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Special holonomy: geometric flow and boundary value problems

特殊完整：几何流和边值问题

基本信息

批准号：
EP/K010980/1
负责人：
Jason Lotay
金额：
$ 29.81万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK010980%2F1
关键词：
Special holonomy geometric flow boundary

项目摘要

A fundamental notion in my research area is a Riemannian metric: this allows us to describe how the curvature of a geometric object varies from point to point. For example, on the surface of a round sphere like a football the metric is the same at every point, whereas on a rugby ball the metric is much more curved near the ends than the middle. The football and the rugby ball are the same basic geometric shape (i.e. a sphere) with different metrics, because you can imagine squashing and stretching to transform one to the other. Intuitively the round metric is, in some sense, the "best" metric on the sphere. One of the biggest problems in geometry is to find "optimal" metrics and has been studied for more than a hundred years, yet continues to be at the forefront of modern research. The quest for optimal metrics has led to pioneering research in mathematics and to the development of major new techniques.An important piece of data associated with a Riemannian metric is its holonomy group. A natural class of optimal metrics are those with so-called special holonomy groups. Two particular examples of geometric objects with metrics with special holonomy are called hyperkaehler and G_2 manifolds (whose dimension has to be a multiple of four or be seven, respectively). Some of the greatest problems in the field are to find examples of these metrics and to determine necessary and sufficient conditions on a given geometric object which ensure the existence of such a metric. The aim of the proposed project is to shed light on these problems using two completely different approaches.To tackle the problem for G_2 manifolds we intend to follow an elegant approach using a geometric flow. Geometric flow techniques have been employed to prove celebrated results in geometry and topology but are also used in engineering applications, for example Mean Curvature Flow is used as a robust means to remove noise from empirical data such as occurs when obtaining brain images from various scanners. The flow allows us to start with a simpler metric and evolve it so that it approaches the G_2 holonomy metric, in a similar way to how heat dissipates from a heat source. The general equation is very complicated, so we consider the simpler situation where the seven-dimensional objects have symmetries.The other half of the project, for hyperkaehler and G_2 manifolds, is to consider the boundary value problem. Such boundary value problems arise throughout geometry and analysis, but also occur naturally in physical applications such as modelling bending beams in engineering and interactions between molecules and cells in biology. These problems are typically substantially more challenging than so-called initial value problems, like geometric flows, so we again simplify the problem, now by considering perturbations of a known solution. In this way we aim to identify which deformations of the boundary metric can be extended to define special holonomy metrics and thus hopefully find new examples of such metrics.

我的研究领域中的一个基本概念是黎曼度量：它允许我们描述几何对象的曲率如何随点变化。例如，在像足球这样的圆形球体的表面上，度量值在每个点上都是相同的，而在橄榄球上，度量值在两端比中间弯曲得多。足球和橄榄球是相同的基本几何形状(即球体)，具有不同的度量，因为您可以想象挤压和拉伸将两者转换为另一个。从某种意义上说，圆形度量是球面上的“最佳”度量。几何学中最大的问题之一就是找到“最优”的度量标准，这一问题已被研究了一百多年，但仍处于现代研究的前沿。对最优度量的追求导致了数学上的开创性研究和重大新技术的发展。与黎曼度量相关的一个重要数据是它的完整群。一类自然的最优度量是那些具有所谓特殊完整群的度量。具有特殊完整度规的几何对象的两个特殊例子称为Hyperkaehler和G_2流形(它们的维度必须分别是4的倍数或7)。该领域的一些最大的问题是寻找这些度量的例子，并确定在给定的几何对象上确保这种度量存在的充要条件。这个项目的目的是用两种完全不同的方法来阐明这些问题。为了解决G_2流形的问题，我们打算采用一种优雅的方法，使用几何流。几何流技术已被用于证明几何和拓扑学中的著名结果，但也被用于工程应用，例如，平均曲率流被用作从经验数据中去除噪声的稳健方法，例如从各种扫描仪获取脑图像时发生的噪声。这个流允许我们从一个更简单的度规开始，并将其演化为接近G_2完整度规，类似于热量从热源散失的方式。一般方程非常复杂，所以我们考虑较简单的情况，即七维物体具有对称性。对于Hyperkaehler流形和G_2流形，方案的另一半是考虑边值问题。这类边值问题不仅存在于几何学和分析领域，而且在物理应用中也自然存在，如工程中的弯曲光束建模和生物学中分子与细胞之间的相互作用。这些问题通常比所谓的初值问题，如几何流，具有更大的挑战性，所以我们再次简化问题，现在考虑已知解的扰动。通过这种方式，我们的目标是确定边界度量的哪些变形可以扩展以定义特殊的完整度量，从而有望找到此类度量的新示例。