Geometric Analysis and special Lagrangian geometry

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My research proposal focuses on special Lagrangian geometry, an important part of modern Differential Geometry and Geometric Analysis. Special Lagrangian submanifolds are high-dimensional geometric objects, discovered by geometers in 1982, that exist within special types of spaces called Calabi-Yau manifolds. Because of this, special Lagrangians are difficult to describe in immediately intuitive ways. However, they are exotic cousins of the everyday soap film. Mathematicians have studied the soap film or minimal surface equations since the 1700s and the tools they developed have gone on to play important roles across maths and the physical sciences. To give one prominent example, Lagrange invented the Calculus of Variations largely to study soap films.Initially, mathematicians studied special Lagrangians solely because of their remarkable geometric properties. However, in an unexpected development, in the mid 90s they appeared in String Theory, as a special type of brane---a higher-dimensional membrane-like object, as opposed to a 1-dimensional string. Based on physical intuition about branes, string theorists made surprising predictions about special Lagrangians, giving their mathematical study further impetus and stimulating work aimed at verifying these predictions.However, major mathematical obstacles arise because families of smooth special Lagrangians can be become badly behaved and form singularities. A smooth geometric object, like the sphere, when viewed at ever-increasing magnification begin to look flatter and flatter, approaching a fixed plane called the tangent plane. When a geometric object has singularities, there may be regions which, however much they are magnified, never become flat like a plane; the tip of an ordinary cone is a good example.This proposal aims to study the properties of singular special Lagrangians in order to resolve (a) whether the predictions from String Theory are correct and (b) whether it is possible to define an invariant of Calabi-Yau spaces by counting the number of certain kinds of special Lagrangians. If the singularities of special Lagrangians are too badly behaved then it will not be possible to ``count'' special Lagrangians in a useful way.A crucial aspect of the proposal is to develop a theory of typical k-dimensional families of special Lagrangians in typical (almost) Calabi-Yau manifolds and to understand what kinds of singularities can occur in these typical families. Recent research has shown that the singularities of special Lagrangians are very varied indeed and so the 'typical' assumption is crucial to help us cut down the number of ways that singularities form. A major technical problem we must overcome is that prior to making the `typical' assumption there are classes of singular special Lagrangians we might have to consider that are not currently under good geometric or analytic control. We must eventually either establish better geometric and analytic control of very general special Lagrangian singularities or else find a way to argue that special Lagrangians singularities that behave very badly are very far from `typical'. We expect that such a theory of typical singularities would have a big impact not just in special Lagrangian geometry but also in many other neighbouring parts of Geometry and possibly beyond.