Mark Haskins, Professor
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.
I am currently the Deputy Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics. My colleague here at Duke, Robert Bryant, is the Collaboration Director and currently Chair of the Mathematics department.
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.
- Contact Info:
Teaching (Fall 2023):
- MATH 620.01, SMOOTH MANIFOLDS
- Physics 205, TuTh 10:05 AM-11:20 AM
|Ph.D.||University of Texas, Austin||2000|
- Differential equations, Elliptic • Einstein manifolds • Geometric analysis • Geometric measure theory • Geometry, Riemannian • Holonomy groups • Parabolic equations and systems • Partial differential equations on manifolds; differential operators
- Recent Publications
- Haskins, M; Nordström, J, Cohomogeneity-one solitons in Laplacian flow: local, smoothly-closing
and steady solitons
(December, 2021) [abs]
- FOSCOLO, L; HASKINS, M; NORDSTRÖM, J, Complete noncompact g2-manifolds from asymptotically conical calabi-yau 3-folds,
Duke Mathematical Journal, vol. 170 no. 15
pp. 3323-3416 [doi] [abs]
- Foscolo, L; Haskins, M; Nordström, J, Infinitely many new families of complete cohomogeneity one G2-manifolds: G2analogues of the Taub-NUT and Eguchi-Hanson spaces,
Journal of the European Mathematical Society, vol. 23 no. 7
pp. 2153-2220 [doi] [abs]
- Foscolo, L; Haskins, M, New G2-holonomy cones and exotic nearly Kahler structures on S6 and S3 x S3,
Annals of Mathematics, vol. 185 no. 1
pp. 59-130 [doi] [abs]
- Mark, H; Hein, HJ; Johannes, N, Asymptotically cylindrical Calabi-Yau manifolds,
Journal of Differential Geometry, vol. 101 no. 2
pp. 213-265 [doi] [abs]
- Recent Grant Support
- Special Holonomy In Geometry, Analysis and Physics, Simons Foundation, 2020/07-2023/06.