Dynamics on homogeneous spaces with applications to number theory.

度量丢番图逼近试图通过局部域上的有限维向量空间中的点的逼近性质来表征真实的数。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。丢番图近似除了其内在的美之外，还有着广泛的应用。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。的确，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，证明曲面上的点满足这些例外定律要困难得多，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其高维类似物具有这样的性质，即具有良好近似性质的点集是稀疏的。这类问题的研究被称为流形上的丢番图逼近，是高希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幂流。另一方面，Ghosh和他的合著者M.Einsiedler也对幂幺流理论做出了基本贡献。这项研究有两个核心目标。为流形上度量丢番图逼近的研究，特别是欧氏空间子流形的丢番图性质的继承问题做出贡献。预计该项目的一部分将与S.Velani（约克）合作进行。S.Velani是丢番图近似的主要专家。此外，Ghosh和Velani对类似问题有不同的方法，人们希望结合方法将为困难问题提供一个新的视角。第二个目标是为研究被称为局部场的结构上的齐次动力学做出贡献，特别是那些具有正特征的结构。项目的这一部分将与M.Einsiedler（俄亥俄州）合作进行。这对应于Ghosh研究的动力学部分，该项目的目的是更好地，希望完整地了解这些系统，这些系统被证明是非常良好的系统，即使它们看起来非常复杂。这也与第一个目标有关，因为在项目的这一部分中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.