CAREER: Diophantine Analysis of Dynamical Systems

职业：动力系统的丢番图分析

• 批准号: 0956209
• 负责人: Yitwah Cheung
• 金额: $ 40万
• 依托单位: San Francisco State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0956209&HistoricalAwards=false
• 关键词: CAREER; Diophantine; Analysis; Dynamical; Systems

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.

首席研究员将研究与实数丢番图性质相关的动力系统理论问题。该项目的第一部分涉及开发半简单流的组合模型，该模型概括了由连续分数给出的模表面上测地流的符号描述。这些模型将为解决丢番图近似中的几个开放问题提供重要的工具，包括著名的利特伍德猜想、施密特猜想（关于晶格的连续最小值），以及确定奇异向量集的豪斯多夫维数。项目的这一部分可能还涉及一些数值研究。该项目的第二部分重点关注与理性台球遍历理论相关的问题。一个目标是测试每一个非遍历 Teichmuller 测地线都源自 Masur-Smillie 型构造的猜想图的有效性。 另一个目标是了解非遍历方向集的豪斯多夫维数如何取决于底层平移表面或有理台球的丢番图性质。统一该项目两个组成部分的主题是在线性群作用下从欧几里得空间离散子集的演化中提取有用信息的技术。 首席研究员所采用的方法受到动力系统观点的启发，并且已经在数论方面取得了突破。预计进一步的开放问题将通过这种方法得到解决。该项目第一部分对所谓的半简单流的更好理解可能会被用来开发有效的算法，用于生成无理向量的有理逼近并在格中查找短向量。这些问题因其众多的应用而引起了计算机科学家的极大兴趣，尤其是密码学。该项目第二部分对非遍历方向的研究部分是由于最近（由主要研究者和他的合作者）发现了一种被称为“豪斯多夫维度二分法”的惊人现象，这种现象以前从未在台球动力学中观察到过（这里的术语“台球”指的是某种类型碰撞的数学模型，而不是人们在台球中观察到的活动） 厅）。更好地理解产生这种现象的机制可能会为解释关键现象的新模型提供基础，并且可能会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数学家。