Recently, two distinct teams of mathematicians and artificial intelligence experts announced significant progress in determining whether the equations governing fluid motion guarantee regular, predictable behavior or if they can fail. One of these results directly relies on the 'layer cascade of vorticity' mathematical strategy, devised by Diego Córdoba, a research professor at CSIC's ICMAT, and Luis Martínez-Zoroa, a professor at CUNEF Universidad. The second team also acknowledges the influence of these prior works.
The Euler and Navier-Stokes equations, used to predict weather or design aircraft, describe the movement of fluids like water, oil, or air. Despite centuries of practical application, fundamental questions about their behavior, such as the potential emergence of singularities (points where velocity becomes infinite), remained unanswered in three dimensions. The Clay Mathematics Institute offered a million-dollar prize for solving this enigma, which OpenAI has now addressed.
Córdoba and Martínez-Zoroa's strategy, developed over nearly a decade, involves constructing the fluid through an infinite succession of increasingly smaller vorticity layers. Each layer is regular, but they interact precisely: larger layers deform the fluid, amplifying the smaller ones. This mechanism, likened to a 'series of gears,' activates layers at progressively shorter time intervals, accumulating towards a final instant where velocity may become infinite.
Unlike other methods seeking singularities in rigid scenarios, their approach enabled a more flexible and localized amplification mechanism. Javier Aramayona, director of ICMAT, praised the work, highlighting Córdoba's career and Martínez-Zoroa's brilliance, reaffirming the institute's commitment to fundamental research.
The truly disruptive idea, stemming from Martínez-Zoroa's doctoral thesis, was to fabricate the singularity from infinite regular solutions organized to amplify each other. They have successfully applied this mechanism to the Euler equations and the incompressible porous medium equation, as well as to a version of the Navier-Stokes equations, publishing results on various scientific platforms.
The primary challenge was to construct a singularity in velocity without the external force acting on the fluid becoming infinite. This required the divergent terms of the equation to cancel precisely. Tristan Buckmaster and Levent Alpöge explicitly acknowledged the Spanish program as a starting point for their extension, aided by AI tools like Claude and Codex.
OpenAI utilized thousands of AI agents in parallel, taking approximately 88 hours for the Navier-Stokes demonstration and an additional 17 hours for its formalization in Lean. Although these results await peer review, Buckmaster publicly suggested that Luis Martínez-Zoroa deserves the Fields Medal.
Córdoba and Martínez-Zoroa reflect on a new era in mathematical research, where human creativity in formulating scenarios merges with machines' capacity for exploration and computation. They emphasize the need to rethink authorship, idea attribution, and the training of future mathematicians.
Javier Aramayona stresses that mathematics extends beyond problem-solving to the deep understanding of structures and theories. ICMAT, a joint center of CSIC and three Madrid universities, reaffirms its dedication to cutting-edge research.
