Suppose X is a smooth projective variety, E and F are vector bundles on X, and M: E —> F is a map of vector bundles. For a positive integer k, we can define the kth determinantal variety of M to be the locus of points x in X for which the linear map M_x has rank at most k. Such varieties give some of the simplest examples of subvarieties of X which are not complete intersections.

Determinantal varieties are almost always singular, however there are several natural desingularizations defined using basic concepts from linear algebra. It is natural to ask what the relationship is between these resolutions. In this talk I will describe two such correspondence results. The first is related to mutations of quiver varieties and the second is related to wall crossing for Grassmann flops. This is based on joint work with Nathan Priddis and Yaoxiong Wen.