Volume-collapsed manifolds in Riemannian geometry and geometric inference

黎曼几何中的体积塌陷流形和几何推理

• 批准号: MR/W01176X/1
• 负责人: John Harvey
• 金额: $ 131.35万
• 依托单位: CARDIFF UNIVERSITY
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=MR%2FW01176X%2F1
• 关键词: Volume; collapsed; manifolds; Riemannian; geometry

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.

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，那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。如果我们能无限地把它卷起来，(I)它的表面积将为零，(Ii)它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，其面积的类比总是被称为“体积”。可以随心所欲地卷起的流形，如圆柱体，可以称为体积折叠流形。我将研究他们行为的两个方面，目的是解决几何学中长期存在的问题。产生的新知识将为支持其他数学家的工作提供新的理论见解，并在医疗保健、金融和工业等不同领域具有实际技术应用首先，似乎我们可以从它正在接近的较小限制空间中了解很多关于流形的信息。紧紧缠绕的圆柱体与线条有关；我们说圆柱体是线条上的“纤维束”。对于直线上的每一点，圆柱体上都有一个对应的小圆圈(纤维)。所有这些圆圈“捆绑”在一起，就构成了圆柱体。然而，这个例子很简单。一个简化的方面是没有曲率(圆柱体是由一张扁纸制成的)。另一个是只有两个维度。理解三维流形如何在保持曲率界限的同时坍塌，在证明庞加莱猜想方面发挥了重要作用。庞卡莱猜想是克莱数学研究所迄今为止唯一一个得到解决的千禧年奖问题。项目A的目标是对与给定极限空间相对应的体积塌陷流形进行分类。这将大大有助于理解曲率和形状如何相互作用的目标，这是几何学的主要研究领域之一。第二个方面是统计。给出一个未知流形中的随机点样本，我们拥有的点越多，我们就越有信心识别这个流形。然而，当歧管是体积折叠的时候，显然很难从它的有限空间中区分它；在这个例子中，区分圆柱体和直线。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表现得如何。当我们对几何对象进行统计时，体积折叠流形显然扮演着重要的角色。理解这些问题是将数学的全部严谨性带到令人兴奋和成功的用于海量数据集的新拓扑数据分析技术的唯一途径。B项目将开发具有强大数学保证的统计工具，并将寻找最优的程序，这意味着它们在最坏的情况下表现得和理论上可能的一样好。在这个项目中，我还将追求更多的实用目标，开发测试所用方法有效性的程序，将自我验证机制集成到数据分析工具中，并开发与企业的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，但就像从两个角度对同一对象进行任何研究一样，目的是每个研究都将相互启发，新的联系将被揭示。将时间和资源投入到这两个问题上是我们发现这些联系的唯一途径，而这一奖学金将使这样一个冒险的研究计划成为可能。

项目成果

Epidemiological waves - Types, drivers and modulators in the COVID-19 pandemic.

• DOI: 10.1016/j.heliyon.2023.e16015
• 发表时间: 2023-05
• 期刊: HELIYON
• 影响因子: 4
• 作者: Harvey, John;Chan, Bryan;Srivastava, Tarun;Zarebski, Alexander E.;Dlotko, Pawel;Blaszczyk, Piotr;Parkinson, Rachel H.;White, Lisa J.;Aguas, Ricardo;Mahdi, Adam
• 通讯作者: Mahdi, Adam

Topological Inference of the Conley Index

康利指数的拓扑推理

• DOI: 10.1007/s10884-023-10310-1
• 发表时间: 2023
• 期刊: Journal of Dynamics and Differential Equations
• 影响因子: 1.3
• 作者: Yim K