Expository writing following an explicit three-dimensional Keller counterexample.
The Jacobian Conjecture is False Per Anthropic (Link in Description)
945 votes, 354 comments. Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable! The two-dimensional…
levent on Twitter / X
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,…— levent (@__alpoge__) July 20, 2026
Jacobian Conjecture -- from Wolfram MathWorld
The Jacobian conjecture asserts that a polynomial map F:C^n->C^n having a nonzero constant Jacobian determinant is an automorphism. In the plane, first stated by Keller (1939), it says that a ring map F of C[x,y] (the polynomial ring in two variables over the complex numbers) to itself that fixes C and sends x, y to f, g, respectively, is an automorphism iff the Jacobian f_xg_y-f_yg_x is a nonzero constant. The condition is easily shown to be necessary. There have been at least five...
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
Higher Dimensional Syntax - Eric Finster
The Kochen-Specker Theorem
The Kochen-Specker theorem is an important and subtle topic in the foundations of quantum mechanics (QM). The theorem demonstrates the impossibility of a certain type of interpretation of QM in terms of hidden variables (HV) that naturally suggests itself when one begins to consider the project of interpretating QM.We here present the theorem/argument and the foundational discussion surrounding it at different levels. The reader looking for a quick overview should read the following sections and subsections: 1, 2, 3.1, 3.2, 4, and 6. Those who read the whole entry will find proofs of some non-trivial claims in supplementary documents.
An OpenAI model has disproved a central conjecture in discrete geometry
An OpenAI model solved the 80-year-old unit distance problem, disproving a major conjecture in discrete geometry and marking a milestone in AI-driven mathematics.
Notes for a Theory | Alberto Garutti
Some explicit counter-examples to Weibel's conjecture
We give two related but different methods for constructing rings $R$ of Krull dimension 1 and $K_{-d}(R) \neq 0$. The first works for $d = 2$, and the second works for any $d \geq 2$.
Manifold hypothesis
The manifold hypothesis posits that many high-dimensional data sets that occur in the real world actually lie along low-dimensional latent manifolds inside that high-dimensional space.[1][2][3][4] As a consequence of the manifold hypothesis, many data sets that appear to initially require many variables to describe, can actually be described by a comparatively small number of variables, linked to the local coordinate system of the underlying manifold. It is suggested that this principle underpins the effectiveness of machine learning algorithms in describing high-dimensional data sets by considering a few common features.
General Theory of Natural Equivalences
Samuel Eilenberg, Saunders MacLane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society, Vol. 58, No. 2 (Sep., 1945), pp. 231-294
Three.js From Zero — Creative Web 3D for Artists · MasterAllArts
Learn Three.js as part of the MasterAllArts creative 3D roadmap: turn drawings, models, shaders, and portfolio ideas into interactive Web 3D scenes. Free articles with live demos.
