Title: Counting with quadratic forms
Abstract: Many theorems in enumerative geometry are restricted to algebraically closed fields. For example, a conic and cubic in the plane intersect 6 times over the complex numbers, but some of these intersections go missing over the reals. In this talk, I will discuss how to use tools from motivic homotopy theory to do enumerative geometry over arbitrary fields. Instead of integer-valued counts, these tools produce equations of quadratic forms. Invariants of these quadratic forms recover classical theorems from complex and real enumerative geometry and imply new theorems over other fields.