New G<inf>2</inf>-holonomy cones and exotic nearly Kahler structures on S<sup>6</sup> and S<sup>3</sup> x S<sup>3</sup>
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2017-01-01
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© 2017 Department of Mathematics, Princeton University. There is a rich theory of so-called (strict) nearly Kahler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kahler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: The metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kahler 6-manifold. A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kahler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kahler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kahler structures in six dimensions.
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Foscolo, L, and M Haskins (2017). New G2-holonomy cones and exotic nearly Kahler structures on S6 and S3 x S3. Annals of Mathematics, 185(1). pp. 59–130. 10.4007/annals.2017.185.1.2 Retrieved from https://hdl.handle.net/10161/19606.
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Scholars@Duke
Mark Haskins
My research concerns problems at the intersection between Differential Geometry and Partial Differential Equations, particularly special geometric structures that arise in the context of holonomy in Riemannian geometry.
In Fall 2024 I was the lead organizer of the program Special Geometric Structures and Analysis, at the Simons Laufer Mathematical Institute in Berkeley California. Many of the lectures given at the program are available to watch.
From 2016-2024 I was the Deputy Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics. My colleague here at Duke, Robert Bryant, was the Collaboration Director. During the course of the Collaboration we organized over 35 research meetings. Most of the lectures from these meetings are available to watch.
Currently, I am particularly interested in special types of 7-dimensional spaces called G2-holonomy manifolds, or G2-manifolds for short. These spaces also arise naturally in modern theoretical physics in the 11-dimensional theory known as M theory. To get from 11 dimensions down to 4 dimensions it is necessary to 'compactify' on a 7-dimensional space and to preserve the maximal degree of (super)symmetry this 7-dimensional space should have G2-holonomy. In fact, realistic 4-dimensional physics appears to demand singular G2-holonomy spaces and trying to construct compact singular G2-holonomy spaces is one of my current research projects.
Manifolds with special holonomy also come equipped with special submanifolds, called calibrated submanifolds, and special connections on auxiliary vector bundles, called generalised instantons. I am particuarly interested in associative and coassociative submanifolds in G2-holonomy spaces and special Lagrangian submanifolds in Calabi-Yau spaces. In the past I have also studied singular special Lagrangian n-folds.
Recently I have become interested in using geometric flow techniques to try to construct G2-holonomy manifolds. This has led me to study singularity formation in Laplacian flow and the structure of solitons in Laplacian flow. I have found new types of shrinking , steady and expanding solitons in Laplacian flow.
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