Lex Fridman Podcast Episode #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI

Release Date: June 15, 2025

In this illuminating episode of the Lex Fridman Podcast, Lex engages in a deep and comprehensive conversation with Terence Tao, one of the most celebrated mathematicians of our time. Widely regarded as the “Mozart of Math,” Tao shares his insights on some of the most challenging problems in mathematics and physics, the evolving role of artificial intelligence (AI) in mathematical research, and his perspectives on collaboration and creativity in the mathematical community.

Early Encounters with Complex Problems

[09:57] Terence Tao: "What are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%?"

Tao recounts his experience as a PhD student grappling with the Kakeya problem, a puzzle originally posed by the Japanese mathematician Soji Kakeya in 1918. The problem investigates the minimal area required to rotate a needle (infinitely maneuverable) by 180 degrees on a plane. His work on this problem elucidated connections between geometry, partial differential equations, number theory, and combinatorics, highlighting the intricate interplay of mathematical disciplines.

Navier-Stokes Equations and Finite-Time Blow-Up

[16:32] Lex Fridman: "Can you speak to the Navier-Stokes? So the existence and smoothness, like you said, Millennial Prize Problem."

The conversation delves into the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. Tao explains the delicate balance between dissipation (viscosity) and transport (fluid movement) forces within these equations. He discusses his groundbreaking work in 2016, where he demonstrated finite-time blow-up for an averaged version of the three-dimensional Navier-Stokes equations. This work provided valuable insights and ruled out certain approaches, guiding mathematicians towards more promising strategies.

The Role of Supercriticality in Partial Differential Equations

[24:37] Terence Tao: "The key phenomenon that my technique exploits is what's called supercriticality."

Tao introduces the concept of supercriticality in partial differential equations (PDEs), where nonlinear transport terms dominate dissipation at increasingly small scales, leading to unpredictable and turbulent behaviors. He contrasts this with criticality and subcriticality, where the balance between forces allows for more predictable and controllable outcomes. Understanding supercriticality is pivotal in both fluid dynamics and wave propagation, influencing the stability and potential singularities within these systems.

Mathematics vs. Physics: Bridging Disciplines

[49:32] Lex Fridman: "What do you think is a more powerful way of discovering truly novel ideas about reality?"

Tao explores the symbiotic relationship between mathematics and physics, emphasizing the necessity of both top-down (theoretical) and bottom-up (experimental) approaches. He highlights how advancements in mathematical modeling and theoretical frameworks have historically propelled physical theories forward, such as how Riemannian geometry became essential for Einstein’s general relativity. Tao expresses optimism about the potential for unifying existing physical theories through mathematical innovation.

Universality and the Central Limit Theorem

[56:32] Lex Fridman: "What's the difference between infinity and the finite world? How do we use infinity in mathematics?"

Tao discusses universality in mathematics, the phenomenon where complex systems exhibit consistent behaviors across different scales and conditions. He uses the Central Limit Theorem as a prime example, explaining how it accounts for the ubiquitous appearance of Gaussian distributions in nature. Tao acknowledges the limitations of universality, such as systemic correlations that deviate from expected random behaviors, as seen in the 2008 financial crisis where hidden systemic risks led to catastrophic outcomes despite reliance on Gaussian models.

Collaboration and the Future of Mathematical Proofs

[89:25] Terence Tao: "Lean is a computer language much like sort of standard languages like Python and C and so forth, except that in most languages, the focus is on producing executable code."

The conversation shifts to computer-assisted proofs and the use of the Lean formal proof programming language. Tao explains how Lean differs from traditional programming languages by allowing users to create certified proofs that ensure mathematical arguments are error-free. He highlights the Equational Theories Project, which formalized 22 million algebraic problems, showcasing the immense potential for scalability and collaboration in modern mathematics through tools like Lean and AI integration.

AI in Mathematics: Current Capabilities and Future Prospects

[104:30] Terence Tao: "We've had human generate mathematics that's very low quality, like submissions people who don't have the formal training and so forth. But if a human proof is bad, you can tell it's bad pretty quickly."

Tao assesses the current state of AI in mathematical research, noting that while AI can assist in tasks like autocomplete in Lean, it struggles with verifying complex proofs due to a high rate of subtle errors. He envisions a future where AI tools become more reliable collaborators, capable of generating viable proof strategies and assisting in literature reviews. However, substantial advancements are needed for AI to autonomously contribute to high-level mathematical breakthroughs akin to those recognized by the Fields Medal.

Prime Numbers and Unsolved Conjectures

[145:00] Terence Tao: "The primes behave like a random set...what is mysterious is the mechanism that really forces the randomness to happen."

Tao delves into the enigmatic nature of prime numbers, often referred to as the "atoms of mathematics." He discusses their seemingly random distribution and the challenge of proving conjectures like the Twin Prime Conjecture and the Riemann Hypothesis. Tao explains the parity barrier, a significant obstacle that prevents current mathematical techniques from advancing these conjectures, emphasizing the need for novel mathematical tools and insights.

Solving the Collatz Conjecture and Mathematical Persistence

[163:09] Terence Tao: "It's like trying to solve a computer game where there's unlimited cheat codes available."

Tao touches upon his work on the Collatz Conjecture, a deceptively simple yet profoundly difficult problem. He outlines his probabilistic approach, demonstrating that statistically, most starting numbers in the Collatz sequence tend to decrease over time. However, proving that all sequences eventually reach 1 remains elusive due to potential "conspiracies" or outlier cases that defy statistical trends.

The Legacy of Grigori Perelman and the Ethics of Mathematical Recognition

[175:11] Lex Fridman: "But you were able to make, as I understand, actually, you've done, you've made so much progress towards the hardest problems in the history of mathematics."

Tao reflects on Grigori Perelman, the mathematician who famously declined the Fields Medal and the Millennium Prize for solving the Poincaré Conjecture. He discusses the personal and professional challenges faced by mathematicians who choose to distance themselves from traditional recognition, highlighting the diverse motivations and ethical considerations within the mathematical community.

Advice for Aspiring Mathematicians and the Future of Mathematical Education

[190:18] Terence Tao: "So I think people have to get skills that are transferable, like learning one specific programming language or one specific subject of mathematics or something that itself is not a super transferable skill. But knowing how to reason with abstract concepts or how to problem solve when things go wrong. These are things which I think we will still need even as our tools get better."

Tao offers pragmatic advice to young students interested in mathematics, emphasizing the importance of transferable skills such as abstract reasoning and problem-solving. He advocates for leveraging modern tools like proof assistants and AI to enhance mathematical research and education. Tao envisions a future where experimental mathematics becomes more accessible and collaborative, encouraging broader participation from enthusiasts and professionals alike.

Final Reflections: The Intersection of Human Intelligence and AI

[198:55] Terence Tao: "Mathematics is so good at spotting connections between what you might think of as completely different problems."

In concluding the conversation, Tao shares his optimistic view of the future interplay between human mathematicians and AI. He believes that while AI can significantly aid in formalizing and verifying proofs, the creative and intuitive aspects of mathematical discovery remain distinctly human domains. Tao underscores the potential for meaningful collaboration, where AI tools amplify human capabilities without supplanting the irreplaceable aspects of human creativity and ingenuity in mathematics.

Notable Quotes:

• [09:57] Terence Tao: "What are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%?"

• [24:37] Terence Tao: "The key phenomenon that my technique exploits is what's called supercriticality."

• [49:32] Terence Tao: "Every time there's a symmetry in a physical system, there is a conservation law."

• [89:25] Terence Tao: "Lean is a computer language much like sort of standard languages like Python and C and so forth, except that in most languages, the focus is on producing executable code."

• [176:25] Terence Tao: "Human civilization, I think the younger generation is always really creative and enthusiastic and inventive."

Terence Tao's discussion provides a profound glimpse into the complexities of modern mathematical research, the transformative potential of AI in this field, and the enduring human spirit of inquiry and collaboration. His insights not only shed light on some of the most formidable challenges in mathematics and physics but also inspire a forward-thinking approach to integrating technology with human creativity.