Geometric Analysis and special Lagrangian geometry

我的研究方向是特殊拉格朗日几何，它是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高维几何物体，由几何学家于1982年发现，它们存在于特殊类型的空间中，称为Calabi-Yau流形。正因为如此，特殊的拉格朗日量很难用直观的方式来描述。然而，它们是日常肥皂电影的异域表亲。自18世纪以来，数学家们一直在研究肥皂膜或最小表面方程，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，拉格朗日发明了变分法，主要是为了研究肥皂膜。最初，数学家研究特殊的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，它们出现在弦理论中，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测，这进一步推动了他们的数学研究，并刺激了旨在验证这些预测的工作。然而，主要的数学障碍出现了，因为光滑的特殊拉格朗日族可能变得很糟糕并形成奇点。一个光滑的几何物体，如球体，在不断增大的放大率下观察时，会变得越来越平坦，接近于一个固定的平面，即切平面。当一个几何物体具有奇点时，可能存在一些区域，无论它们被放大多少，都不会变得像平面一样平坦；普通锥体的尖端就是一个很好的例子。本文旨在研究奇异特殊拉格朗日的性质，以解决(a)弦理论的预测是否正确，(b)是否可能通过计算某些特殊拉格朗日的数量来定义Calabi-Yau空间的不变量。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。该建议的一个关键方面是发展典型（几乎）Calabi-Yau流形中特殊拉格朗日的典型k维族的理论，并了解这些典型族中可能出现何种奇点。最近的研究表明，特殊拉格朗日量的奇点确实变化很大，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，在做出“典型”假设之前，我们可能不得不考虑一些奇异的特殊拉格朗日量，它们目前还没有得到很好的几何或解析控制。我们最终必须要么对非常一般的特殊拉格朗日奇点建立更好的几何和解析控制，要么找到一种方法来证明行为非常糟糕的特殊拉格朗日奇点离“典型”很远。我们期望这样的典型奇点理论不仅会在特殊的拉格朗日几何中产生巨大的影响，而且还会在几何的许多其他邻近部分甚至可能超出。

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.