Scalar curvature and its implications on topology have been a focus of intense study in recent years. In this thesis, we compute Yamabe invariants of certain compact manifolds, including $\mathbb {RP}^n\setminus D^n$ in all dimensions. We provide a new computation of the Yamabe invariant of $\mathbb{RP}^3$ via harmonic functions and a monotonicity formula along the level set of the harmonic functions; this is a joint work with Liam Mazurowski. In collaboration with Dongyeong Ko, we establish a scalar curvature comparison and rigidity theorem in dimension $3$ via capillary minimal hypersurfaces in compact manifolds with non-empty boundary. We also formulate a conjecture concerning $m$-intermediate Ricci curvature that interpolates between the Cheeger–Gromoll splitting theorem and the $K(\pi,1)$ conjecture, highlighting a new perspective on the interplay between Ricci and scalar curvature. We prove the conjecture in dimensions $n \in {3,4,5}$ and verify most cases in dimension $6$. As a corollary, we prove that $6$-dimensional aspherical manifolds does not admit a metric with positive $4$-intermediate curvature, which is only slightly stronger than scalar curvature, providing evidence on the $K(\pi,1)$ conjecture in dimension $6$; this is a joint work with Liam Mazurowski and Tongrui Wang.