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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.

Isadore Singer connected analysis, topology and differential geometry. With Michael Atiyah, he proved the Atiyah-Singer index theorem, linking the analytic index of an elliptic operator to topological data.^[ref]

With Daniel Ray, he introduced analytic torsion using spectral information from Laplacians on differential forms. Their construction uses derivatives of spectral zeta functions to regularize otherwise divergent products of eigenvalues. The original paper proved invariance results and proposed equality with combinatorial torsion; that equality was a conjectural step in their paper.^[ref]

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]