Navier–Stokes Problem, AI, and the Future of Mathematics

Klas Modin

WAF annual meeting, Stockholm, 2026

Navier–Stokes Problem, AI, and the Future of Mathematics

Klas Modin

WAF annual meeting, Stockholm, 2026

  • The Navier-Stokes Millenium Problem
     
  • The events leading up to last week
     
  • What is mathematical research about, really?

(c) OpenAI

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

Euler and Navier–Stokes Equations

Leonhard Euler

Claude Navier

George Stokes

\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}

Claude Navier

\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)
\frac{\partial \mathbf{v}}{\partial t} + \nabla_{\mathbf{v}}\mathbf{v} = \mathbf{f} - \nabla p + \nu\Delta\mathbf{v}
\mathrm{div}(\mathbf{v}) = 0
\mathbf{v}(x,0) = \mathbf{v}_0(x)

What is the Navier–Stokes Millenium Problem?

Prove or disprove the following:

For \(\nu>0\), smooth initial data \(\mathbf{v}_0\), and smooth forcing \(\mathbf{f}\), prove that there exists a smooth solution for all \(x\) and \(t\)

Events Leading up to Last Week

  • …
  • 2019: Tarek Elgindi proves blow-up of 3-D Euler equations for "nearly" smooth data
  • 2024: Diego Córdoba and Luis Martínez Zoroa proved forced blow-up for "nearby" equations
  • about a year ago: Tristan Buckmaster joins Levent Alpöge (Anthropic) to look for forced blow-up using AI tools
  • 15 August: serious progress for Buckmaster and Alpöge
  • 1 September: rumour that a millenium problem is solved
  • 1-4 September: OpenAI, with tremendous resources, obtain forced blow-up proof in 88 hours
  • 4-6 September: Buckmaster contacted by OpenAI
  • 7 September: Buckmaster releases "statement" and a preprint
  • 8 September: OpenAI announces solution to the NS Millenium Problem (proof of blow-up via smooth forcing)

What is the Controversy?

  1. Cynic "offer" from OpenAI to Buckmaster
    (remove Alpöge as author)
  2. OpenAI: "...we cannot rule out that de-identified data derived from their usage of our products helped improve our models"
  3. Leiden declaration from June 2026:
    "... seeks publicity for new results on market timelines before the accepted processes of community evaluation ... overemphasizing the significance of automated tools and undervaluing the prior human contributions which have made those tools possible"

Where does this Leave Mathematics?

Extremely fast change:

2024: AI gave nonsense

June 2025: Math Olympiad problems 

spring/summer 2026: real research problems 

Sep 2026: Millenium problem

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

William Thurston, "On proof and progress in mathematics", 1994

the world