Motivic homotopy theory gives a way of viewing algebraic varieties and topological spaces as objects in the same category, where homotopies are parametrised by the affine line. In particular, there is a notion of $\mathbb A^1$ contractible varieties. Affine spaces are $\mathbb A^1$ contractible by definition. The Koras-Russell threefold KR defined by the equation $x + x^2y + z^2 + t^3 = 0$ in $\mathbb A^4$ is the first nontrivial example of an $\mathbb A^1$ contractible smooth affine variety. We will discuss this example in some detail, and speculate on whether one can use motivic homotopy theory to distinguish between KR and $\mathbb A^3$.
- Algebraic Geometry Seminar