AI-assisted Math Discovery — Course Overview

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High-level program:

Part 1. LLMs and agents as research assistants

Part 2. ML: approximation as a discovery tool

Part 3. A deeper look into LLMs and agents

Part 4. Accelerating search in large spaces

Lecture Descriptions

• Week 1: What AI is. AI-assisted theorem proving and verification -- 25.02

• Week 2: Coding agents. Orchestrating math experiments -- 4.03

  Papers of the week:

  • CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs. Brief version and Long version
  • PrIncipal quiver Grassmannians: conjectures

• Week 3: Linear models -- 11.03

  Papers of the week:

  • Murmurations of elliptic curves

• Week 4: Gradient descent & Decision trees -- 18.03

  Papers of the week:

  • Machine-Learning Number Fields
  • Machines Learn Number Fields, But How? The Case of Galois Groups

• Week 5: Neural networks -- 25.03

  Papers of the week:

  • Machine learning the dimension of a Fano variety
  • Machine learning detects terminal singularities

• Week 6: Physics-informed neural networks & Dimensionality reduction -- 1.04

  Papers of the week:

  • Physics-informed learning of governing equations from scarce data
  • Discovery of Unstable Singularities
  • Structure of the chromatic polynomial

• Week 7: Autoencoders & RNNs -- 8.04

  Papers of the week:

  • Geometric Learning of Knot Topology

• Week 8: Transformers -- 15.04

  Papers of the week:

  • Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers
  • Learning Euler Factors of Elliptic Curves

• There was a break here

• Week 9: LLMs & Reasoning -- 29.04

  Papers of the week:

  • Recognizing and generating knotted molecular structures by machine learning
  • And a whole lot of papers on reasoning

• Week 10: Reinforcement Learning, part 1: Basic concepts & Policy Gradient -- 6.05

• Week 11: Reinforcement Learning, part 2: PPO -- 13.05

• Week 12: RL in Math. Cross-entropy method and PatternBoost -- 20.05

  Papers of the week:

  • What makes math problems hard for reinforcement learning: a case study
  • Resolution of AutoGraphiX conjectures relating the index and matching number of graphs
  • Constructions in combinatorics via neural networks
  • PatternBoost: Constructions in Mathematics with a Little Help from AI
  • Generating Hadamard matrices with transformers
  • Flow-based Extremal Mathematical Structure Discovery
  • Spectral Structure in Finite Free Information Inequalities and p-Stam Phase Transitions

• Week 13: Algorithmic optimization & AlphaEvolve. AI Agents -- 27.05

  Papers of the week:

  • Bruhat intervals that are large hypercubes
  • Mathematical exploration and discovery at scale
  • ShinkaEvolve: Towards Open-Ended And Sample-Efficient Program Evolution
  • Aristotle: IMO-level Automated Theorem Proving
  • Olympiad-level formal mathematical reasoning with reinforcement learning
  • Advancing Mathematics Research with AI-Driven Formal Proof Search