Geometric analysis of special structures in high dimensions inspired from physics; including singularities, torsion, and geometric evolution

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My research is in higher dimensional geometry inspired by theoretical physics. Since Einstein's work in 1915, physicists have been searching for a theory that mathematically unifies gravity with quantum mechanics. A promising candidate is M-theory, that describes the universe using a 7-dimensional shape that is curved in a special way. Such shapes are called G2 manifolds. For the physical theory to be consistent with reality we need these G2 manifolds to have certain cone-like points. These are called G2 conifolds. Although we know thousands of examples of smooth G2 manifolds (without cone-like points), there is still no proof that proper G2 conifolds really exist. They definitely are expected to exist in abundance, both from physical arguments and rigorous mathematical work of myself and Lotay. The long-term goal is to understand the properties and structure of G2 manifolds as well as we understand Calabi-Yau manifolds, which are 6-dimensional shapes with similar properties that are much better understood. Both objects are candidates for grand unified theories in physics. Mathematically, G2 manifolds are very interesting because although they share many common properties with Calabi-Yau manifolds, for technical reasons G2 manifolds cannot be studied using the same tools that have been successful for Calabi-Yau manifolds, namely classical algebraic geometry. This is because, rather than being locally modelled by the complex numbers as are the Calabi-Yau manifolds, the G2 manifolds are instead locally modelled by an exceptional number system that can exist only in 7 dimensions. Because classical tools are not available, we must instead study G2 manifolds using methods of analysis, such as nonlinear partial differential equations. It is precisely for this reason that the mathematical analysis of G2 manifolds and G2 conifolds is so technically difficult. One objective of my research is to construct the first ever examples of G2 conifolds, providing rigorous proof of their existence. This is a very important problem to solve, as it would give mathematical justification for the feasibility of M-theory as a model of our physical universe. The method I propose to use is a generalization of a recently published method of constructing smooth G2 manifolds of myself and Joyce, which involves glueing onto the shape a particular family of spaces that are solutions to Einstein's equations of relativity. Another objective of my research is to understand the set of all possible G2 manifolds (the moduli space), which is itself a shape of high dimension. Studying the ways a smoothly deforming G2 manifold can develop cone-like points involves considering curves on the moduli space that reach the boundary. I plan to investigate this question by analyzing the curvature of the moduli space itself. Establishing upper bounds on this curvature gives quantitative information about the formation of cone-like points and imposes restrictions on the associated physics.