I will discuss the geometric construction and properties of an unramified endomorphism of $\mathbb A^3$ that is not an isomorphism, and the construction of an exotic $\mathbb A^4$ whose product with $\mathbb A^1$ is isomorphic to $\mathbb A^5$. The first gives a counterexample to the Jacobian conjecture in every dimension greater than 2; the 2-dimensional case remains open. The second gives a counterexample to Zariski’s cancellation conjecture in dimension 4 and yields nonlinearizable $\mathbb G_m$-actions on $\mathbb A^n$ for $n$ greater than 4. Both examples were discovered by AI.