  Topological data analysis (TDA) has had enormous success in science and
engineering in the past decade. Persistent topological Laplacians (PTLs)
overcome some limitations of persistent homology, a key technique in TDA, and
provide substantial insight to the behavior of various geometric and
topological objects. This work extends PTLs to directed flag complexes, which
are an exciting generalization to flag complexes, also known as clique
complexes, that arise naturally in many situations. We introduce the directed
flag Laplacian and show that the proposed persistent directed flag Laplacian
(PDFL) is a distinct way of analyzing these flag complexes. Example
calculations are provided to demonstrate the potential of the proposed PDFL in
real world applications.
