$\mathbb{A}^1$-connectivity of schemes

Abstract : $\mathbb{A}^1$-homotopy theory, as developed by Morel-Vovodsky is a way of adapting techniques from algebraic topology to algebraic geometry, with the affine line playing the role of the unit interval for defining a notion of homotopy. We will present a very brief introduction to this topic and focus on the study of the sheaf $\pi_0^{\mathbb{A}^1}(X)$ of $\mathbb{A}^1$-connected components of a simplicial Nisnevich sheaf $X$. This object, which is the $\mathbb{A}^1$-homotopy theoretic analogue of the set of connected components of a space, is extremely difficult to compute in general. However, when $X$ is a sheaf of sets, more more specifically a scheme, one can compare this object to the sheafification of the presheaf of ``naive" $\mathbb{A}^1$-connected components. We will outline the progress made by following this approach. Key results include the verification of Morel's conjecture for non-uniruled surfaces, a counter-example to a conjecture of Asok-Morel and the relationship between the notion of $R$-equivalence and $\mathbb{A}^1$-connectedness on algebraic groups.