The number of degree d rational plane curves through 3d-1 generally chosen points is independent of the generally chosen points over the complex numbers. (There is 1 line through 2 points, 1 conic through 5, 12 rational degree 3 curves through 8...) An early success of Gromov--Witten theory was to give a recursive formula for these invariants. Over the real numbers, there are invariants due to Jean-Yves Welschinger, Cheol-Hyun Cho and Jake Solomon giving an open Gromov-Witten invariant equal to a signed count of real curves. It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. We develop an A1-degree, following Fabien Morel, which in certain cases is the pushforward in Hermitian K-theory. We compute the A1-degree of an evaluation map on the Kontsevich moduli space of stable rational maps to obtain a count of genus 0 curves on certain del Pezzo surfaces through the appropriate number of marked points. This count is valid for any field k of characteristic not 2 or 3. In particular, we define and compute some Gromov--Witten invariants over a finite field. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon. Time permitting, the talk will include joint work with Erwan Brugallé.
De Alain Connes
De Teresa Krick
De Teresa Krick
De Shintaro Nishikawa
![[1237] Moments de fonctions et $L$ stabilité homologique](/media/cache/video_light/uploads/video/Bourbaki.png)
