seminars & talks

2025

1. 

   Noise Sensitivity on Equivalence Relations

   StuKon, Heidelberg, Germany, Jul 2025

   *Talk from another meeting

   Abs Paper talk* Slides Program

   Noise sensitivity was first introduced by Benjamini, Kalai and Schramm in their seminal work on Boolean functions. We propose a construction of a similar taste on binary relations. By flipping every relation with a small probability p, a natural question on recoverability arises, to which we give a positive answer in the case of an equivalence relation (X,∼). We prove that equivalence relations are noise-stable under the prescribed model. In particular, we propose a simple reconstruction algorithm, and show that it achieves an asymptotically zero misclassification error.

2. 

   An Algebraic Construction of Complete Regular Maps via Prime Ideals

   Joint Mathematics Meetings, Seattle, US, Jan 2025

   Abs Slides Program

   A complete regular map is a highly symmetric embedding of a complete graph Kₙ into a compact oriented surface. Norman Biggs [1972] constructed such maps as Cayley maps associated to finite fields. James and Jones [1985] proved Biggs’ construction gives all complete regular maps. For odd p, we give an alternative number theoretic construction of complete regular maps with n = pᶠ vertices. In particular, we establish a bijection between prime ideals lying over p in the (n-1)ᵗʰ cyclotomic field and complete regular maps with n vertices. Furthermore, we prove that this bijection is compatible with the natural actions of the absolute Galois group of ℚ on both of these sets.

2024

1. 

   Gedankenexperimente! Mathematical Thought Experiments for Young Students

   meerMINT Bremen-Nord, Dec 2024

   Program

2. 

   Fibrations & Applications to Computations of Higher Homotopy Groups

   Seminar in Algebraic Topology, May 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.

3. 

   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. 

   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.