Seminar

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This talk will give an overview of the current landscape of research in Bryant's Laplacian flow, a geometric flow of closed $G_2$-structures in the direction of their Hodge Laplacian. This flow, it is hoped, will help us to understand manifolds with special holonomy and calibrated geometry. We will sketch a few of these motivating goals.



The rest of the talk will have two aims. The first will be to describe the features of Laplacian flow by direct comparison to other geometric flows, especially Ricci flow, and in this way to give a sense of the progress of Laplacian flow over the last few years. The second is to collect and summarize the various approaches taken to studying the flow, and to describe their relative successes and limitations. Finally, we plan to discuss open problems that are hopefully accessible to researchers in the two SLMath semester programs, without requiring deep expertise in special holonomy.