Equivariant curve counting and the local equivariant degree

Enumerative geometry asks for integral solutions to geometric questions, such as how many rational curves of degree d pass through through 3d-1 marked points lie on a surface. Recently, motivic and equivariant homotopy have been used to generalize classical enumerative results to non-closed fields and under the presence of a group action respectively. I will give an introduction to equivariantly enriched enumerative geometry by presenting a count of nodal orbits in an invariant pencil of plane conics, enriched in the Burnside Ring of a finite group. I will also discuss joint work with Kirsten Wickelgren on defining a global and local degree in stable equivariant homotopy theory, which we use to give an application to counting orbits of rational plane cubics through 8 general points invariant under a finite group action on CP^2. This defines the first equivariantly enriched count of rational curves valued in the representation ring and Burnside ring, which also recovers a Welchinger invariant when Z/2 acts on CP^2 by conjugation.