Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
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?
- Cynic "offer" from OpenAI to Buckmaster
(remove Alpöge as author) - OpenAI: "...we cannot rule out that de-identified data derived from their usage of our products helped improve our models"
- 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
The Navier-Stokes millenium problem, AI, and the future of mathematics
By Klas Modin
The Navier-Stokes millenium problem, AI, and the future of mathematics
Presentation given at the WAF annual meeting in Stockholm, Sep 2026.
