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Dynamics on homogeneous spaces with applications to number theory.

齐次空间动力学及其在数论中的应用。

基本信息

批准号：
EP/E045308/1
负责人：
Anish Ghosh
金额：
$ 24.17万
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE045308%2F1
关键词：
Dynamics homogeneous spaces applications number

项目摘要

Metric Diophantine approximation seeks to characterize the real numbers, or more generally points in a finite dimensional vector space over a local field, by their approximations properties. Specifically, one seeks to determine which properties are exceptional, in that they hold for a sparse set. In addition to its intrinsic beauty, Diophantine approximation has a wide variety of applications. For instance, numbers which possess certain ``exceptional Diophantine properties play an important role in various engineering and physical problems, for instance in KAM theory and celestial mechanics. Indeed, the study of Diophantine approximation can be traced back to the ancient Greeks who studied it precisely for its applicability. It turns out that showing that points on a surface satisfy these exceptional laws is much more difficult and has been one of the central themes in this area of research. In other words, one wants to show that a surface, or its higher dimensional analogue has the property that the set of points which have good approximation properties is sparse. This study of such problems is referred to as Diophantine approximation on manifolds and is one of Ghosh's key research areas. Ghosh uses an approach inspired by Kleinbock and Margulis which uses powerful tools from the theory of dynamical systems, in particular certain unipotent flows which are extremely well behaved. On the other hand, Ghosh and his co-author M.Einsiedler have also made basic contributions to the theory of unipotent flows. The proposed research has two core objectives. To make contributions to the study of metric Diophantine approximation on manifolds, particularly to the problem of inheritance of Diophantine properties by submanifolds of Euclidean space. It is expected that a part of this project will be undertaken in collaboration with S.Velani (York). S.Velani is a leading expert on Diophantine approximation. Moreover, Ghosh and Velani have different approaches to similar problems and one hopes that combining methods will provide a fresh perspective on difficlt problems.The second objective is to make contributions to the study of homogeneous dynamics over certain structures called local fields-particularly those with positive characteristic. This part of the project will be undertaken in collaboration with M.Einsiedler (Ohio State). This corresponds to the dynamical part of Ghosh's research and the aim of this project will be to gain a better, hopefully complete understanding of these systems which are conjectured to be extremely well behaved systems even though they appear very complicated. This is also related to the first objective because it is highly likely that toold developed in this part of the project will be applicable to number theoretic problems.

度量丢番图近似试图通过实数的近似属性来表征实数，或更一般地表征局部域上的有限维向量空间中的点。具体来说，人们试图确定哪些属性是例外的，因为它们适用于稀疏集。除了其内在美之外，丢番图近似还具有广泛的应用。例如，具有某些“特殊丢番图性质”的数在各种工程和物理问题中发挥着重要作用，例如在 KAM 理论和天体力学中。事实上，丢番图近似的研究可以追溯到古希腊人，他们正是为了它的适用性而研究它。事实证明，证明表面上的点满足这些特殊定律要困难得多，并且一直是该研究领域的中心主题之一。换句话说，人们想要证明一个表面或其高维类似物具有这样的特性：具有良好近似特性的点集是稀疏的。对此类问题的研究被称为流形上的丢番图近似，是 Ghosh 的重点研究领域之一。 Ghosh 采用了受 Kleinbock 和 Margulis 启发的方法，该方法使用动力系统理论中的强大工具，特别是某些表现极其良好的单能流。另一方面，Ghosh 和他的合著者 M.Einsiedler 也对单能流理论做出了基础贡献。拟议的研究有两个核心目标。为流形上的度量丢番图近似的研究，特别是欧几里得空间子流形继承丢番图性质的问题做出贡献。预计该项目的一部分将与 S.Velani（约克）合作开展。 S.Velani 是丢番图近似方面的领先专家。此外，Ghosh和Velani对类似问题有不同的方法，人们希望结合方法能为解决困难问题提供新的视角。第二个目标是为研究某些称为局域场的结构——特别是那些具有正特性的结构——的齐次动力学做出贡献。该项目的这一部分将与 M.Einsiedler（俄亥俄州）合作进行。这对应于戈什研究的动力学部分，该项目的目的是更好地、希望更全面地理解这些系统，尽管它们看起来非常复杂，但它们被认为是表现得非常好的系统。这也与第一个目标相关，因为项目这部分开发的工具很可能适用于数论问题。