GoKawiil
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GoKawiil's math section posted a tutorial-style article walking through roots of unity, starting with simple real-number square and cube roots of 1 and extending the concept into complex numbers. The piece explains how polar coordinate representation, using Euler's formula, is used to derive the full set of complex solutions for equations like z^n=1.
A tutorial-style article demonstrates how a seemingly ordinary system of two equations has no solution within real numbers, using both trial substitution and algebraic methods. It shows that rearranging the equations into a quadratic and applying the quadratic formula produces a negative value under the square root, exposing the limits of the real number system and setting up the need for complex numbers.
OpenAI released hundreds of papers containing AI-generated mathematical proofs, following earlier controversy over its models' handling of the Navier-Stokes equations. The Association for Human Mathematics issued a statement condemning the release, which was shared and endorsed by Fields Medalist Terence Tao, arguing the dump of files demonstrates power rather than scholarship. OpenAI's announcement cited consultations with an independent group, the Advisory Group on Mathematics and Artificial Intelligence, but did not mention that the group had actually opposed continued testing of its proprietary models on advanced problems.
OpenAI recently claimed its AI agents produced a solution addressing the Navier-Stokes equations, one of math's unsolved Millennium Problems, after running 10,000 agents for 88 hours. Mathematicians including Oxford's James Maynard and Brown's Javier Gómez-Serrano say the resulting proof, while apparently technically correct, is written in a form that is extremely difficult for humans to interpret or learn from.
A personal essay recounts the author's attempt to build intuitive, hands-on understanding of physics laws like Coulomb's law rather than relying on rote memorization or lookups. The writer describes citing advice from physicist Ramamurti Shankar to test physical laws with real instrumentation, and notes the practical difficulty of creating and measuring point charges at home, such as with a charged pen and hair.
OpenAI announced that its most advanced AI model produced a mathematical proof that the Navier–Stokes equations, which describe fluid flow, can generate a singularity — a point of physically impossible infinite speed. Mathematician George Karniadakis calculated that for air this breakdown occurs at a vortex width of about 70 nanometres, roughly the distance a molecule travels before colliding with another.
Terence Tao published a blog post detailing his reasons for not adding his name to a letter signed by fellow Fields Medal winners. The post opens with an extended personal anecdote about his childhood fascination with Fermat's Last Theorem before turning to his actual rationale for abstaining from the collective statement.
OpenAI announced it had solved part of the Navier–Stokes Millennium Prize Problem using an AI system, but the announcement via press release drew criticism for lacking transparency about verification and prior human contributions. Hours earlier, mathematicians Tristan Buckmaster and Levent Alpöge said they had reached a partial solution using AI tools from both OpenAI and Anthropic, raising questions about who deserves credit for the breakthrough.
A discussion thread examines how medieval mathematicians like Scipione del Ferro hid discoveries such as cubic-equation solutions to protect their livelihoods, using secrecy as leverage in public problem-solving contests. The post traces this culture of concealment through the Newton–Leibniz calculus dispute, contrasting it with today's norm of open publication despite paywalls and peer-review issues.
OpenAI announced Tuesday that a swarm of 10,000 AI agents, run for 88 hours at a cost of millions of dollars, had found a scenario where the Navier-Stokes equations describe fluids reaching infinite velocity, effectively breaking the famous equations. NYU mathematician Tristan Buckmaster then publicly accused the company of producing a solution suspiciously similar to unpublished work he had been developing for years and was close to finishing.
NYU's Buckmaster and his collaborator Alpöge published a proof that a simplified version of the Navier-Stokes equations can break down, after nearly a year of work using public AI models. Hours later, OpenAI released its own proof covering the full equations using an unreleased internal model, but Buckmaster alleges OpenAI tried to pressure him into excluding his co-author, who works at Anthropic, and dodged questions about whether their models were trained on his private work transcripts.
OpenAI published a preprint claiming to show that the Navier-Stokes equations, which govern fluid motion, can break down under certain conditions, describing a spinning fluid system reaching infinite speed in finite time. The claim addresses part of one of mathematics' seven Millennium Prize Problems, but mathematicians have raised concerns over how the result was announced and over authorship disputes involving researchers connected to rival company Anthropic.