Enumerating Subspaces Relative to Linear Operators

Abstract

This thesis serves three broad purposes. It delves into understanding the positivity phenomena observed in solutions to classical and contemporary enumeration problems in finite vector spaces, such as counting subspaces and flags of various types in a vector space. It offers a non-standard and elementary approach to grasping the geometry of Grassmannians and flag varieties. Finally, it serves as an exposition for the intriguing problem of comprehending the behavior of subspaces under repeated application of a linear map through the study of subspace profiles.