Research Interests

My research has mainly centered on the field of differential geometry, which applies the techniques of calculus to the study of curved objects in various different contexts. In early developments, it was used by Gauss to understand how the curvature of the Earth affected the accuracy of survey maps and, later, by Einstein in his formulation of general relativity. Today, differential geometry plays an important role in computer graphics, in general relativity, in theoretical high energy physics, and in analysis and topology, having been essential in the recent solution of Poincare's conjecture. My own work has involved the study of the systems of differential equations that naturally arise in geometric contexts and the proof of existence and uniqueness theorems for the solutions of these systems of equations. Notably, I proved the existence of geometries with exotic or exceptional holonomy and showed how they were related to various other geometric constructs of interest in both mathematics and physics. I am particularly interested in problems that have hidden symmetries or degeneracies caused by invariance under groups of diffeomorphisms. Currently, I am interested in the geometry of almost complex structures, nonholonomically constrained variational problems, conservation laws, integrable systems, and Finsler manifolds.

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Section 11: Mathematics