Recent advances in the theory of holonomy

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2000-12-01

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Abstract

After its introduction by Élie Cartan, the notion of holonomy has become increasingly important in Riemannian and affine geometry. Beginning with the fundamental work of Marcel Berger, the classification of possible holonomy groups of torsion free connections, either Riemannian or affine, has continued to be developed, with major breakthroughs in the last ten years. I will report on the local classification in the affine case, Joyce's fundamental work on compact manifolds with exceptional holonomies and their associated geometries, and some new work on the classification of holonomies of connections with restricted torsion, which has recently become of interest in string theory.

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Scholars@Duke

Bryant

Robert Bryant

Phillip Griffiths Professor of Mathematics

My research concerns problems in the geometric theory of partial differential equations.  More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems.  Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.


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