Introduction to algebraic topology and category theory

These notes developed as I taught the corresponding course in Utrecht. They are intended for bachelor students that have already learnt point-set topology. The idea behind them is threefold:

• The primary goal is to study the fundamental group; this is done somewhat along the lines of Hatcher’s “Algebraic Topology”, but we depart from it in many places and generally include more details. The largest difference is that we make constant use of the fundamental groupoid.
• The secondary goal is to introduce the language of category theory as it becomes relevant, and to emphasise categorical thinking.
• The last goal is to tackle the classification of surfaces, which is done combinatorially.

Utrecht Summer School on Geometry

I have taught at the school a number of times. You can find here the lecture notes for the courses I have taught:

• A gentle introduction to Morse theory
• In how many ways can you prove the Whitney-Graustein theorem? These are still a work-in-progress. They contain enough material for a 1-day minicourse (for bachelor students), but they only cover the classic approach to the theorem. Ideally, I would like to expand them so that they cover other strategies of proof.

GQT School 2022

I taught teaching a one-day minicourse on “transversality and its applications” at the GQT Graduate School. The exercise sheet can be found here.

Wrinkling Summer School

I taught a 10 hour course on the wrinkling philosophy at the “RET Summer School on distributions and h-principles” in 2017. I prepared several sets of somewhat detailed (but handwritten) notes where I go over the content of the classes:

• Holonomic approximation
• Wrinkled submersions
• Wrinkled embeddings
• S-immersions

The notes contain several little mistakes here and there and are often not quite streamlined. Hopefully the pictures and the (sometimes not completely rigorous) arguments can help the reader navigate the original Eliashberg-Mishachev papers.