In the nineties, Voevodsky proposed a radical unification of
algebraic and topological methods. The amalgam of algebraic geometry and
homotopy theory that he and Fabien Morel developed is known as motivic
homotopy theory. Roughly speaking, motivic homotopy theory imports
methods from simplicial homotopy theory and stable homotopy theory into
algebraic geometry and uses the affine line to parameterize homotopies.
Voevodsky developed this theory with a specific objective in mind: prove
the Milnor conjecture. He succeeded in this goal and won the Fields
Medal for his efforts in 2002.
In this talk, I will start by recalling some facts in motivic homotopy
theory, and then present some results in motivic enumerative geometry
(Euler characteristic, trace formula, ramifications, computation of some
Chow-Witt groups, birational invariance). This is joint work with
Frédéric Déglise and Fangzhou Jin (see arXiv:2210.14832), and Tasos
Moulinos.Zoom notes: Meeting ID: 968 2220 8331