Gilles Castel
Mathematics PhD student at KU Leuven. Interested in mathematics, software development, Vim, Unix, design.


This page lists some of the talks I gave recently in the context of my master's degree. I made them with the yet to be released presentation framework Immersion, which I developed for making mathematical presentations.

master's thesis • May, 2020 • English

Master's Thesis Seminar: Morse Theory – A visual guide from handlebodies to the generalized Poincaré conjecture (slides)

My Master's thesis was on Morse Theory and more specifically discusses a proof of the generalized Poincaré conjecture. I gave a one hour seminar discussing the contents of my thesis: Morse functions, handlebody decompositions, cobordisms, Morse homology, Morse inequalities, cancellation theorems resulting in the h-cobordism theorem, which finally implies the generalized Poincaré conjecture. The video below is me going through the slides.
differential topology • June, 2020 • English

Connected sums and the Mazur swindle (slides only)

This talk was part of my exam for Differential Topology. It discusses the subtleties of defining the connected sum for topological and differential manifolds, the monoid structure in the case of surfaces, and finally the irreducibility of the sphere. For this last topic, we follow a proof in the PhD thesis of Mazur, nowadays referred to as the Mazur swindle.
contact geometry • November 3rd, 2020 • English

What does a vector field know about volume?

In this talk, I discuss the origins and developments of contact geometry, only assuming basic knowledge of differential geometry. After important concepts and theorems are introduced, I discuss the similarly titled research paper written by Hansjörg Geiges in 2020.
computer science • May 5th, 2020 • English

Sensitivity conjecture

The sensitivity conjecture is a conjecture in computer science that was unsolved for over 30 years. In 2019, Hao Huang proves that the conjecture is in fact true, and does so in an extremely elegant way. This talk was in collaboration with two of my fellow students. We discuss the contents of the conjecture, developments and finally the proof of Huang, which was my part. The talk should be accessible to anyone with basic knowledge of linear algebra.
Riemannian geometry • June, 2020 • Dutch

Theorem of Hadamard (slides only)

This short talk was actually part of an exam on Riemannian geometry, which is why no recording is available. I introduce Jacobi vector fields, derive the Jacobi equation and prove two lemmas which eventually lead to the proof of the theorem of Hadamard: ‘Any complete simply connected Riemannian manifold with negative sectional curvature is diffeomorphic to the Euclidian space’. The video below is me going through the slides.
Riemann surfaces • June, 2020 • Dutch

Groups of Möbius transformations (slides only)

Just like the previous talk, this one was part of an exam on Riemann surfaces. I classify automorphisms of the unit disk, introduce the modular group PSL(2, ℤ), prove that it is generated by translation and circle reflection and deduce its fundamental region. I discuss an interpretation of its orbit space and end my talk with a brief discussion of Fuchs groups.
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