Geometric Analysis and special Lagrangian geometry

我的研究计划侧重于特殊拉格朗日几何，现代微分几何和几何分析的重要组成部分。特殊拉格朗日子流形是1982年由几何学家发现的高维几何对象，存在于称为卡-丘流形的特殊类型空间中。正因为如此，特殊的拉格朗日函数很难用直观的方式描述。然而，它们是日常肥皂膜的异国表亲。自18世纪以来，数学家们一直在研究肥皂膜或最小曲面方程，他们开发的工具在数学和物理科学中发挥了重要作用。举一个突出的例子，拉格朗日发明变分法主要是为了研究肥皂膜。最初，数学家研究特殊拉格朗日量仅仅是因为它们显著的几何性质。然而，在一个意想不到的发展中，在90年代中期，它们出现在弦论中，作为一种特殊类型的膜-一种高维的膜状物体，而不是一维的弦。基于对膜的物理直觉，弦理论家对特殊拉格朗日量做出了令人惊讶的预言，这给他们的数学研究带来了进一步的动力，并激发了旨在验证这些预言的工作。然而，主要的数学障碍出现了，因为光滑的特殊拉格朗日量族可能变得行为不良并形成奇点。一个光滑的几何物体，如球体，当在不断增加的放大倍数下观察时，开始看起来越来越平，接近一个被称为切平面的固定平面。当一个几何物体有奇点时，可能会有一些区域，无论它们被放大多少，永远不会像平面一样平坦;这个建议的目的是研究奇异特殊拉格朗日量的性质，以解决（a）弦论的预测是否正确和（B）是否可能定义卡拉比不变量。通过计算某些特殊拉格朗日量的个数，我们可以得到Yau空间的一些结果。如果特殊拉格朗日量的奇点表现得太糟糕，那么就不可能以一种有用的方式“计数”特殊拉格朗日量。该提案的一个关键方面是发展一个典型（几乎）卡-丘流形中特殊拉格朗日量的典型k维族的理论，并理解在这些典型族中可能发生什么样的奇点。最近的研究表明，特殊拉格朗日函数的奇点确实是多种多样的，因此“典型”假设对于帮助我们减少奇点形成的方式至关重要。我们必须克服的一个主要技术问题是，在作出“典型”假设之前，我们可能不得不考虑目前没有得到良好的几何或分析控制的奇异特殊拉格朗日函数类。我们必须最终建立更好的几何和分析控制非常一般的特殊拉格朗日奇点或其他找到一种方法来论证特殊拉格朗日奇点的行为非常糟糕的是非常远离“典型”。我们预计，这样一个理论的典型奇点将有很大的影响，不仅在特殊的拉格朗日几何，但也在许多其他邻近部分的几何和可能超越。

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.