# Solitons in Bryant's G_2-Laplacian flow. Duke University Geometry and Topology seminar. Duke University. September 27, 2021

Invited Talk

The geometry of 3-forms in dimension 7 is very special, because of the existence of stable 3-forms, i.e. 3-forms whose orbit under GL(7) is an open subset of the space of all 3-forms. Such stable 3-forms turn out to be intimately related to the exceptional Lie group G_2, to the octonions and to so-called exceptional holonomy metrics, that is, Riemannian metrics whose holonomy group is contained in the group G_2. I will describe a geometric flow on closed 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 recent constructions of non-compact shrinking, steady and expanding solitons in Laplacian flow. In other better-understood geometric flows, e.g. Ricci flow and mean curvature flow, solitons have played a key role in understanding singularity formation and hence in understanding the long-time behaviour of these flows. Time permitting I will make some comparisons with known solitons in Ricci flow and Lagrangian mean curvature flow. This is joint work with Johannes Nordström and also in part with Rowan Juneman (both at Bath).

### Service Performed By

- Haskins, Mark Professor of Mathematics

### Date

- September 27, 2021

### Service or Event Name

- Duke University Geometry and Topology seminar

### Host Organization

- Duke University

### Location or Venue

- Duke University Math dept