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Daniel Ray coauthored the 1971 analytic torsion paper with Isadore Singer. The original prints D. B. Ray; the author-hosted account by Singer's former student John Lott explicitly identifies him as Daniel Ray.^[ref]

Their construction extracts a geometric invariant from spectra of Laplacians on differential forms. A spectral zeta function makes it possible to define a regularized determinant, which contributes to analytic torsion. Their paper establishes metric-independence under its hypotheses and motivates comparison with a combinatorial invariant.^[ref]

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]