Professor of Mathematics
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. 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
-holonomy spaces and special Lagrangian submanifolds
in Calabi-Yau spaces. In the past I have also studied singular
special Lagrangian n-folds.
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.
Current Research Interests
- Riemannian geometry
- Systems of elliptic partial differential equations
- Einstein manifolds, especially Ricci-flat manifolds
- Riemannian metrics with special or exceptional holonomy.
- Minimal submanifolds, particularly calibrated submanifolds
- Calibrated currents and their singularities and regularity, especially special Lagrangian and associative currents
- Singular spaces with special or exceptional holonomy
- Gauge theory on spaces with special and exceptional holonomy
- Riemannian collapse under Ricci-curvature bounds
Current Appointments & Affiliations
Education, Training, & Certifications
Advising & Mentoring
Presentations & Appearances
Service to the Profession
Academic & Administrative Activities
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