### September 11, 2022

TITLE: Geometric Flows of 3-Sasakian Structures

ABSTRACT: Geometric flows of G_2-Structures are expected to be valuable tools for determining when a G_2-Structure with torsion may be deformed to one which is torsion-free. Several flows of G_2-Structures have been proposed to provide insight into this question, including the Laplacian flow and the Laplacian coflow. Here we consider an alternative application of these geometric flows to the study of Nearly Parallel G_2-Structures, specifically those originating from 3-Sasakian geometry. We write down an ansatz for co-closed G_2-Structures given in terms of the 3-Sasakian data and consider how scaled versions of the Laplacian flow and coflow behave when we start the flows at one of these structures. These results provide us with insight into the stability/instability of the Nearly Parallel G_2-Structures which are special co-closed G_2-Structures in this ansatz. We then can compare these stability results with the analogous conclusions for the scaled Ricci flow starting at a G_2-metric corresponding to our ansatz for the G_2-Structure. This is joint work with Jason Lotay.