I will introduce some basic features of $G_2$-geometry, a geometry peculiar to 7 dimensions defined in terms of the exceptional compact simple Lie group $G_2$. I will then describe a geometric flow on 3-forms, due to Robert Bryant, called Laplacian flow that aims to produce Riemannian manifolds with holonomy group $G_2$. My talk will concentrate on certain special solutions to Laplacian flow called solitons: in particular I will describe a recent construction of noncompact shrinking, steady and expanding solitons in Laplacian flow all with asymptotically conical geometry. In other better-understood geometric flows (e.g. Ricci flow and mean curvature flow) such solitons have played a key role in understanding singularity formation and hence in understanding the long-time behaviour of these flows.

This is joint work with Johannes Nordström and also in part with Rowan Juneman (both at Bath).

Bio: Mark Haskins is a British mathematician. He earned his Doctorate from the University of Texas at Austin in 2000, under the supervision of Abel Prize winner Karen Uhlenbeck. He held postdoc positions at Johns Hopkins and IHES, before he joined Imperial College London in 2005. He remained at Imperial until 2017 when he moved as a Professor to University of Bath. In 2019 he took up his current position as Professor at Duke University.

He is currently the Deputy Collaboration Director for the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics and a member of the Collaboration's Scientific Steering Committee. In his capacity as Deputy Collaboration Director he has overseen the organization (both scientific and logistical) of many of the Collaboration's regular Collaboration meetings.

His research is in differential geometry and geometric analysis. Haskins has made important contributions to the areas of calibrated geometry and special and exceptional holonomy metrics.