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

Motivating example

Complex manifolds, homology and cohomology

Motivating example, redux

Leray's theory of residues

Stratified bundles, Landau varieties, Picard-Lefschetz theorem

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Singularity Theory

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 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Arnold, Gusein-Zade, Varchenko,

Golubitsky, Guillemin

Hirsch

Hwa, Teplitz,

Pham,

Savin, Sternin,