In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. One research direction of continued interest has been the estimation of the smallest singular value. A measurement of matrix’s ``invertibility’’, quantitative bounds on the smallest singular value are important for a variety of tasks including establishing a circular law for a non-Hermitian random matrix and for proving stability of numerical methods. In view of the universality phenomena of random matrices, one tries to prove these estimates for more general matrix ensembles satisfying weaker assumptions.

In the geometric approach to proving smallest singular value estimates a key ingredient is the use of a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace. In this talk we will discuss a new distance theorem and its application to proving smallest singular value estimates for inhomogeneous random rectangular matrix with independent entries. We will also discuss how the recent resolution of the Slicing Conjecture, due to Klartag, Lehec, and Guan, implies smallest singular values estimates for a number of log-concave random matrix ensembles. In some cases, independent entries are no longer necessary!