My name is Othmane Oukrid, and I have been doing my ARIA research project under the supervision of Prof. Mikaël Pichot from the Mathematics and Statistics Department of McGill University. Throughout the summer, we have been working on the classification of the so-called “odd” ring puzzles associated to Moebius-Kantor Complexes. We have been able to complete the classification and finish up writing a proof for it, which will most likely appear in a larger paper with complementary results.
On the one hand, Moebius-Kantor Complexes are non-positively curved spaces defined by local constraints (we impose that the space behaves in a certain way at the “local” level), and which occur quite naturally in a field of mathematics called Geometric Group Theory. On the other hand, ring puzzles are tessellations of the Euclidean plane by a given set of shapes (a tessellation is essentially a “filling” or a “tiling” by a given shape –in the same way pages of a notebook are usually “tiled” by little squares). As it turns out, we can associate to Moebius-Kantor Complexes a set of ring puzzles, which help us understand the Euclidean planes that may be sitting inside of it.
As alluded to above, this research project is in Geometric Group theory (GGT), where my mathematical interests lie. GGT is an active contemporary field of mathematics that ties geometry and algebra in novel –sometimes odd—ways. The general approach is to study infinite groups (a usual algebraic structure) through the geometric objects that they are associated with, essentially doing algebra through geometry. As such, the project of classifying odd ring puzzles associated with Moebius-Kantor Complexes was ideal insofar as it allowed me to convert my interest in GGT into actual mathematical research in the field.
Reflecting upon the past weeks, I would take home three main points regarding mathematics and the process by which it is produced:
Firstly, mathematics is about learning. Throughout the summer, I have been able to dissect the literature on metric spaces of non-positive curvature and on buildings, thus deepening my understanding of Geometric Group Theory, which was my first objective. I am eternally grateful to Prof. Pichot for all the knowledge he has been able to communicate in a very accommodating and encouraging manner.
Secondly, mathematics is about trial and error. While the output of mathematical research is often desired to be as clean and elegant as possible, it is often said that the process is quite a messy one. On this front, the project certainly did not disappoint: many leads were pursued, but few of them resulted in interesting material. Nonetheless, the large number of unsuccessful attempts made coming up with the complete classification –my second objective—all the more exhilarating.
Thirdly, mathematics is as much about mathematics as it is not. Throughout the summer, one of the main challenges I faced was to communicate my research intelligibly –be it orally or in writing, especially given the geometric nature of the project. Oftentimes, it would be quite challenging to translate into writing what seems to be a very clear idea. Hopefully, writing the proof of the classification with the help of Prof. Pichot and presenting my summer project at the Canadian Undergraduate Mathematics Conference in Québec were of tremendous help in terms of allowing me to develop the communication skills necessary for scientific research.
All in all, this ARIA has been a very significant one in my academic journey and I am grateful for the opportunity. It has been a great learning experience and a chance to get to know and collaborate with truly inspiring people. On a more concrete note, it allowed me to produce my first piece of research and opened the door for possible future collaboration with Prof. Pichot to produce further interesting mathematics (we will be continuing our work during the FALL2022 term and beyond, hopefully).
I would like to thank my supervisor, Prof. Mikaël Pichot for his continual, kind, and thorough guidance. His helpful suggestions and comments throughout the summer were much appreciated. I would like to also warmly thank Mr. Mark W. Gallop for his generous donation, without which this ARIA project would not have been possible. I hope this summer project will be the start of a very rewarding mathematical journey.
