Here are handwritten notes and problem sheets for two courses which I taught at HU Berlin in winter term 18/19 and summer term 20 on parametric integrals and singularity theory, respectively. Comments are very welcome!


Parametric Integrals

Lecture notes:

Motivating example
Complex manifolds, homology and cohomology
Motivating example, redux
Leray's theory of residues
Stratified bundles, Landau varieties, Picard-Lefschetz theorem


Exercise sheets:

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5



Singularity Theory

Lecture notes:

Introduction
Basic notions
Classification by corank
Jet spaces
Whitney topologies
Thom's transversality theorem
Thom's transversality theorem, footnotes
Whitney's immersion and embedding theorems
Classification by corank II
The intrinsic derivative
Stability
Stability II
Maps between two-manifolds


Exercise sheets:

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6



References/Further reading:

Arnold, Gusein-Zade, Varchenko, Singularities of Differentiable Maps, Vol. I and II, Modern Birkhäuser Classics, Birkhäuser Basel, 1985 and 1988
Golubitsky, Guillemin Stable Mappings and Their Singularities, Graduate Texts in Mathematics 14, Springer, 1973
Hirsch Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976
Hwa, Teplitz, Homology and Feynman Integrals, W. A. Benjamin, 1966
Pham, Singularities of integrals - Homology, hyperfunctions and microlocal analysis, Universitext, Springer London, 2011
Savin, Sternin, Introduction to Complex Theory of Differential Equations, Frontiers in Mathematics, Birkhäuser Basel 2017