Saurabh Bhatnagar

This is quite a remarkable result: Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5. The Jacobian conjecture roughly says that a multivariable polynomial F has an inverse function (made out of polynomials) provided the Jacobian is non-singular (i.e. matrix of partial derivatives has non-zero determinant). This condition is neccessary by the Inverse Function Theorem from multivariable calculus. The hard question is whether it is also sufficient. Evidently, Fable found that F(x,y,z) = ((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) has det(J_F) = -2 non-zero. However F is not invertible, since F sends three different pts (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to the same image (-1/4, 0, 0). Beyond the disproof itself, it would be value to know if a suitably refined conjecture is recoverable. Per @Acer, GPT5.6 has proposed: "A constant-Jacobian polynomial local biholomorphism with no loss of sheets at infinity—e.g. a proper Keller map—is an automorphism." I would be interested to know if any algebraists (e.g. @levent @littmath) have a reaction to this... A further twist to the story: Not only was the Jacobian conjecture one of the central open problems in algebraic geometry, it was (a special case of) Yitang Zhang's PhD problem! The catch was that Zhang's advisor had him solve it, assuming a lemma of his advisor. But that lemma turned out to be false! As a result, Zhang's thesis crumbled and he then struggled to get recommendation letters and a permanent academic position. Despite all this, Zhang went on to prove bounded gaps between primes! This is one of the most inspiring stories in modern mathematics, and was a motivation for me to work in the same area for my doctorate.

This is quite a remarkable result: Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5. The Jacobian conjecture roughly says that a multivariable polynomial F has an inverse function (made out of polynomials) provided the Jacobian is non-singular (i.e. matrix of partial derivatives has non-zero determinant). This condition is neccessary by the Inverse Function Theorem from multivariable calculus. The hard question is whether it is also sufficient. Evidently, Fable found that F(x,y,z) = ((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) has det(J_F) = -2 non-zero. However F is not invertible, since F sends three different pts (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to the same image (-1/4, 0, 0). Beyond the disproof itself, it would be value to know if a suitably refined conjecture is recoverable. Per @Acer, GPT5.6 has proposed: "A constant-Jacobian polynomial local biholomorphism with no loss of sheets at infinity—e.g. a proper Keller map—is an automorphism." I would be interested to know if any algebraists (e.g. @levent @littmath) have a reaction to this... A further twist to the story: Not only was the Jacobian conjecture one of the central open problems in algebraic geometry, it was (a special case of) Yitang Zhang's PhD problem! The catch was that Zhang's advisor had him solve it, assuming a lemma of his advisor. But that lemma turned out to be false! As a result, Zhang's thesis crumbled and he then struggled to get recommendation letters and a permanent academic position. Despite all this, Zhang went on to prove bounded gaps between primes! This is one of the most inspiring stories in modern mathematics, and was a motivation for me to work in the same area for my doctorate.

@_varunnair We train our encoder and policy from scratch! These models use a ResNet-50 backbone for the visual encoder, with a Vector-Quantized Behavior Transformer (VQ-BeT) as the policy head. The model is small enough that it runs at 3Hz on an Intel NUC CPU.