Dynamics on homogeneous spaces with applications to number theory.

度量丢芬图近似试图通过实数的近似性质来描述实数，或者更一般地说，是局部域上有限维向量空间中的点。具体地说，我们试图确定哪些属性是例外的，因为它们对于一个稀疏集是成立的。丢番图近似除了其固有的美之外，还有各种各样的应用。例如，具有某些特殊丢芬图性质的数在各种工程和物理问题中起着重要作用，例如在KAM理论和天体力学中。的确，丢番图近似法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，并且一直是这一研究领域的中心主题之一。换句话说，人们想要证明一个曲面，或者它的高维类似物具有这样的性质即具有良好近似性质的点的集合是稀疏的。这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使用了一种受Kleinbock和Margulis启发的方法，该方法使用了动力系统理论中的强大工具，特别是某些表现非常好的单能流。另一方面，高希和他的合著者艾因西德勒也对单幂流理论做出了基本贡献。拟议的研究有两个核心目标。对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对丢番图性质的继承问题有贡献。预计该项目的一部分将与s.w elani（约克）合作进行。S.Velani是丢番图近似的权威专家。此外，Ghosh和Velani对类似的问题有不同的方法，人们希望结合这些方法将为解决难题提供一个新的视角。第二个目标是对某些结构（称为局部场）上的均匀动力学研究做出贡献，特别是那些具有正特征的结构。项目的这一部分将与m . einsedler（俄亥俄州立大学）合作进行。这与Ghosh研究的动力学部分相对应，这个项目的目的是更好地，有希望地全面理解这些系统，这些系统被推测为行为非常良好的系统，尽管它们看起来非常复杂。这也与第一个目标有关，因为在这个项目的这一部分开发的工具很可能适用于数论问题。

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.