If ten things make a problem hard, Terence Tao turns nine of them off

Lex Fridman asks Terence Tao for the first research-level problem that gave him pause. Undergraduate lists of famous impossibilities—the Riemann hypothesis, the twin-prime conjecture—do not count, Tao says, because you can make problems arbitrarily hard, and some are even known to be unsolvable. What caught him as a PhD student was a question sitting on the boundary: existing techniques already did about 90 percent of the work. That question was the Kakeya problem, and it has now actually been solved.

The puzzle goes back to the Japanese mathematician Soichi Kakeya around 1918. An infinitely thin needle lies in the plane; you must reverse its direction while sweeping as little area as possible. Spinning it about its center traces a disk of area π/4. A three-point turn, the kind driving schools teach, uses π/8 and for a while looked optimal. Then Besicovitch showed you can turn the needle using as little area as you want—0.001, or smaller—by a fancy sequence of back-and-forth maneuvers that still pass through every intermediate direction.

In three dimensions the picture changes. Imagine the Hubble Space Telescope as a tube that must point at every star, except that space is somehow at a premium. If the tube has zero thickness, a modification of Besicovitch still lets you use as little volume as you like. If it has a tiny thickness delta, the volume should fall as delta shrinks, but very slowly, roughly logarithmically. After a great deal of work that conjecture was proved. The reason anyone cared is that Kakeya tubes reappear in wave packets, number theory, geometry, and combinatorics. A dispersed wave can focus onto a point, defocus, and in a negative Kakeya world might concentrate so violently that the governing equation stops being a wave equation at all.

That focusing picture leads straight into a Clay Millennium Prize problem. The incompressible Navier-Stokes equations govern water; the prize asks whether a smooth velocity field can ever concentrate until the speed becomes infinite at a point. Nobody sees that in a bathtub. In recent years, Tao says, expert consensus has drifted toward the belief that special initial data might form a singularity, but nobody has proved it. Of the seven million-dollar problems, only the Poincaré conjecture, settled by Grigori Perelman, has been solved.

This is the difference between mathematicians and almost everyone else. If something holds 99.99 percent of the time, that is good enough for most work. Mathematicians still want to know whether 100 percent of cases are covered. Compressible Navier-Stokes, which governs air, is already at the center of weather prediction; the incompressible prize problem is the cleaner mathematical cousin of a computation that agencies run every day. The short reason the regularity question is hard, Tao says, is Maxwell's demon.

Imagine oxygen and nitrogen mixing in a box. They should stay mixed, yet in principle a microscopic conspiracy at every collision could sort them apart again. The digits of π look patternless, and everyone believes they are, but a demon in the expansion could still bias one digit over another. Fluid energy is usually spread out and eaten by viscosity; a demon that kept packing it into smaller, faster eddies could produce a self-similar cascade, each transfer taking half as long, until everything blows up in finite time. Real water is turbulent: one eddy splits into three or four, not one, and viscosity wins. The question is whether a sufficiently clever initial blob can keep the energy together long enough to explode.

Many papers have claimed that conservation of energy plus viscosity should give global regularity not just for Navier-Stokes but for a whole family of equations. They failed, usually on a sign error that could not be patched. Tao could not engineer a blowup in the true fluid equations, which were too complicated, so in 2016 he published *Finite Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation*. He averaged the interactions: if a large eddy could dump energy into this small eddy or that one, he closed one channel and kept the other, while still conserving energy. He made the fluid explode by changing the laws of physics, which mathematicians are allowed to do.

The construction is an obstruction, not a solution. Turning interactions off usually makes an equation *more* regular. Tao found a carefully chosen set of deletions that still forced a finite-time blowup. Anyone who wants to prove the real Navier-Stokes equation stays smooth must therefore use some feature his fake equation does not have. Hard problems often come with a dozen methods that look applicable; counterexamples on nearby equations save years by showing which methods can never work.

The feature he was exploiting is supercriticality. Navier-Stokes is a tug of war between linear viscous damping and nonlinear transport. At smaller and smaller scales, transport dominates viscosity, which is why the three-dimensional problem is hard. In two dimensions Olga Ladyzhenskaya proved in the 1960s that there is no blowup; there the equation is critical, and the two forces stay comparable even at tiny scales. Critical and subcritical equations have a large regularity toolkit. Once nonlinear effects dominate at fine scales, Tao and later authors constructed blowups for many supercritical cousins. That is also why planetary orbits can be forecast for ages—planets are almost point masses—while weather two weeks out remains a bet. You cannot describe Los Angeles by one temperature and one wind speed; the fine-scale information is the equation.

A naive averaged blowup just shoves energy to the next smaller scale as fast as possible. That works in five and higher dimensions. In three dimensions Tao found that the energy smears across many scales at once, which gives viscosity an opening. He needed airlocks: push the energy down one scale, wait until the larger scale is empty, then open the next gate. The nonlinearity he built to enforce that delay looked like an electronic circuit. He thanks his wife, trained as an electrical engineer, for talking him through capacitors, resistors, and clocks. The result was a Rube Goldberg machine written as mathematics, and it worked for the averaged equation.

The same logic suggests a program for the true Navier-Stokes equations. If pulses of water could collide like bits, you might build AND gates, then a Turing machine, then a fluid von Neumann machine: a water robot whose job is to assemble a smaller copy of itself, dump all its energy into the copy, power down, and repeat. The equation has a scaling symmetry, so each generation would run smaller and faster, and in finite time you would have a blowup. Tao calls this a pipe dream. He does not have the gates, analog error correction is nasty, and he does not know how to shut the large machine off cleanly. Nothing in the construction contradicts physics, which is evidence that blowup is possible. Other groups are chasing much simpler, nearly self-similar scenarios that might work without the steampunk.

The precedent is Conway's Game of Life. Amateur communities found gliders, glider guns, logic gates, Turing machines, and even self-replicating monsters, all from three or four local rules. You cannot dump those digital gadgets into a continuum, but they show that emergence of computation from simple dynamics is possible. The catch, Tao says, is preparation. Random cells do not build glider guns, and typical fluid data will not compute. The recurring challenge is a dichotomy between structure and randomness: most objects look random, a few have patterns, and inverse theorems test which is which. Either the object has no structure, or it is close to something structured, and in both cases you can move.

Szemerédi's theorem, proved in the 1970s, is the dichotomy in action. Any set of integers with positive density contains arithmetic progressions of every finite length. The odd numbers make this obvious because they are structured; a random coin-flip subset of the integers has no visible pattern and still contains those progressions by fluctuation. That is the infinite-monkey theorem in number-theory clothes: infinitely many monkeys at typewriters will almost surely produce *Hamlet*. Infinity, Tao says, absorbs a lot of sins.

Nothing in life is truly infinite. Mathematicians treat infinity as an idealization of a finite quantity with no available bound—spherical-cow thinking that often makes the algebra cleaner. It also creates traps. Finite sums may be rearranged freely; infinite series can be shuffled into a different sum, which is why undergraduates spend a year on epsilons and deltas. In recent years people have started finitizing infinite theorems: not just “eventually,” but “how long.” A large finite troop of monkeys waits a time exponential in the length of the target text, which is why you never see *Hamlet* and might see a four-letter word.

Once an infinite statement is finitised, Tao finds it much more intuitive. The cost is mess. The infinite version is usually discovered decades earlier; the quantitative rate comes later, uglier, and closer to the finite world we actually compute in.

Science, as Tao frames it, is an interaction among three things we never quite collapse: the real world, our incomplete observations, and the mental models we write down. Mathematics lives inside the model. It asks what the axioms imply, what the model predicts, and whether those predictions fit the data. Almost every other discipline starts from a desired conclusion—build a bridge, make money, forecast an asteroid—and searches for a path. Speculative fiction is one of the few other places that routinely asks “what if.” Physics still wants tomorrow’s weather; mathematics also runs the arrow from hypotheses outward.

Inside mathematics itself, theory has dominated. Tao puts the split at something like 99 percent theoretical and a sliver of experiment, even though Gauss was already hiring human computers to tabulate primes and guess the prime-number theorem. Combinatorial explosion is why brute force still fails: the subsets of {1, …, 1000} number 2^1000, more than any machine will enumerate, and chess has the same disease. AI now explores those spaces without guarantees. Chess engines have overturned opening dogma empirically, without a formal explanation of why a position is good. Tao hopes mathematics will grow a larger experimental wing, perhaps powered by the same tools.

Reality, observations, and models remain distinct, he thinks, but they can approach one another if we are willing to throw away first intuitions. Astronomy is the exhibit: a flat earth fitting the first glance, then a spinning ball, then an expanding universe whose acceleration may not even be constant. A useful theory is a compression. Ten parameters fitting ten data points is overfitting; a dark-matter model with about 14 parameters fitting petabytes is the opposite. Einstein’s line, as Tao recalls it, is that the most incomprehensible thing about the universe is that it is comprehensible. Universality is a mathematical hint at why: quintillions of particles in a box collapse to temperature, pressure, and a handful of knobs. The central limit theorem explains the bell curve when inputs are weakly dependent, and the 2008 financial crisis is what happens when they are not. Beautiful Gaussian copulas did not save anyone from a systemic shock.

Tim Gowers, a Fields Medalist and a friend, has called Tao the closest thing we have to Hilbert. Asked whether mathematics has an underlying structure, Tao points to unification as the actual engine of progress: geometry and number theory were separate until Descartes parameterized the plane; algebra and geometry later met in algebraic geometry. He identifies as a fox. Hedgehogs know one thing extremely well; foxes know many things a little and arbitrage, carrying a trick from a field that has it into a field that does not. Ideal collaborations mix the two. When Tao sees a theorem whose proof uses tools he dislikes, he often reproves it with his own, usually worse, until he can see what the original argument was trying to do.

As a Princeton graduate student he heard John Conway lecture on “extreme proofs.” Conway treated the space of all proofs of a theorem as a shape, and hunted the extreme points: the shortest, the most elementary, the most elegant, each at the expense of everything else. That, Tao says, taught him that a proof is craft. Undergraduate homework rewards any argument that ticks the box. Influential mathematics also has to be a pleasure to read, motivated, and ready to generalize, the same way good code is not spaghetti. Code golf looks frivolous and still produces insights you would not find without the artificial constraint.

Beauty, for Tao, is connection rather than a pile of famous constants. Euler’s identity is cute as a reunion of e, π, and i; what he loves is that exponential growth, rotation, and a right-angle twist in the complex plane become the same motion. Collision of notation is confirmation that you named the right objects. Physics had to learn that lesson with energy. Newton billed force, mass, and acceleration; only later did conserved quantities take center stage, and Hamiltonian mechanics made the Hamiltonian the object that specifies the entire dynamics. That hidden centrality is why the jump to quantum mechanics was possible: Schrödinger evolution is again a Hamiltonian, now an operator, and Noether’s link between symmetry and conservation survives the change of costume.

Unifying quantum mechanics with general relativity, Tao believes, will happen because physics has always unified—Maxwell, Newton, the rest—but the two current theories already cover 99.9 percent of observations. You have to go to insane accelerators or the early universe to see a discrepancy. When Einstein needed curved space, Riemann’s geometry was waiting; Wigner’s “unreasonable effectiveness” is that good physical theories and good mathematical objects keep organizing data the same way. String theory, a leading candidate for decades, is in Tao’s view slowly falling out of fashion because it is not matching experiment. Analogies become the only way back to intuition: a basketball, a golf ball, and a lamp still explain eclipses. With Grant Sanderson he recently made a video on how the ancient Greeks measured the distance to the moon with techniques you can copy at home. Modern science is a victim of its own success; the further it moves from first glance, the more it needs grounding.

Tao’s own work on Einstein’s neighborhood is the wave-maps equation, a sigma-model sitting lower on the hierarchy than gravity itself. Think of arrows planted on a sphere at every point of space, swaying like wheat. In 2000 he proved a global-regularity result for this critical equation: energy cannot be forced entirely onto a point, and once it disperses a little it stays smooth. The nonlinearity coming from the sphere’s curvature was non-perturbative; it looked larger than the linear wave effects even at small energy. He invented a gauge transformation, imagining tiny cameras that ride with the motion until the remaining flow looks almost linear. He found that transformation while visiting his aunt in Australia, eyes closed on the floor, pretending to be the vector field. She walked in, asked what he was doing, and decided not to inquire further.

The general method is what he calls cheating strategically. An engineer told to bridge a river cannot relocate the river or build in paper; a mathematician can change the problem at will, like a game with unlimited cheat codes. Set a large dimension to one. Assume the error term is zero. If ten things make your life difficult, turn nine of them off and keep one, then rotate. As a child he watched Hong Kong action movies in which a hundred goons somehow fight the hero one at a time. If they had swarmed, the cinema would have been worse and the hero would have lost. You do not solve a reasonable math problem in Iron Man mode.

He still works mostly with pen, paper, and four giant blackboards, covering them with symbols and doodles only he can read, then sitting on the couch to see the whole wall. Computers are creeping in because AI has collapsed the friction of small coding tasks. Plotting a moderately iterated function used to cost two hours of Python archaeology; it now takes ten or fifteen minutes. The explorations are still simple. The blackboards remain the place where a problem becomes visible.

Lean looks like Python or C, except that besides running it can emit a proof object. A Python program may tell you that three plus four is seven; Lean can attach a certificate of every step. Proof assistants are not new, but mathematicians helped design Lean so that a line of code resembles a line of argument. In practice it is like explaining a proof to an extremely pedantic colleague: did you mean this, what if this is zero, what type is X? The type inference is old-fashioned tree-matching, not a language model, and the kernel is kept small so that several compilers can check the same development. Modern AI sits on top, mostly as fancy autocomplete. GitHub Copilot, scanning Tao’s Lean buffer, is exactly right about 25 percent of the time, close enough to fix another 10 to 15 percent, and rubbish about half the time. He estimates that formalizing a proof still costs about ten times the effort of writing it by hand.

The pedantry pays when a constant moves. One formalized theorem produced a 12 that had to be consistent through the whole argument; a later paper improved it to 11. On paper you recheck every line, terrified of a hidden use of the old number. In Lean they changed the headline 12 to 11, recompiled, and watched 90 percent of thousands of lines stay green. The red islands told them exactly what to patch. With decent programming hygiene the repair took a day or two instead of three weeks and twenty people. Hovering a cursor also beats flipping back ten pages for a definition. The real social change is atomic collaboration. Tao can get stuck at line 67, paste three failing lines, and receive a trick from someone he has never met. Lean supplies the trust, so he can do mathematics with unreliable strangers.

The Green–Tao theorem was the old style of collaboration: Ben Green had length-three progressions in the primes; Tao had ergodic-theory machinery for longer ones; they bargained until they found a randomness hypothesis Green could prove and Tao could use. Lean blueprints are a different story, a supply chain in which each node of a graph is a self-contained lemma. Most current Lean work still formalizes theorems humans already proved. The Equational Theories Project shows what happens when you scale. About 4,000 algebraic laws generate some 22 million implication questions, far past the literature’s previous record of around 15 equations. No person can check that many proofs, so the project had to be computerized. About 50 contributors—huge by mathematics standards—have settled all but two pairs, and those two already have pen-and-paper proofs being formalized. A decade earlier the Polymath projects used a Bourbaki-style pseudonym, DHJ Polymath, which then failed both tenure committees and credit: outsiders named the most famous participant and forgot the other nineteen. This time everyone is an author, with a self-reported contribution matrix, and Tao does not want GitHub line counts anywhere near a tenure file. Goodhart’s law still applies.

DeepMind’s AlphaProof trained by reinforcement learning on successful and failed Lean proofs of IMO problems—very high-level high school mathematics—and reached the equivalent of a silver medal. Tao is impressed and unsentimental. Difficulty explodes with the number of steps: a 10 percent chance of a wrong turn at each of 20 steps almost never reaches the end. Mapping informal English onto a formal language is itself unsolved, and even translating between Lean, Coq, and Isabelle is unsolved. The system burned on the order of three days of Google server time per high-school problem, used extra time and human help to formalize statements, and does not scale. Numerical-answer contests are a better RL fit because the grade is a bit; long-form informal proofs would need armies of human graders.

What humans still do, Tao argues, keeps shifting. Computing Easter, spherical navigation, and a pile of undergraduate tasks were automated long before language models; Wolfram Alpha already ate a lot of that. Asking a system to try twenty standard PDE techniques on a hundred nearby problems is the kind of scaling he expects to work. Where models fail is noticing a wrong turn. Human crank proofs smell bad immediately. LLM mathematics can look superficially flawless—reinforcement learning trained it to look correct—and then fail in a stupid way no person would attempt. Programmers know the same odorless code. AlphaZero grew a sense of smell for chess and Go positions it cannot explain. If an assistant could tell a helpful split from two subproblems harder than the original, it might start competing with research mathematicians. Random attempts almost always make a problem worse.

Tim Gowers, in 2000, already imagined a 25-years-later dialogue in which human and assistant trade ideas and the machine checks a hundred cases or flags n = 46. Tao has tried that conversation on problems whose answers he already knew. It felt like herding cats: the model wandered back to its favorite tool, he had to argue with line 17, and the labor of prompting plus checking was more exhausting than working alone. He still expects a phase shift. Formalization effort is falling from ten times, to nine, to eight; the day it drops below one, writing in Lean first becomes rational, journals can expedite refereeing of certified papers, and Mathlib can grow in a virtuous cycle, the way LaTeX displaced every other typesetter in a few years.

His printed prediction was that by 2026 there would be research-level papers with AI in the loop—not Fields Medal work, but real computations and verifications. That has already happened, with credit hard to disentangle. Tools copy perhaps 30 or 40 percent of the skill stack: coding friction, a noisy literature review that still hallucinates four references out of six. A correct related-work section would be another phase shift, the way self-driving needs a safety margin the last mile does not yet have. This decade he can imagine an AI proposing a genuine conjecture linking two things people thought unrelated. Training data is missing the negative space: the failed conjectures, the modifications, the embarrassing tries mathematicians never publish. His joke is that the models will have to go to graduate school, sit in office hours, and get the assignments wrong.

Fridman raises Grigori Perelman, who proved the Poincaré conjecture by working largely alone for seven years. The two-dimensional story is familiar: a sphere is the only closed surface on which every loop contracts, unlike a torus. Poincaré asked the same question for a bounded three-dimensional space. Oddly, dimensions four and five were easier and fell first. Combinatorial and algebraic attacks stalled in three dimensions until Richard Hamilton proposed Ricci flow, a PDE that inflates a crumpled balloon until the wrinkles smooth, or until a singularity forms—the same global-regularity-versus-blowup fork as Navier-Stokes. Hamilton could finish the two-dimensional case. In three dimensions the flow is supercritical. Neck pinches can be snipped; a knotted nightmare might not.

Perelman’s move, Tao says, was to introduce new quantities—Perelman’s reduced volume and entropy—that play the role energy played for Hamilton’s mechanics, making the problem critical so the nonlinearities looked less terrifying. He then classified the singularities of that critical flow and showed how to perform surgery on each of them. Nothing in that chain is what a language model would reliably pick from a hundred dead ends. Perelman must have had a sense that the track was right, because years sit between A and B.

Tao does not work that way. He is a fox: if a problem sours, he switches problems, or he cheats by assuming the bad case does not occur and scouting whether the rest of the argument survives. Sometimes mistakes are productive. With four collaborators he once thought he had solved a hard PDE in two months, planned champagne, and then found a thirteenth term in a lemma that was worse than the other twelve combined. Two years of increasingly desperate attempts produced a different strategy that avoided the term. Had they known the true length at month two, they might never have started—Columbus selling a route to India. Mathematical diseases, as he calls problems that swallow a career, occasionally pay; he does not recommend them without the right fortitude. Grant proposals promise progress, not a proof of every listed conjecture.

Asked which problem haunts him, Tao does not pick the Riemann hypothesis. Even with every cheat code he knows, there is still no path; something must break in another field first. Twin primes sit closer. The natural numbers are easy additively and easy multiplicatively; mixing the two is where undecidable statements hide. The twin-prime conjecture asks how often a prime plus two is still prime. Green–Tao showed that the primes contain arithmetic progressions of every length, a robust pattern: you can delete 99 percent of the primes and the progressions survive, “like cockroaches.” Twin primes are sparse enough that a tiny, well-chosen redaction would erase them while leaving the prime-number theorem intact. Any proof that only uses aggregate statistics must also fail on that censored list. One conspiracy is almost impossible to kill; several kill each other. That is why bounded gaps succeeded: infinitely many prime pairs differ by at most 246, because you cannot run fifty conspiracies at once. The remaining wall is the parity barrier. Almost-primes are well understood; if you could get the density of true primes inside a good almost-prime set above 50 percent you would have twins, and current techniques cannot cross that line. Tao would love to breach it. Goldbach and a stack of neighbors sit behind the same wall.

Collatz is the other simple monster. Odd numbers go to 3n+1, even numbers halve, and every starting value is supposed to fall to the 4-2-1 cycle. The paths look like a random walk with a slight downward drift, a casino whose odds are against you. Tao proved a statistical theorem: something like 99 percent of inputs eventually become much smaller. Almost-sure theorems always leave an outlier, the same Maxwell demon that might blow up a fluid. Alex Kontorovich and others have suggested a Collatz trajectory could encode a self-replicating machine. Conway had already shown that more elaborate Collatz-like maps can simulate Turing machines; he even designed a language, Fractran, that is Turing complete. Riemann, if false, would shock number theory and rattle cryptography built on the belief that primes look random. P versus NP is the meta-problem: a positive answer could unlock the others, the evidence leans toward no, and complexity theory is unusually rich in no-go theorems that kill whole approaches.

The Fields Medal, Tao found, does not solve the problems on your desk; it does put you in the establishment. People want opinions, free time shrinks, and the social contract shifts from proving technical theorems toward mentoring and steering. He still declines a lot. Perelman declined both the medal and the million dollars, saying that if the proof is correct no other recognition is needed, and went off to collect mushrooms near St. Petersburg. Tao has never met him and treats the choice as allowed. He also warns against a history that names only the person who took the last step. At Princeton as a student he watched Andrew Wiles’s mailbox overflow when Fermat’s Last Theorem was announced. Kevin Buzzard now has a five-year grant to formalize that proof in Lean down to facts a 1980 number theorist would have known, a tower of algebraic objects far from the primes and functions Tao can define for a high-school audience. Wiles’s seven secret years are a romantic hedgehog story. Tao’s own productivity comes from leaving a problem that is not yielding, then eating his words when a stubborn collaborator’s fifth try works.

Humans, he thinks, do not have a math organ. They repurpose vision, language, or the gaming circuit, which is why mathematicians take different paths to the same lemma and why a single classroom style arrives too late for many students. His advice is to find enrichment outside school—YouTube, puzzle forums, the probability questions poker players actually need—and, with Lean, to let amateurs contribute atomic lemmas the way citizen scientists already find comets. Programming feels more open than modern mathematics mostly because you can print “hello world” this weekend. Careers will not be one profession for life; transferable skills are abstract reasoning and debugging when tools fail. He learned Lean because he had evangelized the combination of language models and proof assistants and then felt obliged to walk the walk, after a workshop with Kevin Buzzard and Jordan Ellenberg that ran just before ChatGPT. Structured procrastination still works: invent a worse task and do the original to avoid it. Euclid remains a contender for greatest mathematician, Hilbert’s 23 problems still shape the century, and the right next move when no technique applies is to try something stupid on purpose. What gives him hope is younger students, and the fact that navigation—once a problem people died over—now lives in a pocket. On its good days, the mathematical community is a superintelligence no individual comes close to replicating.