There had been a lot o buzz about Levent Alpoge’s counter example of the Jacobian conjecture. I myself had been introduced to it by Juergen Moser, who was prominently mentioning it in a dynamical systems course. It had been a beautiful conjecture. Unfortunately it has now been gone. There will be certainly more adjustments coming and some things which have been ranked high will be solved by machines. I had mentioned the Jacobian conjecture in this unit in Math 22 of Fall 2018 and even given it as a homework: (see problem 23.5 from Fall 2018):
Levent used Claude Fable (a particularly strong AI model) to find the following counter example. Running the following mathematica code verifies it:
{u,v,w}={ (1+x y)^3 z + y^2(1+x y)(4+3x y),y+3x(1+x y)^2 z+3 x y^2(4+3x y),2x - 3x^2 y - x^3 z};
({u,v,w} /. x ->0 /. y->0 /. z->-1/4 )== ({u,v,w} /. x->1 /. y->-3/2 /. z->13/2) (* is not injective *)
Det[{{D[u,x],D[u,y],D[u,z]},{D[v,x],D[v,y],D[v,z]},{D[w,x],D[w,y],D[w,z]}}] == -2 (*still has constant det *)