Mathematics of AI

Lead: Joan Bruna

This area studies large language models as mathematical objects. The central question is how what we call “intelligence” emerges in them: which capabilities appear, at which scale, and by which mechanism.

Some of these questions arise only at scale. In order to study phase transitions in learning one needs a large model whose weights can be examined and whose further training can be controlled. The core provides this experimental access to its own post-training data and runs; access to the starting model’s pretraining data and history remains dependent on the provider.

Questions we will pursue include:

• Emergence and reasoning features. When and how do reasoning capabilities, such as following or producing a long proof, appear during training, and can their appearance be predicted? What are the mechanisms of feature learning in this setting?
• Post-training. What does training with verifiable rewards change inside a model, and why does it improve some abilities and not others?
• Optimization and sampling. The large-scale optimization and sampling problems behind training and generative modeling, studied together with our applied mathematics area.

Mathematics is an unusually clean setting for these questions, because ground truth is often available: whether a model’s answer is correct can be checked, not guessed.