Questions around scalar curvature link geometric analysis, differential geometry, and algebraic topology, and they are closely tied to the mathematical foundations of general relativity. Scalar curvature is the trace of the Ricci curvature and measures how volumes deviate from Euclidean geometry at small scales, but its global geometric meaning and rigidity consequences remain subtle. Two main toolkits dominate the subject: spinorial Dirac/index methods going back to Lichnerowicz (powerful, but requiring a spin structure) and minimal-hypersurface methods initiated by Schoen and Yau (facing dimension and singularity issues). Recent progress has been sparked in part by Gromov's quantitative comparison and rigidity questions and by ongoing attempts to reach a deeper geometric understanding of lower scalar-curvature bounds.
The project treats classical landmark results, such as the torus obstruction to positive scalar curvature, together with newer quantitative problems from a conceptually unified standpoint: comparison principles for scalar and mean curvature along maps between Riemannian manifolds. Guided by this perspective, it aims to bridge long-standing gaps between existing techniques via generalizations of Dirac-operator methods (building on recent refinements of these methods) and complementary analytic ideas (including Bochner-type arguments where applicable). The objectives include a higher-index spinorial framework that replaces traditional degree hypotheses, new boundary-value tools for Dirac-type operators (including C*-algebra coefficients), extensions to open, incomplete, and singular settings, and a systematic study of almost-rigidity phenomena near the equality cases of such comparisons. The expected impact is a flexible toolkit that yields sharp rigidity, extremality, and comparison results, and contributes to synthetic characterizations of lower scalar-curvature bounds.
