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

Tao explains how the Kakeya problem extends from two to three dimensions, raising questions about minimal volume needed to rotate objects in space, and how this problem unexpectedly relates to wave propagation and the formation of singularities in nonlinear wave equations. This connection highlights the intricate interplay between pure mathematical puzzles and physical phenomena, such as fluid dynamics and wave concentration, which are central to understanding stability and blowup in equations like Navier-Stokes.

The Navier-Stokes Regularity Problem and Mathematical Physics

A significant portion of the discussion focuses on the Navier-Stokes equations, which govern fluid flow and remain one of the seven Millennium Prize Problems. Tao elaborates on the fundamental question of whether smooth initial conditions can lead to singularities—points where velocity or energy becomes infinite—in finite time. While everyday experience with fluids like water suggests such blowups do not occur, the mathematical possibility remains open and deeply challenging.

Tao shares his own research contributions, where he constructed an averaged version of the Navier-Stokes equations that does exhibit finite-time blowup. This result does not solve the original problem but serves as an obstruction, showing that any proof of global regularity must exploit features absent in the averaged model. He explains the delicate balance between dissipative forces like viscosity and nonlinear transport effects, and how the supercritical nature of Navier-Stokes at small scales complicates the analysis. This work has helped clarify why the problem is so difficult and guides mathematicians away from approaches doomed to fail.

Computational Universality and Fluid Dynamics

One of the most fascinating ideas Tao discusses is the conceptual leap from fluid dynamics to computation. Inspired by cellular automata like Conway’s Game of Life, Tao envisions the possibility of encoding computation within fluid flows, effectively creating a “liquid computer.” He describes how, by carefully engineering interactions in an averaged Navier-Stokes system, one can simulate logic gates and even a Turing machine, leading to a scenario where fluid dynamics could, in principle, perform arbitrary computations.

This analogy extends to the idea of a fluid-based von Neumann machine capable of self-replication, a concept with profound implications for both mathematics and physics. While this remains a theoretical construct far from practical realization, it provides a roadmap for understanding the complexity and unpredictability inherent in nonlinear PDEs and highlights the deep connections between computation, physics, and mathematics.

Infinity, Randomness, and Structure in Mathematics

Tao delves into the philosophical and technical challenges posed by infinity and randomness in mathematics. He discusses how infinity is an abstraction that simplifies mathematical reasoning but can also lead to paradoxes and pitfalls if not handled carefully. The infinite monkey theorem serves as a metaphor for randomness and pattern emergence, illustrating how infinite sequences almost surely contain every finite pattern, yet finite approximations require careful quantitative analysis.

He also explores the dichotomy between structure and randomness, a recurring theme in modern mathematics. Tao explains how inverse theorems help identify when an object exhibits genuine structure versus when it behaves like a random object. This framework underpins many results in additive combinatorics and number theory, such as Szemerédi’s theorem on arithmetic progressions, which holds both for highly structured sets like the odd numbers and for random subsets, revealing a profound unity in seemingly disparate phenomena.

Mathematics, Physics, and Engineering: Different Ways of Understanding

Tao offers a nuanced perspective on the relationship between mathematics, physics, and engineering as disciplines. He frames science as an interaction between reality, observations, and mental models, with mathematics focusing on the logical consequences of these models. Unlike other fields driven by goals and applications, mathematics often proceeds from axioms to conclusions, exploring hypothetical scenarios and their implications.

He highlights the symbiotic relationship between theory and experiment in physics, where models guide observations and observations refine models. Mathematics plays a crucial role in formalizing and understanding these models, but the ultimate test lies in their empirical adequacy. Tao also notes the growing role of experimental mathematics, facilitated by computers, which complements traditional theoretical approaches and opens new avenues for discovery.

The Unreasonable Effectiveness of Mathematics and Universality

A recurring theme in the conversation is the “unreasonable effectiveness” of mathematics in describing the physical world. Tao reflects on how complex systems often exhibit universal behavior at macroscopic scales, independent of microscopic details. He cites the central limit theorem as a fundamental example explaining the ubiquitous appearance of the bell curve in nature.

This universality allows for remarkable data compression, where vast amounts of observational data can be explained by simple mathematical models with few parameters. Tao emphasizes that understanding when universality holds and when it breaks down is crucial, citing the 2008 financial crisis as an example where systemic correlations invalidated Gaussian assumptions, underscoring the importance of rigorous mathematical modeling in real-world applications.

The Beauty and Elegance of Mathematical Proofs

Tao shares his appreciation for the aesthetic dimension of mathematics, particularly the elegance and craftsmanship involved in constructing proofs. He recounts how a talk by John Conway on “extreme proofs” influenced his view of proofs as objects with qualities like simplicity, length, and elementary nature, which can be optimized and appreciated beyond mere correctness.

He draws analogies between mathematical proofs and programming, noting that well-written proofs, like well-structured code, are easier to understand, generalize, and build upon. Tao also highlights the role of notation and conceptual clarity in revealing deep connections, exemplified by Euler’s identity, which unites fundamental constants and operations from different areas of mathematics in a single elegant formula.

Formal Proof Assistants and the Lean Programming Language

The conversation turns to the emerging role of formal proof assistants, with Tao describing Lean, a programming language designed to write and verify mathematical proofs with computer assistance. Unlike traditional programming languages focused on executable code, Lean produces formal certificates that guarantee the correctness of proofs, providing unprecedented rigor and reliability.

Tao explains how Lean requires explicit typing and detailed justification of every step, making the formalization process more pedantic but also more precise. He discusses the challenges and benefits of using Lean, including improved collaboration, error detection, and the ability to update proofs efficiently. Tao also notes the growing integration of AI tools to assist with lemma search and proof automation, signaling a transformative shift in mathematical practice.

AI and the Future of Mathematical Discovery

Tao offers a candid assessment of the current capabilities and limitations of AI in mathematics. He acknowledges impressive achievements like DeepMind’s AlphaFold and AlphaZero but points out the exponential difficulty in scaling AI to solve complex, multi-step proofs due to error accumulation. He highlights the challenge of translating informal mathematical language into formal code and the subtlety required to detect when AI-generated proofs go astray.

Despite these hurdles, Tao is optimistic about AI’s potential to assist mathematicians by generating conjectures, suggesting proof strategies, and automating routine tasks. He envisions a future where human mathematicians collaborate seamlessly with AI oracles that can verify, generate, and explore mathematical ideas, accelerating discovery while preserving human creativity and insight.

Iconic Mathematical Problems: Twin Primes, Riemann Hypothesis, and Collatz Conjecture

Throughout the discussion, Tao revisits some of the most famous open problems in mathematics. He explains the twin prime conjecture, which posits infinitely many prime pairs differing by two, and contrasts it with the Green-Tao theorem, which proves the existence of arbitrarily long arithmetic progressions of primes. Tao emphasizes the subtlety of the twin prime problem, noting that it is vulnerable to “conspiracies” where primes could be selectively removed to eliminate twin pairs, making the problem resistant to current statistical and combinatorial methods.

He also touches on the Riemann Hypothesis, describing it as a deep statement about the distribution of primes with far-reaching implications, including cryptography. Tao admits that this problem remains out of reach without a major breakthrough in mathematics. The Collatz conjecture is presented as a deceptively simple iterative problem involving halving even numbers and tripling odd numbers plus one, yet it remains unsolved despite extensive numerical evidence and partial results. Tao highlights the complexity and unpredictability of such problems, which often encode computational universality and undecidability.

Reflections on Mathematical Creativity, Collaboration, and Career

Tao reflects on the diverse styles of mathematical thinking, contrasting the “hedgehog” who specializes deeply in one area with the “fox” who explores many fields and seeks connections. He identifies himself as more of a fox, enjoying the challenge of adapting techniques across disciplines and finding unifying themes. Tao stresses the importance of collaboration, strategic problem simplification, and the willingness to “cheat” by turning off difficult aspects temporarily to gain insight.

He also discusses the emotional and psychological aspects of mathematical research, advocating for flexibility and the ability to move on from problems that become too frustrating. Tao admires mathematicians like Grigori Perelman, who pursued a solitary and intense path to solve the Poincaré conjecture, but he personally prefers a more balanced approach. He encourages young mathematicians to find their own workflow and to embrace the variety of ways in which mathematical talent can manifest.

Conclusion: The Future of Mathematics and Human Understanding

In closing, Tao expresses hope and excitement about the future of mathematics and human knowledge. He acknowledges the challenges posed by the increasing complexity of mathematical problems and the limitations of human cognition but sees great promise in the integration of AI, formal proof systems, and collaborative platforms. Tao envisions a future where mathematics becomes more accessible, inclusive, and dynamic, fueled by new tools and a global community.

He reminds us that mathematics is a language with which we describe the universe, and its power lies in its ability to reveal deep connections and universal truths.