Seminars & Colloquia Calendar

Abstract:  Families of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zero-dimensional varieties (configurations of points on the Riemann sphere) and one-dimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such families: the problem of (non)existence of continuous sections of "universal" families. Informally, these results give answers to the following sorts of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1, but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n-1)(n-2).

These results are joint work with Lei Chen.