This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.
On the Navier–Stokes Millennium Prize Problem
We’re sharing an AI-generated solution to the Navier–Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean.
NIK on Twitter / X
>sam altman: "we tried this because there were rumor on the internet">rumor: math professor close to solving navier-stokes>the professor: using codex for a year>openai: has all his logs>checked every codex session tagged to N-S>found the most promising one>spun up 10,000… pic.twitter.com/RInlP88gmA— NIK (@ns123abc) September 8, 2026
Shallow Water Equations - RDycore: A River / Flooding dynamical core for E3SM
The two-dimensional shallow water equations can be written in the conservative form
Shallow water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. The shallow-water equations in unidirectional form are also called (de) Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant.
GitHub - openai/NavierStokesAndEuler at 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538
Lean certificates accompanying Navier-Stokes and Euler results - openai/NavierStokesAndEuler
tristanbuckmaster (@tristanbuckmaster@mastodon.social)
Today, Levent Alpöge and I have made public three results: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler. https://cims.nyu.edu/~tristanb/statement.pdf https://cims.nyu.edu/~tristanb/euler.pdf https://cims.nyu.edu/~tristanb/ipm.pdf https://cims.nyu.edu/~tristanb/boussinesq.pdf https://github.com/tristanbuckmaster/fluid_lean
AI's solution to 87-year-old riddle takes mathematicians by surprise | New Scientist
Mathematicians have been trying to prove the Jacobian conjecture for nearly a century, but now the Claude Fable 5 AI has apparently found a counterexample that disproves it
Pauls Online Math Notes
Welcome to my math notes site. Contained in this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, II and III) as well as Differential Equations at Lamar University. The notes contain the usual topics that are taught in those courses as well as a few extra topics that I decided to include just because I wanted to. There are also a set of practice problems, with full solutions, to all of the classes except Differential Equations. In addition there is also a selection of cheat sheets available for download.
interfluidity » Quick thoughts about airline economics
If you’ve not already read this piece by Matt Stoller challenging the conventional wisdom that airline deregulation in the 1970s has been a great success, you should. See also this 2012 piece by Phillip Longman and Lina Khan, and this 2015 piece by Longman. Thanks to Matt Ygesias and Marshall Steinbaum for pointers.
Leiden Declaration on Artificial Intelligence and Mathematics
This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research.
Leiden Declaration on Artificial Intelligence and Mathematics
This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research.
Leiden Declaration on Artificial Intelligence and Mathematics
This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research.
Leiden Declaration on Artificial Intelligence and Mathematics
This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research.
Host a PDS via a Cloudflare Tunnel - Hitchhiker's Guide to the Atmosphere
Learn how to use a Cloudflare Tunnel to host your PDS on a local network.
The Viscous Frontier
open.substack.com