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...
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
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…
Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture
Expository writing following an explicit three-dimensional Keller counterexample.
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
Theorems for free! | Proceedings of the fourth international conference on Functional programming languages and computer architecture
The polymorphic blame calculus integrates static typing, including universal types, with dynamic typing. The primary challenge with this integration is preserving parametricity: even dynamically-typed code should satisfy it once it has been cast to a ...
The Black Jacobins: Toussaint L'Ouverture and the San D…
A powerful and impassioned historical account of the la…
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 $d \geq 2$.
Chris Peikert on Twitter / X
WOW!! 🤯Among many jaw-dropping results, this proves NP-hardness of the Closest Vector and Nearest Codeword Problems for *polynomial* approximation factors, for the first time ever, and via a totally new approach (Reed-Solomon techniques). Amazing! https://t.co/3yh3scLX5R— Chris Peikert (@ChrisPeikert) August 1, 2026
Computational Geometry in C (Second Edition)
Homepage for textbook on Computational Geometry
Polynomial Functors: A Mathematical Theory of Interaction
This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical theory from the ground up, highlighting pictorical techniques and concrete examples to build intuition and provide applications.
Przemek Chojecki | PC on Twitter / X
The Growing Map of Open Mathematical Problems.We mapped 15,000+ conjectures from UnsolvedMath to show potential links between concepts.It also shows how under formalized the frontier is (less than 10%). pic.twitter.com/nm0PCpXVPf— Przemek Chojecki | PC (@prz_chojecki) August 28, 2026
Limits and Colimits in a Category of Lenses
Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to be extensive, and it has an image factorisation system.
