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，如果我们将其滚成圆柱体，则在数学家的眼中，该圆柱体仍然是二维的。我们向上卷起纸的越紧，圆柱体的表面积越小。如果我们能够无限紧密地将其卷起，它将（i）表面积为零，并且（ii）不再是圆柱体，而是一条线。该项目如何研究这种行为对更复杂形状的概括，从而占据了更多的尺寸。这些形状称为歧管，一旦它们至少具有三个维度，类比与其区域的类比总是称为“音量”。可以按照我们喜欢的方式“卷起”的歧管，例如圆柱体，可以称为“体积汇总的歧管”。我将研究其行为的两个方面，目的是解决几何学中的长期问题。所产生的新知识都将提供新颖的理论见解，以支持其他数学家的工作，并在医疗保健，金融和行业之类的各种领域中具有实际的技术应用，似乎我们可以从较小的限制空间中告诉很多关于歧视的信息。紧密包装的圆柱与线路有关。我们说气缸是线上的“纤维捆”。对于线上的每个点，气缸上都有一个小圆圈（“纤维”），所有这些电路都“捆绑”在一起，加在一起。但是，这个示例很简单。一个简化的方面是缺乏曲率（圆柱体由平坦的纸制成）。另一个是只有两个维度。了解三维流形在维持曲率绑定的同时如何崩溃，在庞加莱猜想的证明中发挥了重要作用，这是迄今为止解决的克莱数学学院的“千年奖”问题。这将极大地有助于理解曲率和形状相互作用的目标，这是几何学研究的主要研究领域之一。第二个方面是统计。给定一个从未知歧管中的随机示例，我们就越拥有识别歧管的信心就越多。但是，当歧管被体积汇合时，显然很难将其与极限空间区分开。在此示例中，将圆柱与线路区分开。在数学统计中，我们试图了解统计程序在最挑战的情况下的表现。当我们执行几何对象的统计数据时，体积胶囊的歧管显然具有重要作用。理解这些问题是将数学的全部严格性带入令人兴奋而成功的新拓扑数据分析技术中使用的唯一方法。项目B将开发具有强大数学保证的统计工具，并将搜索最佳的过程，这意味着它们在最坏情况下以及在最坏情况下以及在最坏情况下的表现也是如此。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方法的有效性，将自然验证机制整合到数据分析工具中，并与企业建立联系，以使用几何学意识到的数据分析。项目A和B将使用截然不同的方法，但是，从两个观点中对同一对象的任何研究都可以揭示出其他研究的信息，并且将会揭示其他连接。致力于追求这两个问题的时间和资源是我们发现这些联系的唯一途径，这一奖学金将使这样的冒险研究计划正是如此。

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