A classical result in differential topology states that there are no nowhere-vanishing vector fields on the 2-sphere. One may ask an analogous question in algebraic geometry: does the tangent bundle of the sphere defined by the equation $x^2+y^2+z^2=1$ over a field $k$ admit a nowhere-vanishing section? Equivalently, can the vector $(x,y,z)$ be completed to an invertible $3\times 3$ matrix over the ring $k[x,y,z]/(x^2+y^2+z^2−1)$?

More generally, one can ask the same questions for an arbitrary affine quadric defined by an equation $q=1$, where $q$ is a homogeneous degree 2 polynomial. In joint work with Marc Levine, we gave an essentially complete answer to the first question under the assumption that the equation $q=1$ has a a solution in $k$. In particular, for the 2-sphere the answer is positive if and only if $−1$ is a sum of four squares in $k$.

Our proof combines techniques from motivic homotopy theory with constructions from the classical theory of quadratic forms. However, it is inherently non-constructive and yields no explicit formulas. Recently, Peter Müller found an explicit formula in the case of the 2-sphere. In this talk, I will discuss these questions, some of the key ingredients of the proofs, and possible directions for future research.