In this talk, I will speak about an index theoretical approach to a series of Gromov’s questions related to positive scalar curvature. Our approach is to study the index pairing of the Dirac operator and vector bundles. The concrete construction of index pairing reveals a quantitative control of positive scalar curvature, which results from a new uncertainty principle for Fourier transforms. This is based on joint work with Zhizhang Xie and Guoliang Yu.