Collaboration Between Mathematicians and DeepMind: The Role of Machine Learning in Mathematical Discovery

Introduction

• Topic: Collaboration between pure mathematicians and DeepMind using machine learning to make new mathematical discoveries
• Key Points
  • Results of collaboration
  • Use of machine learning in mathematical discovery
  • Future of machine learning in mathematics
• Acknowledgements: University of Oxford and the Maths Institute

Structure of the Talk

1. Introduction to Machine Learning and Mathematics Applications
2. Combinatorial Invariance Conjecture (By Geordie Williamson)
3. Introduction to Knot Theory (By Andres Juhash)
4. Signature Slope Conjecture in Knot Theory (By Mark Lakan B)

What is Machine Learning?

• Field: Computer science focused on systems learning from data
• Applications: Tasks too complex to program explicitly (e.g., image recognition)
• Breakthroughs
  • Image content recognition
  • Language translation
  • AlphaGo beating world champion at Go
  • AlphaFold in protein folding prediction
• Techniques Used in Mathematics
  • Supervised Learning: Learning a function from input-output pairs
  • Attribution: Identifying parts of the input used for predictions

Machine Learning Example

• Image Recognition: Training a model to recognize a tabby cat from images
• Model Validation: Ensuring the model uses relevant parts of the image (pixels of the cat) for predictions

Application in Mathematics

• Patterns in Mathematics: Fundamental to pure math, involves generating examples and finding new structures
• Historical Example: Discovery of prime number patterns leading to the prime number theorem
• Millennium Prize Problems: Example of big conjectures formed from pattern observation
• Mode of Discovery: Machine learning to detect patterns in complex mathematical objects

Rediscovering Euler’s Formula Example

• Problem: Predicting edges of a polyhedron from other measurements (faces, vertices)
• Process
  • Generate data set of polyhedra properties
  • Train a supervised learning model to predict edges
  • Use attribution techniques to identify critical measurements
  • Rediscovered formula: Edges = Faces + Vertices - 2

Case Studies

Combinatorial Invariance Conjecture (By Geordie Williamson)

• Model: Uses Bruhat graph and Kazhdan-Lusztig polynomial
• Approach: Use graph neural networks to predict polynomials from Bruhat graphs
  • Training achieved 97% accuracy
  • Saliency analysis identified important edges in the graphs
  • New conjecture formulated based on hypercube-like structures found in the graph
• Impact: Progress towards solving a 40-year-old conjecture

Knot Theory (By Andres Juhash)

• Introduction to Manifolds: Spaces that locally look like coordinate space
• Key Terms: Knot, genus, hyperbolic surfaces, band sum operations in DNA, classical invariants
• Four-dimensional Poincare Conjecture: An unsolved problem requiring non-slice knots counterexamples
• Applications of Knot Theory
  • Chemistry: Synthesizing molecules as knots
  • Biology: DNA recombination and unlocking
  • Quantum computing: Using knots for qubits

Signature Slope Conjecture (By Mark Lakan B)

• Not Invariants: Mathematical quantities like numbers or polynomials associated with knots
• Machine Learning Application: Using hyperbolic invariants to predict four-dimensional invariants
• Example: Predicting signature of knots using hyperbolic invariants
  • Initial conjectures confirmed by millions of examples but disproved later
  • Revised successful theorem considering additional hyperbolic properties
• Conclusion: Machine learning aided in conjecture formation and proof discovery

Panel Discussion Insights

Genesis of Collaboration

• Started in 2018: Connection between Geordie Williamson and Demis Hassabis, and involvement of Mark and Andres
• Team Formation: Around 10 people on DeepMind’s side contributed

Philosophical and Practical Aspects

• Supervised Learning: Useful but requires mathematician's insight
• Saliency Techniques: Crucial for interpreting machine learning outputs in mathematical contexts

Progress and Noteworthy Anecdotes

• Collaborative Experience: Enriching but challenging, involving many trial-and-error attempts
• AI as an Aid: Machine learning suggested conjectures that took months for human proof
• Skepticism Addressed: Reiterations that these techniques are tools, not magic bullets

Future Directions

• Predicting Mathematical Results: Potential use in larger conjectures like the Riemann Hypothesis or Hodge Conjecture
• Teaching Machine Learning in Math: Possibly integrating ML techniques in mathematical education
• Need for More Examples: Encouraging widespread use and creating introductory resources for mathematicians

Conclusion

• Exciting Potential: Machine learning as a valuable aid in discovering and proving new mathematical theorems
• Interdisciplinary Success: Collaboration between mathematicians and AI researchers can lead to ground-breaking discoveries