Alex Kontorovich
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Lex Fridman Podcast · #503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis &…
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What Did AI Prove About the quasi-Riemann Hypothesis?
Alex Kontorovich says the proof gives an infinite vertical strip with no zeros whose real part lies between seven-eighths and one. The strip is a quasi-Riemann hypothesis. Bringing seven-eighths down to one half would be the full Riemann hypothesis, and the method cannot get there.
He did not think he would see a proof of the quasi-Riemann hypothesis in his lifetime. The drop covered the zeta function and Dirichlet L-functions. A second proof reaches about eleven-twelfths. The width of the known zero-free region had not moved in seventy years. The full hypothesis still needs a new idea. He calls the speed of the progress far outside his model of what was possible.
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Can AI Formalize a Proof a Human Already Wrote?
Alex Kontorovich says autoformalization is basically solved, and not in the way he wants. A sorry-free proof of a theorem human beings have already written on paper is, in his view, a finished problem.
The library work is not finished. People will still put in thousands of hours, in the open, building a library of mathematics, and the byproducts that help a person understand a theorem still come from that work. He still wants that open library work, and he says the work stays very difficult. He had already been through one shock about a month earlier. After Labor Day the next shock was Navier-Stokes. He calls the quasi-Riemann result round three.
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How Long Is Fermat's Last Theorem in Lean?
Alex Kontorovich says the Lean formalization of Fermat's Last Theorem is 13 million lines. He was sure that formalization was at least three years away.
He had just told a colloquium in the Princeton mathematics department that current systems build one or two layers on Mathlib and are weak at canonization, the cleanup that puts a lemma where it belongs in the library. A manuscript that pulls many areas together sat outside that halo. The statement of Fermat's Last Theorem is elementary, so there is no hidden meaning to check, and Lean reports the proof sorry-free. Whole textbooks formalize when they sit one or two steps from Mathlib. Three or four steps out, definitions made for different
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Did AI Solve the Riemann Hypothesis?
No. Alex Kontorovich says the Riemann hypothesis is not solved. He wants its solution, and he did not expect to see, in his lifetime, the kind of progress that arrived a few days ago.
He will grant, for the sake of argument, what he calls the solved axiom. It says any mathematical problem a human can solve together with AI can also be solved by AI alone. He says we are nowhere near that point, and he does not expect to reach it. In that world, the only way to make progress is to press a button. The system thinks about the problem. It does not have to solve every problem.
Even then, he argues, it would not destroy pure mathematics. A theorem matters because it lives in human heads, not only
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What Should Pure Math PhD Students Learn?
Alex Kontorovich says PhD students in pure mathematics should do what mathematicians have always done: learn the most difficult, most abstract mathematics their mind can handle, and learn as much of it as possible. An advisor should guide that process.
At the undergraduate level, grades used to be basically Gaussian. The distribution is now bimodal. Students who use AI to check their understanding, asking what a professor meant in class, handing over notes from the last three weeks, and requesting a quiz before a test, are getting much higher grades. Students who have a problem set due in an hour, ask AI for the answers, and rewrite them in their own words show a decorrelation between homew
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How Many Authors Does a Math Paper Have?
Alex Kontorovich says almost every math paper written until the 20th century had one author. He thinks the median in mathematics is now two. The field is not a lab science, and the work is still a solitary process.
Graduate students receive problems he is not working on, so a success stays theirs even when the idea was his and they carried it out. A paper released a month ago came from ten years of thinking. Five authors made incremental progress until a solution was in hand. At the start they could not give a formal Lean statement with the gap still marked. The statement showed up only after hacking away little by little until the pieces fit.
The project began as a phenomenon he did not u
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What Is the Riemann Hypothesis?
Alex Kontorovich says the Riemann hypothesis predicts that every wave behind the prime numbers has exactly the same amplitude. The Riemann zeta function takes complex numbers in and sends complex numbers out, and it governs how the primes are distributed. A zero is a complex input that makes the output equal zero, the origin in the complex plane.
People want to know how the primes are spread through the natural numbers. Riemann's breakthrough was signal processing on the primes, the same move a microphone makes with sound. A voice box makes pressure waves, Fourier analysis turns the waves into amplitudes and frequencies, and a speaker rebuilds the voice from the stored numbers. The zeta fun
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Is the quasi-Riemann Hypothesis Proof a New Idea?
Alex Kontorovich says the quasi Riemann Hypothesis case is not a genuinely new idea. It is a beautiful combination of things that were already very well known. A lot of people are kicking themselves because they had some idea in that direction 20 years ago.
Two years ago, a big breakthrough on cubic theta functions and cubic Gauss sums became the path the AI took to the solution. He pictures a semigroup as tiles on a board: from one tile you may place another to the left, the right, the front, or the back, and keep going. Existing ideas, pushed that way, might prove a whole lot more than humans have explored. Deep problems in mathematics sometimes fall to a three page solution that is eleme
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Why Is AI Math More Than a Convex Hull?
Alex Kontorovich says AI math is more than a convex hull because known moves, chained together, reach much further than that label allows. Tim Gowers describes the better picture as the semi-group generated by the moves. Putting known moves together is the same notion, but what the space of known moves can accomplish is much bigger than the derogatory convex hull.
People talk about the work dismissively, as if the result already sat in the convex hull. The system does the moves humans already know, does them really well, and runs a really good tree search. On the quasi-RH, the moves taken were all things known to human beings. A very long cascade of the moves is shocking. The sequence looks
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Would Alex Kontorovich Call AI in Mathematics a Crisis?
Alex Kontorovich does not know whether he would call the moment in AI and mathematics a crisis. A lot of opportunity sits in the crisis. The moment is a chance to rethink what the people in the work are trying to accomplish.
The feeling around the historic moment is the chaos of opportunity. Some people are reacting negatively. Some people are reacting positively. Sometimes a negative reaction and a positive reaction are happening in the same person.
The label crisis stays open. What stands is the opportunity to rethink the aim of the work, and the fact that negative and positive reactions can show up across different people and inside one person at the same time.