What is the Navier-Stokes regularity problem and why is it so difficult to solve?

The Navier-Stokes regularity problem (a Millennium Prize Problem) asks if, starting with smooth fluid flow, velocity can ever become infinite (a singularity). It's difficult due to the 'Maxwell's demon' analogy, where unexpected conspiracies of energy transfer at small scales could lead to finite-time blow-ups that current mathematical tools struggle to rule out definitively.

How did Terry Tao use a 'liquid computer' concept to study the Navier-Stokes equations?

Tao engineered a simplified (averaged) Navier-Stokes equation that could be 'programmed' to create a finite-time blow-up. This involved creating a complex nonlinearity that mimicked a computational process, like a self-replicating von Neumann machine made of water, providing a roadmap for how blow-up might occur in the actual equations by ruling out certain proof strategies.

What is the 'structure versus randomness' dichotomy in mathematics that Terry Tao often studies?

This dichotomy refers to the observation that most mathematical objects appear random (like the digits of pi), while a small subset exhibit clear patterns. Tao's work on 'structure theorems' or 'inverse theorems' helps to classify when an object has structure or is related to a structured object, aiding progress in either case.

How do mathematicians 'finitize' infinite statements to gain intuition?

Mathematicians often idealize real-world phenomena by letting quantities go to infinity or zero, which simplifies the math. To gain intuition and avoid pitfalls, they 'finitize' these infinite statements, asking for quantitative rates in a finite context, like: 'how long do a large but finite number of monkeys need to type Hamlet?'

What is the difference between mathematics and physics as disciplines?

Science (including physics) deals with reality, observations, and constructing models. Mathematics, however, operates within the models. It starts from axioms and explores their consequences, tests predictions within the model, and determines its internal consistency, often serving as a 'compression' tool for vast amounts of scientific data.

What is Euler's Identity and why is it considered beautiful to mathematicians?

Euler's Identity (e^(iπ) = -1) is considered beautiful because it unifies seemingly disparate mathematical concepts: exponential growth (e), circles/rotation (π), rotation in a new direction (i, complex numbers), identity (1), and nullity (0). It connects dynamics, geometry, and complex analysis in one elegant statement.

What is the role of the Hamiltonian in physics, from classical to quantum mechanics?

The Hamiltonian represents the total energy in a physical system. While not initially obvious in Newtonian mechanics where force and mass were primary, it became clear to be a fundamental concept capable of completely describing system dynamics. In quantum mechanics, a different type of Hamiltonian (an operator) still governs evolution, serving as a unifying concept and linking symmetries to conservation laws (Noether's theorem).

Is it possible to unify general relativity and quantum mechanics into a 'theory of everything'?

Tao believes it is possible, citing the historical trend of unification in physics (e.g., electricity and magnetism). The main challenge is the extreme accuracy of both theories; deviations only appear at very high energies or in the early universe, making experimental data scarce. A new mathematical language, analogous to the Hamiltonian, is likely needed to describe space-time at quantum scales.

How does Terry Tao approach solving difficult mathematical problems?

Tao employs a strategy he calls 'strategic cheating.' When faced with a complex problem, he simplifies it by temporarily ignoring most difficulties (like setting dimensions to one or assuming error terms are zero). He solves these simplified versions one by one, learning from each, before gradually reintroducing complexity. He likens it to a choreographed fight scene where a hero tackles enemies one at a time.

How is the Lean formal proof programming language changing mathematics?

Lean allows mathematicians to write proofs in a formal language that can be verified by computers, ensuring 100% correctness. This enables unprecedented collaboration (trustless mathematics), atomizes proof steps for easier debugging, and facilitates large-scale projects like the 'Equation Theories Project.' It is catalyzing a shift towards more experimental and collaborative mathematics.

What is the Fields Medal and why did Grigori Perelman decline it?

The Fields Medal is a prestigious award in mathematics. Grigori Perelman famously declined both the Fields Medal and the Millennium Prize (including $1 million) for his proof of the Poincaré conjecture. He stated he was not interested in money or fame, believing that if the proof is correct, no other recognition is needed, reflecting a high ideal of pure mathematical pursuit.

What is the Collatz conjecture and why is it so famously difficult?

The Collatz conjecture states that if you take any positive integer, apply a rule (if even, divide by 2; if odd, multiply by 3 and add 1), and repeat, you will always eventually reach 1. It's notoriously difficult because despite its simple rules, the sequences behave like a random walk with a downward drift but exceptions could exist.

What advice does Terry Tao give to young students struggling with math?

Tao advises students to explore diverse sources of mathematical enrichment outside the classroom, such as YouTube videos, math puzzles, and hobbyist communities (e.g., poker, chess, baseball) where math is applied. He believes that finding practical, enjoyable applications that resonate with different learning styles (visual, symbolic, game-oriented) can make math fun and accessible.

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

Lex Fridman

Science & Technology7 min read195 min video

Jun 14, 2025|1,768,151 views|24,337|1,861

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TL;DR

Terence Tao discusses hard math problems, the future of AI in mathematics, and the nature of discovery.

Key Insights

The Navier-Stokes regularity problem, a Clay Millennium Prize, remains unsolved due to "Maxwell's demon" type conspiracies where fluid energy could theoretically concentrate infinitely, despite not being observed in reality.

Tao engineers a "liquid computer" thought experiment as an obstruction proof for averaged Navier-Stokes, demonstrating how designed nonlinearities can force energy blow-up, thereby guiding future proofs for the real equations.

Mathematics often unifies seemingly disparate fields (e.g., geometry and number theory, algebra and geometry) by finding common underlying structures and representations.

The rise of AI and formal proof assistants like Lean could revolutionize mathematics by enabling large-scale collaboration, automating routine checks, and potentially helping in proposing new conjectures and laws of physics.

Problems like the Twin Prime Conjecture and Riemann Hypothesis remain incredibly difficult because current mathematical tools struggle to prove "randomness" or rule out subtle "conspiracies" in number patterns, highlighting limitations in current methods.

Human intuition, creativity, and the ability to invent new theories and abstractions are still paramount in mathematics, though AI offers significant potential to augment these capabilities through verification and complex computation.

CHALLENGES IN FLUID DYNAMICS AND THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Terence Tao begins by discussing challenging mathematical problems, highlighting the incompressible Navier-Stokes equations that govern fluid flow. He explains the million-dollar Clay Millennium Prize problem of their global regularity: whether smooth initial fluid states can ever develop singularities (infinite velocity). While such blow-ups are not observed in reality, mathematicians seek 100% certainty. The difficulty lies in what he likens to "Maxwell's demon"—the theoretical possibility of highly improbable conspiracies where energy might concentrate infinitely, making absolute proof elusive. These equations are vital for practical applications like weather prediction and understanding turbulence.

ENGINEERING A LIQUID COMPUTER FOR BLOW-UP SCENARIOS

To understand the Navier-Stokes problem, Tao designed a simplified, 'averaged' version of the equations that he could force to blow up. He envisioned a 'liquid computer'—a fluid-based Turing machine capable of self-replication. This hypothetical machine would encode information in water patterns, transfer energy to increasingly smaller, faster versions of itself, and ultimately lead to a singularity. This thought experiment provided an "obstruction" proof: if his engineered equations blow up, then any successful proof for the actual Navier-Stokes equations must leverage features his averaged model lacks, thereby ruling out certain proof strategies.

SUPERCRITICALITY AND THE LIMITS OF PREDICTION

Tao explains that the Navier-Stokes equations are 'supercritical,' meaning that at smaller scales, the nonlinear 'transport' terms (which cause unpredictability and turbulence) dominate the linear 'dissipation' terms (which calm things down). This criticality is key to why weather prediction is limited to about two weeks; the fine-scale interactions become overwhelmingly complex and nonlinear. In contrast, 'critical' or 'subcritical' equations (like 2D Navier-Stokes or planetary motion) are more predictable because forces balance or dissipation dominates, allowing for global regularity proofs. Understanding this distinction helps identify which systems are inherently prone to chaotic behavior and potential blow-ups.

MATHEMATICS, PHYSICS, AND THE ROLE OF AXIOMS

Distinguishing mathematics from physics, Tao highlights that science deals with real-world observations and models, while mathematics focuses on the models themselves, exploring their logical consequences from defined axioms. Mathematics often idealizes concepts (e.g., infinity, zero thickness) to simplify problems and gain clearer insights, though this risks deviating from physical reality. The "unreasonable effectiveness of mathematics" suggests deep connections, often seen in 'universality'—how complex systems exhibit simple, predictable laws at macro scales (like the bell curve in the Central Limit Theorem) due to underlying universal principles, rather than micro-level complexity.

THE UNITY OF MATHEMATICS AND THE HEDGEHOG VS. FOX APPROACH

Mathematics is characterized by ongoing unification, like how Descartes connected geometry and number theory. Tao identifies as a 'fox' mathematician, preferring to learn diverse techniques from one field and apply them to seemingly unrelated problems, arbitrating connections. This contrasts with 'hedgehog' mathematicians, who specialize deeply in a single field. Collaboration often benefits from both styles. His preferred method involves re-proving results he admires using tools he is more familiar with, even if the proof is less efficient, as it aids in understanding and finding new connections—a highly exploratory and often self-teaching process.

THE CRAFTSMANSHIP OF MATHEMATICAL PROOF AND ELEGANT EQUATIONS

Tao recounts John Conway's concept of 'extreme proofs,' optimizing proofs for various qualities like elegance or brevity, which significantly influenced his own approach. Beyond mere correctness, proofs should be readable, well-motivated, and adaptable. Comparing it to coding, he notes the value of writing clear, maintainable code versus 'spaghetti,' illustrating the craftsmanship in proof writing. Euler's identity (e^(iπ) = -1) stands out for its beauty, as it elegantly unifies disparate mathematical concepts: exponential growth (e), rotation (π), and imaginary numbers (i), revealing deep connections between dynamics, geometry, and complex numbers.

FORMAL PROOF ASSISTANTS AND THE FUTURE OF MATHEMATICS WITH AI

Tao extols the potential of formal proof assistants like Lean, which not only compute but also generate verifiable proofs. While currently requiring significantly more effort than traditional pen-and-paper proofs (roughly 10x), Lean's ability to automate verification, pinpoint errors, and facilitate atomic-level collaboration with dozens of contributors is transformative. He envisions a future where formalized proofs are standard, expedited by AI-powered tools (like advanced autocomplete) that assist in lemma discovery and error localization, allowing mathematicians to focus on novel ideas rather than verifying minute details. This fosters a 'trustless mathematics' paradigm, expanding collaboration beyond traditional academic boundaries.

AI'S ROLE IN MATHEMATICAL DISCOVERY: FROM VERIFIER TO CONJECTURE GENERATOR

While current AI systems like DeepMind's AlphaProof can tackle high-level math problems, they still struggle with the exponential complexity of research-level proofs, often making subtle yet 'stupid' errors. Tao believes AI's greatest impact will be in 'experimental mathematics'—large-scale exploration of patterns, data, and even proofs (e.g., his 'equation theories' project). Moving beyond verification, he is hopeful AI could generate novel, meaningful conjectures, acting as a 'sense of smell' for mathematical viability, a capability currently exclusive to humans. This would require AI to learn from both successful and 'failed' human mathematical journeys, data that is rarely recorded.

THE POINCARÉ CONJECTURE AND PERELMAN'S ISOLATED TRIUMPH

Tao explains the Poincaré Conjecture, a Clay Millennium Prize problem solved by Grigori Perelman, which classifies 3D spaces as spheres if every loop within them can be contracted to a point. Perelman's breakthrough leveraged Richard Hamilton's Ricci flow, a process to smooth out curved spaces. He transformed the 'supercritical' Ricci flow into a 'critical' one by introducing new quantities (Perelman's reduced volume and entropy), allowing him to classify and 'perform surgery' on complex singularities. Perelman's seven-year solitary work, declining major awards, highlights a unique, almost monastic approach to profoundly difficult problems, a stark contrast to Tao's collaborative 'fox' style.

THE MYSTERY OF PRIME NUMBERS: RANDOMNESS AND CONSPIRACIES

Prime numbers are 'multiplicative atoms' of mathematics. While their additive behavior is understood, combining addition and multiplication (e.g., in the Twin Prime Conjecture about primes differing by two) leads to extreme difficulty. Tao explains that twin primes are 'fragile'; a minimal 'conspiracy' of edited primes could eliminate them while statistically appearing normal. This means any proof needs extremely delicate tools beyond aggregate statistical analysis. In contrast, 'arithmetic progressions' (sequences of numbers with a constant difference) are 'indestructible,' appearing in both random and structured sets of numbers. His work on the Green-Tao theorem proved that primes contain arbitrarily long arithmetic progressions.

THE COLLATZ CONJECTURE: DECEPTIVE SIMPLICITY AND UNPREDICTABLE PATHS

The Collatz Conjecture (if even, divide by two; if odd, multiply by three and add one) is famously simple to state but incredibly difficult. Iterating numbers often leads to a 'hailstone' sequence that eventually reaches one. Tao's work shows that statistically, a vast majority of numbers trend downwards. However, like Navier-Stokes, the challenge lies in ruling out outlier numbers that might 'fly off to infinity.' He discusses analogies to Conway's Game of Life, suggesting that complex rules in a simplified system can generate undecidable behaviors, implying a similar deep, almost computational complexity within the Collatz problem itself. Bridging this 'parity barrier' to prove randomness remains a significant challenge in number theory.

THE FIELDS MEDAL AND THE EVOLUTION OF A MATHEMATICIAN

Winning the Fields Medal transformed Tao from a researcher to an 'establishment' figure, bringing increased administrative duties and public engagement. Despite this, he continues to embrace new mathematical fields and tools like Lean, viewing them as essential for efficiency. His commitment to learning and adapting is driven by a desire to overcome the 'inefficiencies' of traditional mathematics, such as routine computations or literature review. He contrasts his accelerated, fox-like career path with the more traditional, specialized routes, emphasizing the need for diverse styles and personalized learning in mathematics education. He finds hope in the creativity of younger generations and the potential for science to trivialize previously intractable problems, making complex knowledge accessible.

Mentioned in This Episode

●Software & Apps

●Tools

●Companies

●Organizations

●Studies Cited

●Concepts

●People Referenced

Common Questions

The Kakeya problem asks for the minimum area (or volume in 3D) needed to turn a needle around to face every direction. It's surprisingly connected to partial differential equations (like wave propagation), number theory, and geometry, impacting our understanding of phenomena like wave concentration and singularities.

Topics

Navier-Stokes Equations Kakeya Problem Poincaré Conjecture Prime Numbers Riemann Hypothesis Twin Prime Conjecture Proof Assistants Lean Programming Language Unification Of Forces Philosophy Of Mathematics Hilbert's Problems Euler's Identity Fluid Dynamics Computational Mathematics

Mentioned in this video

Concepts

Fermat's Last Theorem

A famous theorem stating that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2; its proof by Andrew Wiles was a major mathematical event and is being formalized in Lean.

Conway's glider

A small, stable pattern in Conway's Game of Life that moves across the grid, analogous to a vortex ring in fluid dynamics.

Twin primes conjecture

An unsolved problem in number theory which posits that there are infinitely many pairs of prime numbers that differ by two.

Vortex rings

Candidates for basic logic gates in the hypothetical 'water punk' computer, analogous to gliders in Conway's Game of Life.

Kakeya problem

A geometric problem about turning a needle around in the smallest possible area (or volume in higher dimensions). Tao worked on its three-dimensional generalization related to wave propagation.

Von Neumann machine

A self-replicating machine, concept discussed in the context of colonizing Mars, and then envisioned as a fluid analog by Tao for the Navier-Stokes blow-up.

Riemann hypothesis

One of the unsolved Millennium Prize Problems related to the distribution of prime numbers, particularly how they behave multiplicatively as randomly as possible.

Digits of Pi

Used as an example of a sequence of numbers that is believed to have no discernible pattern, representing randomness in mathematics.

Navier-Stokes regularity problem

A famous unsolved Millennium Prize Problem concerning the existence and smoothness of solutions to the Navier-Stokes equations, asking if fluid velocity can ever become infinite (a singularity).

Szemerédi's Theorem

A theorem from the 1970s stating that any sufficiently large set of numbers with positive density contains arithmetic progressions of any length.

Infinite Monkey Theorem

The popular version states that an infinite number of monkeys typing randomly will almost surely produce any given finite text, like the complete works of Hamlet, over infinite time.

Plato's Cave Allegory

An allegory used to illustrate how humans perceive shadows of reality rather than reality itself, prompting a discussion on whether mathematicians and scientists truly access reality through their models.

E = mc²

Einstein's famous equation relating energy and mass, cited as an example of mathematical beauty in simplicity and its 'unreasonable effectiveness' in describing the universe.

Avogadro's number

Used to illustrate the immense number of particles in a system, making direct tracking impossible and highlighting the need for 'universality' in emergent laws.

Central Limit Theorem

A theorem explaining the ubiquitous presence of the bell curve (Gaussian distribution) in natural phenomena, showing how average behavior emerges from many independent random variables.

Hamiltonian mechanics

A reformulation of Newtonian physics where energy (the Hamiltonian) is the central object, which was crucial for the development of quantum mechanics.

Schrödinger's equation

A fundamental equation in quantum mechanics that describes how quantum systems evolve once their Hamiltonian is specified, showing the importance of the Hamiltonian in both classical and quantum physics.

Noether's Theorem

A fundamental theorem connecting symmetries in physical systems to conservation laws, applicable to both classical and quantum mechanics.

Poincaré conjecture

One of the Millennium Prize Problems, solved by Grigori Perelman, asking if every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Ricci flow

A geometric flow used to deform a Riemannian manifold, aiming to smooth out irregularities and eventually yield a standard shape like a sphere; central to Perelman's proof of the Poincaré conjecture.

Perelman's reduced volume and entropy

New quantities introduced by Perelman to transform the supercritical Ricci flow problem into a critical one, simplifying the analysis of singularities in the Poincaré conjecture.

Green-Tao theorem

A theorem proving that prime numbers contain arithmetic progressions of arbitrary length, a significant advancement in understanding the additive structure within primes.

Goldbach Conjecture

An open problem in number theory stating that every even integer greater than 2 is the sum of two prime numbers; mentioned as a 'next-door neighbor' to the Twin Prime Conjecture in difficulty.

Collatz conjecture

A notoriously difficult unsolved problem in mathematics, simple to state (if even, divide by two; if odd, multiply by three and add one; iterate this process) but extremely hard to prove it always reaches 1. Tao has made progress on its statistical behavior.

P vs NP problem

A major unsolved problem in computer science asking whether every problem whose solution can be quickly verified can also be quickly solved; considered a 'meta problem' with wide ripple effects.

Conway's Game of Life

Studies & Research

Finite-Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation

Paper published by Terry Tao in 2016, demonstrating how to engineer a blow-up in a modified (averaged) Navier-Stokes equation to understand obstructions to proving global regularity for the full equation.

People

John Horton Conway

Mathematician known for Conway's Game of Life and for his unique way of thinking about proofs and mathematical systems, including generalizations of the Collatz problem.

Soichi Kakeya

Japanese mathematician who proposed the original Kakeya needle puzzle around 1918.

Tim Gowers

Fields Medal-winning mathematician and colleague of Tao, who called Tao the 'closest thing we get to Hilbert'.

James Clerk Maxwell

Physicist who unified theories of electricity and magnetism, cited as a historical example of unification in physics.

Bernhard Riemann

Mathematician who developed Riemannian geometry, which proved to be exactly what Einstein needed for his theory of curved space-time in general relativity.

DHJ Polymath

A fictional pseudonym used for publications of early Polymath projects, in the spirit of Bourbaki, to make all authors equal in credit.

Nicolas Bourbaki

Pseudonym for a famous group of mostly French mathematicians in the 20th century who wrote a series of influential mathematics textbooks.

Kevin Buzzard

Mathematician known for his work in formalizing proofs with Lean, suggesting it might be the future of mathematics; also leading a project to formalize Fermat's Last Theorem.

Peter Scholze

Mathematician whose formalized theorems were a point of discussion for a workshop Tao helped organize, highlighting developments in computer-assisted proof.

Jordan Ellenberg

Mathematician who collaborated with Tao and Kevin Buzzard on a workshop about computer-assisted proofs.

Richard Hamilton

Mathematician who proposed the Ricci flow approach to solving the Poincaré conjecture, a partial differential equations method to smooth out curved spaces.

Ben Green

Collaborator with Tao on the Green-Tao theorem, known for his work on arithmetic progressions in prime numbers.

Andrew Wiles

Princeton professor who famously proved Fermat's Last Theorem, a significant event during Tao's graduate studies.

Events

2008 Global Financial Crisis

Cited as an example where reliance on Gaussian models for mortgage defaults failed due to systemic correlations, highlighting limitations of mathematical models when assumptions break down.

Software & Apps

Lean

A formal proof programming language that allows mathematicians to write proofs that can be verified by a computer, producing certificates of correctness and enabling atomic-level collaboration.

DeepMind's AlphaProof

An AI system trained using reinforcement learning on formal Lean proofs of IMO problems, demonstrating AI's capability in high-level mathematics.

Wolfram Alpha

A computational knowledge engine that can solve many undergraduate-level math tasks, demonstrating early automation in mathematics.

LaTeX

A typesetting language widely adopted by mathematicians for writing papers, cited as a past 'phase shift' in mathematical workflow.

Organizations

Equation Theories Project

A large-scale crowdsourcing mathematics project led by Tao to classify relationships between 4,000 algebraic laws, generating 22 million problems, with crucial help from Lean for verification.

Polymath projects

Early crowdsourcing mathematics projects without the Lean component, limited by the need for human moderators to verify contributions.

Companies

GitHub

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