Exceptional geometric structures required for string theory and M-theory: moduli spaces and formation of singularities

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My research lies in an area of higher dimensional geometry that is closely linked to theoretical physics. For about one hundred years, physicists have been searching for a theory that mathematically unifies gravity with quantum mechanics. An extremely promising candidate is M-theory, which describes the universe in terms of a 7-dimensional geometric shape (called a manifold) that is curved in a very special way. These shapes are called G2 manifolds. However, for the physical theory to be consistent with reality one requires these G2 manifolds to have certain corners (called singularities) which look like cones. The subset of G2 manifolds that have cone-like points are called G2 conifolds. The problem is that although we know thousands of examples of "smooth" G2 manifolds (those without cone-like points), there is still no mathematical proof that proper G2 conifolds actually exist. They definitely are expected to exist, and in abundance, both from physical arguments and from rigorous mathematical work of myself and Lotay.The principal short-term goal of my proposed research project is to construct the first ever examples of G2 conifolds, thereby providing a rigorous mathematical proof of their existence. This is an extremely important problem to solve, because it would give conclusive 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 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 general relativity.Another important short-term goal of my proposed research project is to understand the set of all possible G2 manifolds (called the "moduli space"), which is itself a geometric shape of very high dimension. Studying the way in which a smoothly deforming G2 manifold can develop cone-like points involves considering curves on the moduli space that reach the boundary. I propose 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 singularities and imposes restrictions on the associated physics.The long-term mathematical goal is to understand the 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 of these types of manifolds are candidates for grand unified theories in physics, particularly superstring theory and M-theory. Mathematically, G2 manifolds are very interesting objects because they share many common properties with Calabi-Yau manifolds, such as special types of submanifolds (smaller shapes sitting inside them) and connections (rules for measuring the rates of change on such shapes). In spite of this, there is sharp contrast, however, because for technical reasons G2 manifolds cannot be studied using the same tools that have been successful for Calabi-Yau manifolds, namely methods of classical algebraic geometry. This is because, rather than being locally modelled by the complex numbers like the Calabi-Yau manifolds are, they are locally modelled by an exceptional number system that exists only in 7 real dimensions. Since tools of algebraic geometry are not available, we need to study such manifolds instead using techniques from analysis, namely nonlinear partial differential equations. It is precisely for this reason that the mathematical analysis of G2 manifolds and G2 conifolds is so technically difficult, and why there are so many fewer mathematicians working in this modern area as opposed to the classical area of Calabi-Yau manifolds.