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 7dimensional spaces called G_{2}holonomy manifolds, or G_{2}manifolds for short. These spaces also arise naturally in modern theoretical physics in the 11dimensional theory known as M theory. To get from 11 dimensions down to 4 dimensions it is necessary to 'compactify' on a 7dimensional space and to preserve the maximal degree of (super)symmetry this 7dimensional space should have G_{2}holonomy. In fact, realistic 4dimensional physics appears to demand singular G_{2}holonomy spaces and trying to construct compact singular G_{2}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 G_{2}holonomy spaces and special Lagrangian submanifolds in CalabiYau spaces. In the past I have also studied singular special Lagrangian nfolds.
 Contact Info:
Teaching (Fall 2023):
 MATH 620.01, SMOOTH MANIFOLDS
Synopsis
 Physics 205, TuTh 10:05 AM11:20 AM
 Education:
Ph.D.  University of Texas, Austin  2000 
 Keywords:
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
(More Publications)
 Haskins, M; Nordström, J, Cohomogeneityone solitons in Laplacian flow: local, smoothlyclosing
and steady solitons
(December, 2021) [abs]
 FOSCOLO, L; HASKINS, M; NORDSTRÖM, J, Complete noncompact g2manifolds from asymptotically conical calabiyau 3folds,
Duke Mathematical Journal, vol. 170 no. 15
(October, 2021),
pp. 33233416 [doi] [abs]
 Foscolo, L; Haskins, M; Nordström, J, Infinitely many new families of complete cohomogeneity one G2manifolds: G2analogues of the TaubNUT and EguchiHanson spaces,
Journal of the European Mathematical Society, vol. 23 no. 7
(January, 2021),
pp. 21532220 [doi] [abs]
 Foscolo, L; Haskins, M, New G2holonomy cones and exotic nearly Kahler structures on S^{6} and S^{3} x S^{3},
Annals of Mathematics, vol. 185 no. 1
(January, 2017),
pp. 59130 [doi] [abs]
 Mark, H; Hein, HJ; Johannes, N, Asymptotically cylindrical CalabiYau manifolds,
Journal of Differential Geometry, vol. 101 no. 2
(October, 2015),
pp. 213265 [doi] [abs]
 Recent Grant Support
 Special Holonomy In Geometry, Analysis and Physics, Simons Foundation, 2020/072023/06.
