There is recent work by Marc Hoyois, Marc Levine, Jesse Kass and myself giving enrichments of enumerative results to the Grothendieck-Witt group of quadratic forms. I will discuss joint work with Jesse Kass along these lines

A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines depends on the surface, but Segre showed that a certain signed count is always 3. We extend this count to an arbitrary field using A1-homotopy theory: we define an Euler number in the Grothendieck-Witt group for a relatively oriented algebraic vector bundle as a sum of local degrees, and then generalize the count of lines to a cubic surface over an arbitrary field. This is joint work with Jesse Leo Kass

One can phrase interesting objects in terms of fixed points of group actions. For example, class numbers of quadratic extensions of Q can be expressed with fixed points of actions on modular curves. Derived functors are frequently better behaved than their non-derived versions, so it is useful to consider the associated derived functor, called the homotopy fixed points. Sullivan's conjecture is an equivalence between appropriately completed spaces of fixed points and homotopy fixed points for finite p-groups. It was proven independently by H. Miller, G. Carlsson, and J. Lannes. This talk will present Sullivan's conjecture and its solutions, and discuss analogues for absolute Galois groups conjectured by Grothendieck.

The solutions in \mathbb{C} to a system of polynomial equations form a nice topological space which is useful even for studying solutions to the polynomials over smaller fields such as R or even Q. To study solutions over Q or characteristic p fields, it is more useful to replace the notion of topological space with an object in a suitable category where one can do homotopy theory, such as the Morel-Voevodsky category for A^1 homotopy theory, and pro-spaces, where one has the étale homotopy type of a scheme. We will define A^1 homotopy theory, étale topological type, and an étale realization between them of Isaksen. We will use this to discuss Grothendieck's anabelian conjectures and obstructions to solutions to polynomial equations