seminars & talks

2024

1. 

   Fibrations & Applications to Computations of Higher Homotopy Groups

   Seminar in Algebraic Topology, Jun 2024

   Abs Paper talk Slides

   This is an expository paper on fibrations in the context of higher homotopy groups. The paper does not assume any pre-requisite knowledge, adopting a rather axiomatic approach. We direct our focus towards fibrations and show how one can apply this powerful machinery to compute homotopy groups. One main result regards the loop-space fibration, which shifts the homotopy group by one degree. Another highly non-trivial result utilisies the Hopf fibration to prove that the homotopy groups of S2 and S3 coincide for n ≥ 3. The goal of this paper is to trivialise and familiarise, and to equip the reader with the right ideas for a more developed treatment of the subject.

2. 

   Zermelo’s Theorem in Game Theory

   Mathematics Undergraduate Seminar at Constructor University, Feb 2024

   Abs Paper

   We already know that tic-tac-toe is a solved game, meaning it is determined if both players play their best. This is, however, not so clear for the game of Chess. We will first start with a good game of tic-tac-toe, then refine the intuition into rigor on our way to proving the first formal theorem in the theory of games, credited to Ernst Zermelo. In full generality, we show that in two-player finite games of perfect information, there is always a strategy to reach a win or draw; with no regard to what the strategy should be. One could then ask the following question. Given a winning position, how quickly can a win be forced? We conclude the talk with a reflection on Zermelo’s findings as well as the powerful, non-intuitive consequences of his results.

2023

1. 

   Mathe Beim Frieseur | Die Russellsche Antinomie

   General Audience Talk on Russell’s Paradox, Dec 2023

   Abs Slides

   Soll ich mich rasieren? Das ist die Frage, wenn es um das behrumte Russell-Paradoxon geht. In diesem Vortrag präsentieren wir eine einfache Analogie zur mengentheoretischen Konstruktion dieses fundamentalen Ergebnisses in der Welt der Mathematik.

2. 

   The Banach-Tarski Paradox

   Mathematics Undergraduate Seminar at Constructor University, Oct 2023

   Abs Paper Slides

   Contrary to common belief, the result by Stefan Banach and Alfred Tarski is not at all paradoxical, and rather challenges our intuition about the notion of infinity. In this article, we introduce a paradoxical decomposition of F2, the free group on two generators, then extend this notion by the embedding of F2 in Euclidean space R3. We conclude with a few notes on non-measurability, and reflect on criticism of the Axiom of Choice.