I will present a series of new developments in the subject of strong convergence, development in joint works: with David Gao, Aareyan Manzoor, Gregory Patchell on a new source of purely finite matricial fields; with David Gao on Toeplitz exactness; and with David Gao and Mahan Mj on strongly converging unitary representations of extensions by exact groups. These works feature a new systematic method for proving strong convergence of unitary representations, and answers various open problems in the field. As an example, we show that for any closed hyperbolic 3-manifold, there exists a sequence of unitary representations with finite images strongly converging to the left regular representation. In light of Antoine Song's approach towards Yau's conjecture on the existence of minimal surfaces of negative curvature in 3-spheres, our work can be accessed towards constructing minimal 3 dimensional submanifolds whose shape is hyperbolic 3-space, inside larger dimensional spheres.