Mathematics and AI Seminar
This fall I am organizing a series of talks on the connections between mathematics and contemporary AI. There are multiple angles, and we will discuss both the mathematical aspects that make AI work and the impact of AI on mathematical life, in research as well as in teaching.A social dimension will also run through these presentations. It is clear that these technologies will continue to transform the world as we know it, and it seems important to use the sometimes illuminating reach of mathematics to try to understand the nature and extent of the changes to come.
Wednesday, September 16; 12:30 to 13:20 (B-3410)
What future for automated mathematics?
Guillaume Roy-Fortin
Dozens of Erdős problems, the unit distance conjecture, a complex structure on the 6-dimensional sphere, a counterexample to the Escobar conjecture... and the list goes on! What list? The list of mathematical problems that have been solved thanks to the dramatic progress of modern large AI models. These models are now capable of autonomously producing mathematical results comparable in level to university research. What are the consequences of such a technological feat? What, then, is mathematics, if it can be automated? In this talk, we will discuss these recent advances and the profound upheavals they are causing in the mathematics profession, notably by revisiting some aspects of Terence Tao's presentation at the ICM last July.Wednesday, September 30; 12:30 to 13:20 (B-3410)
AI alignment: the disinterested scientist approach.
Guillaume Roy-Fortin
The recent rapid advances in AI, together with numerous incidents such as the one involving HuggingFace, have highlighted the importance of developing AI systems whose behavior is aligned with what their designers intend and expect. After a brief review of Bayesian inference, we will present a recent approach developed by Bengio and his collaborators. This approach aims to build an AI capable of providing reliable predictions in response to the questions it is asked, without allowing those predictions to be influenced by their possible consequences.Wednesday, October 14; 12:30 to 13:20 (B-3410)
Superposition and polysemanticity: understanding the internal representations of AI to better secure them
Guillaume Roy-Fortin
Information flows through the neural networks powering modern AI systems layer by layer. Given the success (!) of these models, it is natural to wonder what is stored in the neurons of intermediate layers during and after training. Is it possible to determine what kind of information a neuron ultimately learns to store? This is the fundamental problem of interpretability, which aims to better understand the internal representations of neural networks.
Drawing on the work of Elhage et al. (2022), we will see that the activation vector of a neural network layer can sometimes encode more features than its dimension would seem to allow. A bit like managing to fit 5 types of utensils into a drawer designed to hold only 4, for example. This phenomenon, known as superposition, relies in particular on the fact that not all features are present simultaneously. It also helps explain why a single neuron can play several distinct roles, a phenomenon known as polysemanticity. Linear algebra will be our main tool for illustrating these concepts.Wednesday, October 21; 12:30 to 13:20 (B-3410)
Artificial intelligence and computer algebra systems: what role in mathematics education?
Michel BeaudinNovember, date to be announced
Generative artificial intelligence and the problem of incoherent feedback
Marc-Kevin Daoust, Serge Vicente
This paper examines the relationship between the accuracy and coherence of AI-generated content. When generative artificial intelligence produces incoherent content, that content tends to be unintelligible and unreasonable. Incoherence creates challenges for the use of AI-generated content in professional or industrial contexts, where intelligibility and reasonable acceptability are essential criteria for decision-making. Several companies developing generative AI claim that improving AI performance (accuracy) will be the key to solving the problem of incoherence. Drawing on 1,300 evaluations generated by 65 generative AI systems, we suggest that improving AI accuracy reduces incoherence but does not eliminate it. We focus on three coherence requirements discussed in the literature on structural rationality and decision theory. We also explore the implications of our observations for other debates, such as the epistemic properties of artificial intelligence and the relationship between accuracy and coherence.December, date to be announced
The Lean software
Xavier Provençal
Note: this is a non-technical presentation open to everyone.