Terence Tao at IMO 2024: AI and Mathematics

The speaker introduces Professor Terence Tao, highlighting his early achievements in the International Mathematical Olympiad (IMO), where he earned a bronze medal at age 11, a silver medal the following year, and a gold medal at 13, making him the youngest gold medalist. Tao's transition from IMO participant to a professor at the University of California, Los Angeles (UCLA) is discussed, emphasizing his significant contributions to mathematics. The talk shifts focus to the impact of AI on mathematics, referencing a presentation on AlphaGeometry, an AI product capable of answering IMO geometry questions, and the upcoming AI Math Olympiad. Tao discusses the differences between competition and research mathematics, noting the evolving role of machine assistance in the field, with a historical perspective on the use of machines like the abacus and the evolution of the 'computer' as a human profession to modern electronic computers.

This paragraph delves into the historical and ongoing use of tables and databases in mathematical research. The speaker mentions the prime number theorem, discovered by Legendre and Gauss through the analysis of prime number tables, and the Birch and Swinnerton-Dyer conjecture, which was also initially identified through data tables. The Online Encyclopedia of Integer Sequences (OEIS) is introduced as a modern example of a mathematical table, used by mathematicians to identify patterns and connections in sequences. The paragraph also touches on the use of scientific computation, or number crunching, in mathematics, with examples ranging from the early 20th-century efforts to model fluid dynamics to modern applications in solving complex equations and the utilization of SAT solvers and SMT solvers in logic and algebraic problem-solving.

The speaker discusses the innovative ways computers are currently being used in mathematics, beyond traditional database lookups and number crunching. Three main areas are highlighted: the use of machine learning and neural networks to uncover new correlations in mathematics, the development of formal proof assistants that allow mathematicians to verify the correctness of arguments and proofs, and the application of AI in generating conjectures and solving problems that were previously intractable. The paragraph also touches on the historical use of computers for case analysis and the potential for AI to transform mathematical research by enabling the exploration of a vast space of problems in a way that was not previously feasible.

This section focuses on the history and development of computer-assisted proofs, exemplified by the proof of the Four Color Theorem. The speaker outlines the initial attempts at proving the theorem in the 1970s, which involved a combination of computer and human efforts. The paragraph details the process of proving the theorem, including the identification and checking of special subgraphs, and the subsequent formalization of the proof using proof assistants like Coq. The speaker also discusses the controversy and skepticism surrounding computer-assisted proofs and the efforts to formalize these proofs to achieve complete verification.

The speaker discusses the Kepler conjecture regarding the most efficient packing of spheres in three-dimensional space. The paragraph details the various attempts to prove this conjecture, including the work of Thomas Hales and his team, who eventually succeeded using a combination of innovative strategies and computer assistance. The talk highlights the iterative process of refining the proof, the challenges in getting the proof accepted by the mathematical community, and the subsequent efforts to formalize the proof using the Flyspeck project. The paragraph emphasizes the importance of collaboration between humans and AI in the process of mathematical discovery and proof verification.

This section explores the recent advancements in the application of AI and proof assistants in mathematical research. The speaker mentions Peter Scholze's work in condensed mathematics and the formalization of a key theorem using the Lean proof assistant. The paragraph discusses the process of formalizing proofs, the challenges faced, and the benefits of having a formal proof, including the ability to make changes and improvements more efficiently. The speaker also highlights the indirect benefits of formalization projects, such as the expansion of the Lean math library and the development of tools to aid in future formalization efforts.

The speaker reflects on the current state and potential of AI in solving mathematical problems directly. While acknowledging that AI is not yet capable of solving a wide range of problems reliably, the paragraph discusses the usefulness of AI as an assistant, particularly in generating conjectures and exploring problem spaces. The speaker expresses optimism about the future of AI in mathematics, suggesting that it could lead to new ways of doing mathematics on an unprecedented scale. The paragraph concludes with a call for flexibility and adaptability in the face of AI's growing role in mathematical research.

In this final section, the speaker emphasizes the importance of collaboration in mathematical research, facilitated by the use of AI and proof assistants. The paragraph discusses the speaker's personal experience with collaborative projects, such as the formalization of a combinatorics proof with a large team, and the benefits of breaking down complex proofs into smaller, manageable pieces. The speaker also reflects on the future of mathematics, suggesting that AI could enable mathematicians to tackle larger classes of problems and explore the space of mathematical questions more broadly.

The speaker answers questions about his early academic journey, including his decision to attend university at a young age and the influence it had on his development as a mathematician. He also discusses his approach to selecting research topics, emphasizing the importance of serendipity and conversations with other mathematicians. The paragraph concludes with the speaker's thoughts on the future of mathematics, suggesting that flexibility and adaptability will be key in a field increasingly shaped by AI and new technologies.