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
Floor, ceiling, slopes, and K-theory
We calculate K_*(k[x]/x^e;Z_p) by evaluating the syntomic cohomology Z_p(i)(k[x]/x^e) introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case e=2 and p>2. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for e=2. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.
How I Vibed a Proof of Conway’s Conjecture — overreacted
You can just prove things, apparently.
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 $d \geq 2$.
Junyu Ren on Twitter / X
How we solved the 70-year-old Pierce–Birkhoff conjecture in real algebraic geometry with $400.One thing we kept noticing: models from different families (including the latest public models from GPT and Claude) seemed to complement each other, even when they were independently… pic.twitter.com/7MzWstXfy6— Junyu Ren (@junyu_r) September 9, 2026
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
Formalizing 100 Theorems
There is a "top 100" of mathematical theorems on the web, which is a rather arbitrary list (and most of the theorems seem rather elementary), but still is nice to look at. On the current page I keep track of which theorems from this list have been formalized. Currently the fraction that already has been formalized seems to be
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.
