Josefien Kuijper: The K-theory groups of varieties

Abstract.

The K-theory groups of a category can be seen as an invariant that keeps track of how objects in the category assemble and decompose. Clasically, these groups can be constructed for categories that have enough extra structure, such as abelian or dg-categories. The category of varieties has none of this. Nonetheless, the canonical candidate for the zeroth K-theory group of varieties, also called the Grothendieck group, has been around for a while and is well studied. In recent years, people have attempted and succeeded in constructing the higher algebraic K-theory groups of the category of varieties in different (but equivalent) ways. I will talk about the newest (and arguably the most elegant) of these constructions.