We report on the development of localization methods useful for quadratic enumerative invariants, replacing the classical Gm-action with an action by the normalizer of the diagonal torus in SL2.
We discuss applications to quadratic counts of twisted cubics in hypersurfaces and complete intersections (joint with Sabrina Pauli) as well as work by Anneloes Vierever, and our joint work with Viergever on quadratic DT invariants for Hilbert schemes of points on P3 and on (P1)3.[-]

After inverting 2, the motivic sphere spectrum splits into a plus part and a minus part with respect to a certain natural involution. Cisinsky and Déglise have shown that, with rational coefficients, the plus part is given by rational motivic cohomlogy. With Ananyevskiy and Panin, we have computed the minus part with rational coefficients as being given by rational Witt-theory. In particular, this shows that the rational bi-graded homotopy sheaves of the minus sphere are concentrated in bi-degree (n,n). This may be rephrased as saying that the graded homotopy sheaves of the minus sphere in strictly positive topological degree are torsion. Combined with the result of Cisinski-Déglise mentioned above, this shows that the graded homotopy sheaves of the sphere spectrum in strictly positive topological degree and non-negative Tate degree are torsion, an analog of the classical theorem of Serre, that the stable homotopy groups of spheres in strictly positive degree are finite.[-]