Tristan Buckmaster | A ‘Deep Blue-Kasparov moment for mathematics’

# Tristan Buckmaster: Mathematics Is Approaching Its "Deep Blue–Kasparov Moment" **TL;DR:** NYU's Tristan Buckmaster reviewed the history and latest progress on finite-time blow-up in the Euler equations at a seminar, and discussed how AI agents are changing the way mathematics is done—likening it to a "Deep Blue defeats Kasparov moment" for mathematics. ## The Origin of the Analogy: A "Deep Blue–Kasparov Moment" Buckmaster attends a lunch gathering for professors and postdocs on the 13th floor at NYU, where everyone sits around a table discussing politics or mathematics. Over the past few months, the discussions have revolved almost entirely around one topic: AI. In August, he had some results that he could not yet discuss publicly. During those discussions, some people were skeptical of AI's capabilities, and his response was: mathematics is about to change; you have no idea what is coming; this will be an enormous transformation—and this was before any of the public announcements. He compared this approaching turning point to a "Deep Blue–Kasparov moment," with the central point being: **mathematics can go beyond human intelligence**. ## The Euler Equations: Equations That Gave Rise to an Entire Field The problem Buckmaster studies has deep historical roots. - **In 1747**, d'Alembert formulated the first partial differential equation—the one-dimensional wave equation. - **In 1757**, just ten years later, Euler said: "This is too boring. I'm going to come up with a better equation." He jumped directly from a one-dimensional linear partial differential equation to a three-dimensional fully nonlinear partial differential equation. Only ten years separated these two results, and this was already the second famous partial differential equation to be discovered, which is incredible. Later, a viscosity term was added to obtain the Navier–Stokes equations—here, `u` is the fluid's velocity, a vector field carried by the fluid, which can be derived from Newton's second law. Adding internal viscosity (internal friction) leaves the equations similar in form but changes their properties. Other equations, such as the heat equation, appeared in between, but there were not many—these were still the early days of partial differential equations. The Euler equations essentially gave rise to the entire field of partial differential equations. They have incredible physical significance, but from a mathematical perspective, they have also been profoundly influential. ## A Millennium Prize Problem: Navier–Stokes Existence and Smoothness The Navier–Stokes existence and smoothness problem is one of the seven Millennium Prize Problems. It is divided into four subproblems, which are really just two different problems—the distinction between the torus and R3 is not very substantial. **Problem One:** Given smooth initial data with finite kinetic energy, if it evolves according to the Navier–Stokes equations, does it remain smooth for all time? **Problem Two:** Must this always happen? Is there an initial condition—a particular state of the fluid—that produces unphysical behavior as it evolves according to the equations, such as the velocity becoming infinite at a point, causing the equations to break down? Buckmaster believes that the original intention behind choosing the Navier–Stokes equations instead of the Euler equations was to make the problem more tractable and somewhat easier. At the time, people thought the Navier–Stokes equations should be well-posed and should have smooth solutions for all time, whereas they were less certain about the Euler equations. In practice, however, this made the problem harder, because blow-up does indeed occur. For the blow-up question, the problem's formulators also tried to make things slightly easier by allowing a smooth forcing term, but the forcing must be very regular—if one considers the whole space, it must decay rapidly at infinity. It was the last two subproblems that were solved. ## The Main Result: Finite-Time Blow-Up in the Euler Equations What they proved was an analogous phenomenon with viscosity removed. This is one of their several proofs concerning the Euler equations, and one of the versions that has been made public: > There exist smooth axisymmetric initial data and a forcing term such that finite-time blow-up occurs at time one. There is a slight difference from the Navier–Stokes equations—for the Euler equations, the velocity does not necessarily have to blow up; **the vorticity can blow up** instead. Vorticity is the curl of the velocity. That is precisely what happens here: the vorticity blows up, while the velocity remains bounded. ## The Historical Development of Known Results - **In 1934, Leray** proved the existence of global weak solutions with finite energy. Since these solutions exist, the natural question is: are they smooth? - **In 1976–1977, Scheffer** proved partial regularity: if singularities exist, they must lie in a set with Hausdorff dimension at most 5/3; in particular, it has measure zero. This was later improved by Caffarelli, Kohn, and Nirenberg. - Some people took this as evidence that the solutions were well-posed—because if singularities occur only on a set of measure zero, all that remains is to prove that those singularities do not actually exist, which seems reasonable enough. ### Nightmare Solutions In 2017, Buckmaster and Vicol proved that if one considers only the class of weak solutions—not Leray solutions, but another kind of weak solution—extremely strange solutions can exist: so-called **"nightmare solutions"**. You can start from complete rest, with zero velocity, and then the velocity suddenly becomes nonzero before returning to rest. His analogy is this: you have a glass of water on your bedside table, completely still. You fall asleep, wake up in the middle of the night, and discover that the water is sloshing around. Then you fall asleep again, and the water becomes still again. "I always say that at this point I should promptly leave mathematics and start a company, creating infinite energy out of nothing. Get lots of investors—but I didn't do that." These are certainly unphysical—they create kinetic energy from nothing. ### Singularities at the Boundary - **In 2013, Luo and Hou** showed that if one considers a boundary problem for the Euler equations, singularities can occur at the boundary. This was a numerical result, but a very compelling one. - **Ye and Hou** proved that singularities can occur in the Euler equations with C^{1,α} initial data—quite regular, with more than one derivative, but not meeting the smoothness criterion proposed by Majda and Bertozzi. Buckmaster called this an astonishing result: before it, there had been no clear distinction between smooth data and C^{1,α} data. At the time, the gold standard in singularity research was the Burgers condition, whereas here the vorticity remains bounded before blow-up. ### The Method of Pins and Rigorous Proof Together with postdoc Yanji Wang, Ya Le, who was then at Princeton, and Javi Gomes-Arnaldo, Buckmaster developed the **"method of pins"**—a numerical method for constructing numerical solutions or self-similar solutions, essentially the core of the solution. Chen and Hou then used a computer to prove rigorously that this blow-up scenario occurs. Later, the team, joined by Gonzalo Lava—one of his postdocs, who was then a PhD student at MIT—and Dick M, proved that many such solutions exist, but **all of these solutions have boundaries**. ### Clarifying Media Misconceptions Buckmaster specifically pointed out that many news articles last year suggested that we were on the verge of solving the Navier–Stokes problem, but this was completely untrue. Partly because of those articles, the broader community believes that we are close to solving it—but no one is close, and this has nothing to do with the well-posedness question. "I just wanted to point that out, because the broader community thinks the first Navier–Stokes problem is the one about to be solved, when in fact it's the second—though of course that doesn't mean we're very close. On a human timescale, we may still have ten years to go." ### A Breakthrough in the Case Without Boundaries In 2024, **Diego Córdoba and Luis Martínez** proved that for the forced incompressible porous media equation—which describes fluid moving through porous materials such as sand or rock—singularities can occur even with fairly smooth forcing. "Fairly smooth" means that all spatial derivatives of the forcing are bounded; they could also argue, somewhat roughly, for one additional derivative in time. The only thing missing was smoothness of the space-time derivatives. This was the first blow-up for incompressible fluid equations. Buckmaster noted that for compressible fluids, we know shocks occur, and have known this for a long time. But for incompressible fluids, this was the first case without a boundary. Cases with boundaries can be viewed as nonsmoothness across the boundary—reflecting the solution across the boundary produces a solution on the whole space, essentially resembling a discontinuity across the boundary. The case without boundaries presents an entirely different challenge. ## AI Agents Enter the Picture In 2025, a year earlier, Buckmaster unexpectedly received an email from Levent Szilégi, whom he had never met. Szilégi asked: "Would you like to try solving this problem?" His response was: "Look, we're not going to solve it." Szilégi said: "Great, let's try." And so they began working on it. This is where agents entered the picture. "These agents aren't malicious; they're helpful." Everything has developed so quickly that people have already forgotten how new the idea of using agents to do mathematics really is—the concept itself existed years ago, but only recently has it actually been put into practice. ### One-Shot Prompting vs. Iterative Feedback In one study, they tested different models—no longer frontier models by today's standards—including Gemini 2.5 Pro, Grok 4, and GPT-5. These models differed in quality: - When given just one prompt to solve problems **in one shot**, they could solve zero out of two IMO problems (some models could not solve any, some could solve two, and some fell somewhere between zero and two). - But if they gave the IMO problems to each model, had it answer about 100 times, then fed the outputs back into the same model and did another 100 rounds, **repeating this process**, all the models eventually solved five out of six problems. In other words, despite their differing levels of quality, they all had the same underlying potential. ### Claude Code and Combinatorics as a Testing Ground Claude Code launched on May 22. Agentic coding existed before that, but this was probably the first consumer-facing version that let you launch an agent directly. The first thing they did was try it out on some simple mathematics problems. One problem he particularly likes is associated with Jordi Williamson and collaborators—he recalls Shan Ellenberg being among them, and that these authors had all worked at institutions such as CMS and CSMA: **take a grid and place stones on it; how many stones can you place on the grid without forming a triangle?** From a combinatorial perspective, brute-force enumeration is impossible—the number of combinations is simply staggering. These authors' approach was to train a Transformer model, starting with a large number of random guesses as training data, have it produce a reasonably good guess, then perform local search and repeat the process. Local search generates more training data, and the process keeps iterating—these are the methods that led to the discovery of solutions of this kind. At first, Levent and Buckmaster were just playing around: "They did this, so let's throw a bunch of agents at it, try different algorithms, try different models." They tried replacing the Transformer with a diffusion model and experimented with many things—not especially successfully, but it was fun. They were simply testing the limits of these methods. --- Source: Tristan Buckmaster | A 'Deep Blue-Kasparov moment for mathematics' — SammaVaco (https://www.youtube.com/watch?v=XcYKzHqJuaw)