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By Michael Atiyah and Raoul Bott.

Verified locator: Annals of Mathematics 86(2), September 1967, pp.374–407; introduction, isolated fixed-point formula.

Michael Atiyah connected topology, geometry and analysis. With Isadore Singer, he proved the Atiyah-Singer index theorem, relating an analytic index of an elliptic operator to topological data. Oxford's mathematical memorial explains its connection with Riemann-Roch theorem and Gauss-Bonnet theorem formulas.^[ref]

His work with Raoul Bott provides another local-to-global bridge: a holomorphic Lefschetz fixed-point formula expresses a global alternating trace in terms of contributions at fixed points. Such results make local geometric calculations determine invariants of global spaces of solutions.^[ref]

Raoul Bott made fundamental contributions to topology and differential geometry, with applications to Lie groups and differential operators. Harvard's departmental memorial documents this range of work.^[ref]

With Michael Atiyah, he developed a holomorphic Lefschetz fixed-point formula for elliptic complexes. In the holomorphic setting, global traces on cohomology become a sum of local contributions from nondegenerate fixed points. This connects a map's behavior at isolated points with its action on global solution spaces.^[ref]