## Overview

My research concerns problems at the intersection between

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 7-dimensional spaces called

Manifolds with special holonomy also come equipped with special submanifolds, called

**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 7-dimensional spaces called

**G**, or G_{2}-holonomy manifolds_{2}-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 G_{2}-holonomy. In fact, realistic 4-dimensional 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 Calabi-Yau spaces. In the past I have also studied*singular*special Lagrangian n-folds.## Current Appointments & Affiliations

Professor of Mathematics
·
2019 - Present
Mathematics,
Trinity College of Arts & Sciences

## Education, Training & Certifications

University of Texas, Austin ·
2000
Ph.D.