Conference Agenda

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.

Quantum wreath product algebras are certain generalizations of Hecke algebras of type A, which depend on a choice of an associative algebra. They naturally appear in the study of Lie superalgebras, categorifications of quantum groups, representations of p-adic groups, and more. I will explain how to contruct their Schur versions, and how they give rise to new quantum group-like algebras.