"Solve for \(X\): \(X\times \mathbb{A}^1=\mathbb{A}^n\):" Old and new around exotic affine spaces and Zariski cancellation problem

A great deal of often seemingly elementary questions concerning the geometry of affine algebraic varieties are intimately linked to the central question of how to recognize and characterize affine spaces among algebraic varieties in an effective way, depending on the type of problem addressed. In this talk, I intend to present an overview of some classical results on this question using a variety of techniques, ranging from commutative algebra to differential topology to logarithmic birational geometry, and then focus on some more recent advances and new perspectives opened by ideas and techniques at the interface between algebraic geometry and \(\mathbb{A}^1\)-homotopy theory.