The Mathematical Foundations of Learning to Reason in Large Language Models

Reasoning has become a defining capability of large language models, yet the learning mechanisms behind its emergence and generalization remain poorly understood. In this talk, I will explore how chain-of-thought training on short problems enables length generalization, and how self-improvement can extend this ability further. I will then examine how the reasoning frontier evolves under reinforcement learning with verifiable rewards, revealing an implicit curriculum that emerges from mixed-difficulty training and showing how the difficulty spectrum shapes learning dynamics.