of the Cone
Heat Space Blowup
Heat Trace Asymptotics

We will orient ourselves partially at the "Basics of b-Calculus" by Daniel Grieser, and also use "Operators of Fuchs Type" by Matthias Lesch. Further classical reference are the lecture notes of Richard Melrose on "Atiyah-Singer Index Theorem".

Prerequisites

Familiarity with basic elements of global analysis and partial differential equations is of advantage, but not an excusive precondition.

Literatur

D. Grieser "Basics of b-calculus", arXiv:math.AP/0010314
D. Grieser "Notes on Heat Kernel asymptotics", www.staff.uni-oldenburg.de/daniel.grieser//wwwlehre/heat.pdf
M. Lesch "Differential Operators of Fuchs type, conical singularities, and asymptotic methods", Teubner Texte zur Mathematik Vol. 136 (1997), also available from arXiv:dg-ga/9607005
R. Melrose "The Atiyah-Patodi-Singer Index Theorem", A.K. Peters (1993), also available from www-math.mit.edu/~rmb/book.html