Final Report Summary - DIFFERENTIALGEOMETR (Geometric analysis, complex geometry and gauge theory)
The second component of the project involved the study of 7 and 8 dimensional Riemannian manifolds with exceptional holonomy groups. These also satisfy a version of the Einstein condition and there are important connections with “M-theory” in theoretical physics. The staff involved were the PI, J. Nordstrom and T. Walpuski (who completed a PhD in the area). Nordstrom’s work focused on the refinement of the “twisted connected sum” construction and on existence and singularity formation for associative submanifolds. Walpuski’s work focused on a version of the Yang-Mills equation in this context, and the failure of compactness due to “bubbling” on an associative submanifold. These are relatively undeveloped fields and the output of the project constitutes a major advance in our understanding of these special geometries.