Geometric Analysis and Topology Seminar

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Singularity formation, singularity resolution and solitons in Bryant’s Laplacian flow

Speaker: Mark Haskins, Duke University

Location: Warren Weaver Hall 512

Date: Friday, February 21, 2025, 11 a.m.

Synopsis:

Bryant introduced a geometric flow on closed non-degenerate 3-forms on a compact 7-manifold called Laplacian flow. Stationary points of Laplacian flow are very special: they are so-called torsion-free G_2-structures. Every such torsion-free G_2-structure gives rise to a Ricci-flat metric and these structures are currently the only known source of Ricci-flat metrics on compact simply connected odd-dimensional manifolds. Bryant’s Laplacian flow provides a parabolic PDE approach to the construction of such torsion-free structures if one can understand long-time existence and convergence of the flow. This talk will introduce Bryant’s Laplacian flow describe some of its key geometric and analytic properties and then focus on recent progress in understanding this flow particularly issues related to finite-time singularity formation and singularity resolution and solitons within Laplacian flow. We will see some ways in which gradient Laplacian solitons behave like gradient Ricci solitons but also that they exhibit some behaviours that are impossible for Ricci solitons.