The Role of AI in Modern Mathematics Research

Professor Terence Tao highlights the transformative impact of AI on mathematics, bridging traditional methods with modern computational tools.

### The Evolution of Mathematical Computation: From Ancient Tools to AI Assistance
* 👨 ‍🏫 Professor Terence Tao, an influential mathematician and IMO gold medalist at age 13, reflects on the **significance of the International Mathematical Olympiad (IMO)** in his life and the **fun it brings to participants**.
* 🤖 His talk focuses on the **transformative impact of AI and machine assistance** in mathematics, highlighting tools like DeepMind's **AlphaGeometry**, which can tackle geometry problems from the **IMO**.
* 🧮 Tao explains that while competition mathematics requires **quick problem-solving**, research mathematics involves **prolonged investigation**, often leading to the need for **changing the problem itself**.
* 🕰 The use of machines in mathematics dates back thousands of years, with the **abacus being one of the earliest examples**, while the concept of computers has evolved significantly over the last **300-400 years**.
* 📊 He discusses the role of databases like the Online Encyclopedia of Integer Sequences (OEIS), which helps mathematicians **discover connections between different mathematical problems** through sequences.
* 🧑 Tao emphasizes the importance of **scientific computation**, which has been in use since the **1920s**, and mentions the advent of **SAT solvers** that tackle complex logic puzzles, although they face **scalability challenges**.
* 📜 He cites significant achievements in computer-assisted proofs, such as the **Four Color Theorem** and the **Kepler Conjecture**, illustrating the ongoing evolution of proof verification methods through **formal proof assistants** and **modern programming languages**.
* 🌐 Finally, he highlights recent advancements in formalizing complex mathematical proofs with tools like Lean, showcasing how **collaborative efforts can improve the workflow of formalization** and **enhance the mathematical library for future research**.

### The Future of Mathematics: Interactive Proofs, Machine Learning, and AI Innovations
* 🔍 The evolution of mathematics is being shaped by **interactive proof systems** that allow users to **click on elements for detailed explanations** of hypotheses, variables, and axioms, paving the way for future textbooks to adopt this **engaging format**.
* 🤝 A recent collaborative project in combinatorics, involving around 20 people, formalized a complex proof in just **three weeks**, showcasing the potential of these interactive systems to **enhance collaboration and visualization** in mathematical research.
* 📊 The project utilized a dependency graph to break down large theorems into manageable pieces, allowing contributors from various fields, including **computer programming**, to work on specific segments without needing to **understand the entire proof**.
* 🔄 The Lean compiler used in the project ensures that **only verified contributions are accepted**, which fosters **trust and collaboration** among mathematicians who may not have previously worked together.
* 📈 Machine learning is also making strides in mathematics, particularly in knot theory, where neural networks have been trained on **vast databases to predict knot invariants**, demonstrating the potential for AI to **uncover hidden relationships** within mathematical structures.
* 🧠 Large language models like GPT-4 are emerging as tools for mathematical problem-solving, with some success in addressing problems like those from the **International Mathematical Olympiad**, although they still struggle with **basic arithmetic** and require **careful prompting** to yield accurate results.
* ⚙ AI tools are being developed to assist in formal proof writing, potentially **speeding up the process** and allowing mathematicians to focus on **more complex tasks**, though significant challenges remain in **automating large proofs**.
* 🌟 The future of mathematics may involve AI generating conjectures and exploring problem classes at an **unprecedented scale**, transforming how mathematicians approach research and collaborate on **complex problems**.

Terence Tao at IMO 2024: AI and Mathematics

AIMO Prize

Professor Terence Tao's Talk on AI and Machine Assistance in Mathematics

Professor Terence Tao, a former IMO participant and now a professor at UCLA, discusses the impact of AI and machine assistance in mathematics.

He highlights the long history of using machines for mathematical computations, from the abacus to modern computers.

Tao emphasizes the transformative potential of AI tools in research mathematics and their role in discovering important mathematical results through databases and tables.

The Role of Computers in Solving Mathematical Problems

Computers can transform geometry problems into systems of equations, but their complexity grows exponentially with size.

SAT solvers and SMT solvers are used to tackle logic puzzles and deduce conclusions from a set of hypotheses, but they also struggle with scalability.

Recent advancements in AI and machine learning, along with formal proof assistants, are providing new and creative ways to use computers in mathematics, leading to significant breakthroughs like the computer proof of the Pythagorean triple problem.

Early mathematical proofs, like the one for graph reducibility, relied on tedious manual checks and were later verified by computers in the '90s.

The Kepler conjecture for sphere packing, proposed in the 17th century, also faced challenges in proving its optimality, with a successful proof using a flexible scoring method and computer assistance announced in 1998.

The proof's complexity led to a lengthy refereeing process, ultimately raising questions about the certainty of its computer calculations.

The controversy over computer-assisted proofs led to the Flyspeck project, which formalized a major theorem in 12 years.

Peter Scholze's condensed mathematics aims to solve functional analysis questions using algebraic methods, with a crucial vanishing theorem at its foundation.

The modern proof assistant Lean has successfully formalized the theorem in 18 months, adding to its math library and creating a blueprint workflow for future formalization projects.

The Future of Interactive Textbooks and Collaborative Proofs

Interactive textbooks may become the norm, offering clickable locations for hypotheses, variables, and explanations.

A recent project in combinatorics, involving 20 people, successfully formalized a 33-page proof in just three weeks, using a dependency graph for collaboration.

Formal proof assistants, like Lean, are changing the way mathematics is done, allowing for greater collaboration and faster adjustments to proofs.

Machine learning is being used to study knot theory, which explores the equivalence of knots through invariant numbers and polynomials.

A neural network trained on a database of knots has successfully predicted knot signatures from hyperbolic geometry invariants.

This black box approach has led to the identification of key invariants and the formulation of conjectures, ultimately providing a theoretical explanation for the relationship between knot signatures and geometric invariants.

AI models are struggling with basic arithmetic and problem-solving, often guessing incorrect answers.

Various approaches are being explored to improve their accuracy, such as connecting them to reliable software and using proof assistant languages.

AI is also proving useful as a muse for problem-solving techniques and offering assistance in writing formal proofs, with the potential to generate good conjectures and explore problem spaces in the future.

The speaker discusses the transformative potential of AI tools in mathematics, enabling unprecedented progress.

They highlight the importance of proving theorems the traditional way to guide AI effectively.

The conversation also touches on Voevodsky's impact on homotopy type theory, the social nature of mathematical research, and the speaker's own unconventional path to university at a young age.