CAREER: Diophantine Analysis of Dynamical Systems

主要研究人员将研究动力系统理论中与实数的丢番图性质有关的问题。该项目的第一部分涉及开发半单流的组合模型，该模型推广了由连分式给出的模曲面上测地线流的符号描述。这些模型将为解决丢番图近似中的几个公开问题提供重要的工具，包括著名的Littlewood猜想，Schmidt猜想(关于格的逐次极小)，以及确定奇异向量集的Hausdorff维度。该项目的这一部分还可能涉及一些数值研究。该项目的第二部分集中在有理台球遍历理论的相关问题上。其中一个目标是检验这样一种猜想的正确性，即每一条非遍历的泰希穆勒测地线都是由Masur-Smillie类型的结构产生的。另一个目的是理解非遍历方向集的Hausdorff维度如何依赖于底层平移面或有理台球的丢番图性质。统一项目的两个组成部分的主题是从线性群作用下的欧几里德空间的离散子集的演化中提取有用信息的技术。首席研究人员遵循的方法是受到动力系统观点的启发，并已经导致了数论的突破。预计将通过这种方法解决更多尚未解决的问题。项目的第一部分对所谓的半单流的更好的理解很可能被用来开发高效的算法，用于生成对无理向量的有理逼近并在格中找到较短的向量。这些问题因其广泛的应用而引起计算机科学家的极大兴趣，特别是在密码学方面。在该项目的第二部分中，对非遍历方向的研究在一定程度上是由(首席研究员和他的合作者)最近发现的一种惊人的现象所推动的，这种现象被称为“Hausdorff维度二分法”，以前从未在台球动力学中观察到(这里的术语“台球”指的是某种类型碰撞的数学模型，而不是人们在台球厅观察到的活动)。更好地理解产生这一现象的机制可能会为解释临界现象的新模型提供潜在的基础，并可能引起物理学家的兴趣。在人力资源开发方面，该项目涉及培养研究生成为研究型数学家。

The principal investigator will work on problems from the theory of dynamical systems related to Diophantine properties of real numbers. The first part of the project involves the development of combinatorial models for semisimple flows that generalize the symbolic description of the geodesic flow on the modular surface given by continued fractions. These models will provide important tools for addressing several open problems in Diophantine approximation, including the famous Littlewood Conjecture, Schmidt's Conjecture (on successive minima of a lattice), and determining the Hausdorff dimension of the set of singular vectors. This part of the project may also involve some numerical studies. The second part of the project focuses on problems related to the ergodic theory of rational billiards. One objective will be to test the validity of the conjectural picture that every nonergodic Teichmuller geodesic arises from a Masur-Smillie-type construction. Another objective is to understand how the Hausdorff dimension of the set of nonergodic directions depends on the Diophantine properties of the underlying translation surface or rational billiard. The theme unifying the two components of the project is the technique of extracting useful information from the evolution of a discrete subset of Euclidean space under the action of a linear group. The approach followed by the principal investigator is inspired by a dynamical systems viewpoint and has already led to breakthroughs in number theory. Further open problems are expected to be solved via this approach. The improved understanding of so-called semisimple flows that will result from the first part of the project can likely be used to develop efficient algorithms for generating rational approximations to irrational vectors and finding short vectors in lattices. These problems are of immense interest to computer scientists for their numerous applications, especially to cryptography. The investigation of nonergodic directions in the second part of the project is motivated in part by the recent discovery (by the principle investigator and his collaborators) of a striking phenomenon known as the "dichotomy of Hausdorff dimension" that has never before been observed in the dynamics of billiards (the term "billiards" here refers to a mathematical model for a certain type of collision, not to the activity one observes in pool halls). A better understanding of the mechanism that produces this phenomenon may potentially provide the basis for a new model to explain critical phenomena and may be of interest to physicists. On the human resource development side, the project involves the training of graduate students to become research mathematicians.