Volume-collapsed manifolds in Riemannian geometry and geometric inference

一张纸是一个二维物体，如果我们把它卷成一个圆柱体，在数学家看来，这个圆柱体仍然是二维的。我们把纸卷得越紧，得到的圆柱体的表面积就越小。如果我们能把它无限紧密地卷起来，它将（i）表面积为零，（ii）不再是真正的圆柱体，而是一条线。这个项目研究这种行为如何推广到更复杂的形状，占据更多的维度。这些形状被称为流形，一旦它们至少有三个维度，它们的面积的类比总是被称为“体积”。可以像圆柱体那样尽可能紧地“卷起”的歧管可以称为“体积收缩歧管”。我将研究它们行为的两个方面，目的是解决几何学中长期存在的问题。产生的新知识将提供新的理论见解，以支持其他数学家的工作，并在医疗保健，金融和工业等不同领域具有实际的技术应用首先，我们似乎可以从它正在接近的较小的极限空间中了解很多关于流形的信息。紧密缠绕的圆柱体与线有关;我们说圆柱体是线上的“纤维束”。对于线上的每一点，圆柱体上都有一个小圆圈（“纤维”）。所有这些圆圈“捆绑”在一起，加起来就是圆柱体。然而，这个例子很简单。一个简化的方面是缺乏曲率（圆柱体是由一张平的纸制成的）。另一个是只有两个维度。了解三维流形如何在保持曲率边界的情况下坍缩，对于庞加莱猜想的证明起到了重要作用，庞加莱猜想是迄今为止唯一解决的克莱数学研究所“千年奖问题”之一。项目A的目标是对给定极限空间对应的体积坍缩流形进行分类。这将大大有助于理解曲率和形状如何相互作用的目标，这是几何学研究的主要领域之一。给定来自未知流形的点的随机样本，我们拥有的点越多，我们就越有信心识别流形。然而，当流形是体积收缩的时，显然很难将它与它的极限空间区分开;在这个例子中，区分圆柱与直线是非常困难的。在数理统计中，我们试图了解统计程序在最具挑战性的情况下的表现。当我们对几何对象进行统计时，体积塌陷流形显然起着重要作用。理解这些问题是将数学的全部严谨性引入到用于大规模数据集的令人兴奋和成功的新拓扑数据分析技术中的唯一途径。项目B将开发具有强大数学保证的统计工具，并将搜索最佳程序，这意味着它们在最坏情况下的性能在理论上是可能的。在这个项目中，我还将追求更多的应用目标，开发程序来测试所使用的方法的有效性，将自我验证机制集成到数据分析工具中，并与业务建立联系，以使用几何感知数据分析。项目A和B将使用非常不同的方法，但正如从两个角度对同一物体进行研究一样，其目的是使每一项研究都能为另一项研究提供信息，并揭示新的联系。投入时间和资源来追求这两个问题是我们发现这些联系的唯一途径，而这项奖学金将使这样一个冒险的研究计划成为可能。

A piece of paper is a two-dimensional object and, if we roll it up into a cylinder, that cylinder is still two-dimensional, in the eyes of mathematicians. The more tightly we roll up the paper, the smaller the surface area of the resulting cylinder. If we could roll it up infinitely tightly it would (i) have zero surface area and (ii) not really be a cylinder any more, but rather a line.This project studies how this behaviour generalises to more complicated shapes which take up more dimensions. These shapes are called manifolds, and once they have at least three dimensions the analogy to their area is always called the 'volume'. Manifolds which can be 'rolled up' as tightly as we like, such as the cylinder, can be called 'volume-collapsed manifolds'. I will study two aspects of their behaviour, with the aim of addressing long-standing questions in geometry. The new knowledge produced will both provide novel theoretical insights to support the work of other mathematicians, and have practical technological applications in sectors as diverse as healthcare, finance and industryFirstly, it would seem that we can tell a lot about the manifold from the smaller limit space that it is approaching. The tightly wrapped cylinder is related to the line; we say that the cylinder is a 'fibre bundle' over the line. For every point on the line, there is a little circle (the 'fibre') on the cylinder which corresponds. All these circles 'bundled' up together add up to the cylinder. However, this example is a simple one. One simplifying aspect is the lack of curvature (a cylinder is made from a flat piece of paper). Another is that there are only two dimensions. Understanding how three-dimensional manifolds collapse while maintaining a curvature bound played a significant role in the proof of the Poincaré Conjecture, the only one of the Clay Mathematics Institute's 'Millennium Prize Problems' to have been solved so far.Project A will aim to classify the volume-collapsed manifolds corresponding to a given limit space. This will contribute greatly to the goal of understanding how curvature and shape interact, which is one of the major fields of research in geometry.The second aspect is statistical. Given a random sample of points from an unknown manifold, the more points we have the more confident we can be of identifying the manifold. However, when the manifold is volume-collapsed, it will clearly be very difficult to distinguish it from its limit space; in this example, to distinguish cylinder from line. In mathematical statistics, we seek to understand how well statistical procedures perform in the most challenging cases. When we are carrying out statistics on geometric objects, volume-collapsed manifolds clearly have a major role to play. Understanding these questions is the only way to bring the full rigour of mathematics to the exciting and successful new topological data analysis techniques being used on massive data sets.Project B will develop statistical tools which come with strong mathematical guarantees and will search for procedures which are optimal, meaning that they perform as well as is theoretically possible under the worst case scenario. In this project I will also pursue more applied goals, developing procedures to test the validity of methods being used, integrating self-validating mechanisms into data analysis tools and developing links with business to use geometrically-aware data analysis.Projects A and B will use very different methods, but as with any study of the same object from two points of view the aim is that each study will inform the other and that new connections will be revealed. Dedicating time and resources to the pursuit of both problems is the only way for us to discover those connections, and this Fellowship will enable exactly such an adventurous research programme.