Abstract: The Atiyah-Bott localization theorem allows to express equivariant cohomology classes in terms of their residues on the fixed locus which turns out to be a powerful tool for problems in enumerative geometry: Ellingsrud and Strømme used it to give a count of smooth rational cubic curves on a complete intersection variety. Recently, Marc Levine found an analogue of the Atiyah-Bott localization theorem in equivariant cohomology with coefficients in the Witt ring of quadratic forms.
In the talk I will recall the Atiyah-Bott localization theorem and its quadratic version. Then I will apply it to get some quadratic enrichments of enumerative counts, like the count of lines on a cubic surface or the count of smooth rational cubic curves on a quintic threefold. This is based on joint work with Marc Levine.