# Terence Tao at IMO 2024: AI and Mathematics

TLDRProfessor Terence Tao, a renowned mathematician and IMO gold medalist, delivered a talk at IMO 2024, reflecting on his experiences and discussing the impact of AI on mathematics. He highlighted the historical use of machines in mathematical computations, from the abacus to modern computers, and explored contemporary AI tools like AlphaGeometry and SAT solvers. Tao also touched on the transformative role of AI in research mathematics, emphasizing the potential of machine learning and formal proof assistants in conjecture generation and proof verification, suggesting an exciting future for mathematics where AI and human collaboration could lead to unprecedented discoveries.

### Takeaways

- ๐ Terence Tao is a renowned mathematician, a former child prodigy at IMO, and currently a professor at UCLA.
- ๐ He discusses the integration of AI in mathematical research, highlighting the differences between competition and research mathematics.
- ๐ The use of machines in mathematics dates back thousands of years, from the abacus to modern computers.
- ๐ง AI, particularly machine learning and neural networks, is being used to uncover new correlations and solve problems in mathematics.
- ๐ Reference to AlphaGeometry, a product by DeepMind, capable of answering IMO-level geometry questions.
- ๐ The Online Encyclopedia of Integer Sequences (OEIS) is a valuable resource for mathematicians, often aiding in research.
- ๐งฎ Historically, 'computer' referred to a job role, with human computers performing calculations, especially็ๅฅณๆงๅจไบๆๆ้ด.
- ๐ The process of creating mathematical tables for calculations has evolved into using databases in modern research.
- ๐ Tao mentions the use of AI in proving complex mathematical theorems, such as the Pythagorean triple problem.
- ๐ค Large language models like GPT-4 are being experimented with to solve IMO problems, though with limited success rates.
- ๐ Machine learning has also been applied in knot theory, identifying connections between combinatorial and geometric invariants.

### Q & A

### Who is Terence Tao and what is his significance in the field of mathematics?

-Terence Tao is a renowned mathematician who participated in the IMO at the age of 11, receiving a bronze medal, and subsequently a silver and a gold medal. He is a professor at the University of California, LA, and is considered one of the most influential mathematicians of our time, particularly known for his work in harmonic analysis, partial differential equations, combinatorics, and mathematical Olympiad problems.

### What is the AlphaGeometry product mentioned in the transcript and how does it relate to AI?

-AlphaGeometry is a product by DeepMind that can answer some IMO geometry questions. It is an example of AI being used to tackle complex mathematical problems, showcasing the growing intersection of AI technology and mathematical research.

### How does Terence Tao perceive the impact of AI on research mathematics?

-Tao sees AI as a transformative tool in research mathematics, noting that while it is exciting and beginning to change the field, there is also a sense of continuity as machines have been used to assist in mathematics for a long time.

### What is the history of using machines to do mathematics as discussed by Tao?

-According to Tao, the history of using machines for mathematics dates back thousands of years, starting with simple tools like the abacus. The concept of 'computer' originally referred to a job profession, with human computers performing calculations, especially during times like World War II for tasks like ballistics. The use of mechanical and then electronic computers has evolved over the past 300-400 years.

### What is the significance of the Online Encyclopedia of Integer Sequences (OEIS) in mathematical research?

-The OEIS is a database of integer sequences that has become an essential tool for mathematicians. It allows researchers to compare sequences they encounter in their work with known sequences, potentially revealing connections between different mathematical problems and leading to new research directions.

### How does Terence Tao describe the use of computers in mathematical research before the advent of electronic computers?

-Before electronic computers, the term 'computer' referred to human beings who performed calculations. These human computers were used to create mathematical tables, such as logarithm tables, and to perform complex calculations, especially during significant events like World War II for tasks like computing ballistics.

### What role do databases and tables play in modern mathematical research according to the transcript?

-Databases and tables, evolved from historical mathematical tables, play a crucial role in modern mathematical research. They provide a foundation for research, allowing mathematicians to rely on verified sets of data and previously solved problems to build upon and compare with their own findings.

### Can you explain the concept of 'knot theory' as mentioned in the transcript?

-Knot theory is a branch of mathematics that studies mathematical knots. A knot is a simple closed curve in three-dimensional space that cannot be untangled to produce a simple straight line. The field explores questions of which knots are equivalent (can be deformed into each other without cutting or passing through themselves) and develops invariants that remain unchanged under such deformations.

### What is the role of machine learning in knot theory as discussed by Terence Tao?

-Machine learning, particularly through the use of neural networks, has been used to predict knot invariants, such as the signature, from other geometric invariants. This has helped uncover connections between different types of invariants that were not previously known, demonstrating the potential of AI to assist in discovering new mathematical relationships.

### How does Terence Tao view the future of AI in mathematics?

-Tao envisions an exciting future for AI in mathematics, where it will serve as a valuable assistant beyond brute force computation. He anticipates that AI will become adept at generating conjectures and solving classes of problems, potentially leading to a new scale of mathematical exploration and discovery.

### Outlines

### ๐ Introduction to Professor Terence Tao and AI in Mathematics

The speaker introduces Professor Terence Tao, a renowned mathematician known for his early achievements at the International Mathematical Olympiad (IMO) and his current role as a professor at UCLA. Tao reflects on his enjoyable experiences at the IMO and emphasizes the importance of both competition and social aspects of such events. The talk then shifts focus to the impact of AI on mathematics, with mentions of AI tools like AlphaGeometry and the AI Math Olympiad. Tao discusses the differences between competition and research mathematics, highlighting the evolving nature of mathematical research with the aid of machine assistance.

### ๐ The Evolution of Computational Tools in Mathematics

This section delves into the historical use of computational tools in mathematics, starting with the abacus and evolving through mechanical and human computers. The speaker discusses the role of 'computer' as a job, especially during World War II, and the use of tables for mathematical computations. The importance of tables, now known as databases, in mathematical research is emphasized, with examples like the Online Encyclopedia of Integer Sequences (OEIS) being crucial for pattern recognition and hypothesis generation in mathematical sequences.

### ๐งฎ The Role of Tables and Computational Power in Mathematical Discovery

The speaker continues to elaborate on the use of tables in mathematical research, citing the prime number theorem as an example of a discovery made possible through computational analysis. The talk then explores the concept of scientific computation, or number crunching, which has been a part of mathematical research since the 1920s. Historical examples such as Hendrik Lorentz's work on fluid dynamics and the use of human computers are discussed, illustrating the evolution of computational methods in solving complex mathematical problems.

### ๐ Advanced Computational Techniques and Their Impact on Mathematics

This paragraph introduces more advanced computational techniques such as SAT solvers and SMT solvers, which are used to solve logic puzzles and complex mathematical problems. The speaker discusses the limitations of these solvers, particularly their inability to scale well with problem size. However, the paragraph also highlights a significant achievement in using these methods to solve the Pythagorean triple problem, which was proven with the help of a computer SAT solver.

### ๐ค The Emergence of AI and Machine Learning in Mathematical Research

The speaker discusses the recent and exciting developments in using AI and machine learning for discovering new connections in mathematics. Large language models and formal proof assistants are introduced as tools that are becoming increasingly accessible and useful to mathematicians. The paragraph also touches on the historical use of computer-assisted proofs, such as the Four Color Theorem, and the ongoing efforts to formalize mathematical proofs using proof assistants like Coq and Lean.

### ๐ The Transformation of Mathematical Proofs with Formal Proof Assistants

This section focuses on the transformation of mathematical proofs through the use of formal proof assistants. The speaker describes the process of formalizing proofs and the benefits it brings, such as increased collaboration and the ability to quickly iterate and improve upon proofs. Examples like the Kepler conjecture and the PFR theorem are used to illustrate the practical application of proof assistants in verifying and formalizing complex mathematical arguments.

### ๐ The Future of Mathematics with AI and Collaborative Proofs

The speaker envisions a future where AI plays a significant role in mathematics, particularly in the generation of conjectures and solving classes of problems simultaneously. The paragraph discusses the potential for AI to handle tasks that are currently too tedious for humans, such as formalizing proofs, and the possibility of mathematicians collaborating on a scale not previously possible. The use of AI as a 'muse' to suggest new techniques and approaches to problem-solving is also highlighted.

### ๐ค The Importance of Human Interaction in the Age of AI Mathematics

In the final section, the speaker emphasizes the continued importance of human interaction and collaboration in mathematics, even as AI becomes more integrated into the field. The paragraph discusses the serendipity of mathematical discovery and the role of conferences and discussions in shaping research directions. The speaker also addresses the personal aspects of choosing research topics and the impact of early academic experiences on a mathematician's career.

### Mindmap

### Keywords

### ๐กAI

### ๐กIMO

### ๐กMachine Learning

### ๐กProof Assistants

### ๐กFormal Proof

### ๐กComputational Power

### ๐กNeural Networks

### ๐กSAT Solvers

### ๐กLarge Language Models

### ๐กAlphaGeometry

### Highlights

Professor Terence Tao shares his early achievements at the IMO, including being the youngest participant to receive a gold medal.

Tao discusses the transformative impact of AI on research mathematics, distinguishing it from competition mathematics.

The history of using machines for mathematics is traced back thousands of years, with the abacus as an early example.

Before electronic computers, human computers were used, particularly during World War II for calculations like ballistics.

Tao mentions the use of tables in mathematics, highlighting the importance of the Online Encyclopedia of Integer Sequences.

The application of scientific computation in mathematics is explored, including its use in modeling and solving complex problems.

AI and machine learning are revealed as new tools in mathematics, with potential for discovering correlations and solving problems.

Formal proof assistants are introduced as a way to verify mathematical arguments with the help of computers.

Tao describes the process of computer-assisted proofs, using the Four Color Theorem as a historical example.

The Kepler Conjecture and its proof, which relied heavily on computer assistance, is discussed.

The use of AI in knot theory is highlighted, showcasing how machine learning can predict knot invariants.

Large language models like GPT-4 are noted for their potential in solving mathematical problems, despite current limitations.

Tao envisions a future where AI can generate conjectures and solve classes of problems, expanding the scope of mathematical research.

The importance of collaboration in mathematics is emphasized, with AI potentially facilitating larger and more diverse teams.

Tao reflects on his early entry into university and the individualized nature of educational paths in mathematics.

The selection process for research topics is discussed, with an emphasis on the role of serendipity and collaboration.