Seminars — Winter 2026

Abstract : As part of a research project during the summer of 2025, we explored the use of artificial intelligence to support mathematical research. Specifically, we sought to contribute to the resolution of a complex problem: the generation of maximal snake polyominoes, which are the longest possible chains of cells within a grid. Today, the generation of these mathematical objects is limited due to algorithmic and computational complexity, which becomes insurmountable in large grids. To this end, we tested two deep neural network models: a Transformer model, which generates snakes cell by cell, and a diffusion model, which generates a snake by learning to remove noise from an image. This presentation describes the functioning of these models as well as our primary results.

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Abstract : As classical objects in combinatorics on words, Christoffel words offer a wide range of surprising interpretations that intertwine through equivalent definitions, each providing the opportunity to bridge mathematical subjects that are sometimes distant from one another. More than 15 equivalent definitions have been proposed since the study of these words began. Two new definitions have been introduced over the last two years right here in Quebec. Christoffel inclusion is the most recent among them.

Abstract : Intuitively, a sweater appears to have four openings: two for the sleeves, one for the neck, and one for the waist. However, from a topological point of view, it can be shown that it only has three! This presentation will explain how homotopy theory leads to this conclusion by studying shapes through continuous deformations. The concepts of homotopy and homotopy equivalence will be presented both formally and through several visual examples. Subsequently, cell complexes (CW complexes) will be used as a natural framework to construct (and deconstruct) spaces from elementary cells. Finally, these tools will allow us to simplify various topological spaces, including a model of a sweater. This talk assumes no prior knowledge of topology.

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Abstract : Infinity in mathematics brings its share of paradoxes when not handled properly. As far back as Antiquity, Zeno of Elea raised doubts regarding the treatment of an infinite sequence of operations by formulating his famous paradox: will an arrow ever reach its target if it must always cover half of the remaining distance? Today, thanks to modern definitions of series convergence—notably those of Bolzano-Weierstrass and Augustin Cauchy—we know that the arrow does indeed reach its target. Nevertheless, some series are divergent, yet certain mathematicians manage to assign them a value. In this presentation, I will discuss number series and certain paradoxes related to the values assigned to specific divergent series. More precisely, after a historical review of the early stages of series theory, I will present a summation method developed by Ernesto Cesàro, which allows for the rigorous assignment of a value to certain divergent series. Finally, I will introduce a theory specifically developed for handling divergent series, known as summability theory.

Seminar Video: Watch on YouTube