Symmetric products of curves, scalar curvature, and positivity in algebraic geometry

Abstract: In this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of curves. I demonstrate that these spaces can be utilized to address nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, symmetric products of curves sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and Dranishnikov. As a natural generalization of this circle of ideas, I will also address the Gromov–Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions. This is joint work with Alexander Dranishnikov and Ekansh Jauhari.