Markoff Surfaces and Superstrong Approximation

马尔可夫曲面和超强逼近

• 批准号: 1603715
• 负责人: Alexander Gamburd
• 金额: $ 18万
• 依托单位: CUNY Graduate School University Center
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1603715&HistoricalAwards=false
• 关键词: Markoff; Surfaces; Superstrong; Approximation

The existence of irrational numbers has fascinated mankind since antiquity; their approximation by rational numbers is of great importance in both theory and applications. The Markoff Diophantine equation arose in Markoff's fundamental work (in 1879) characterizing those irrationals that are badly approximable. The equation arises in several fields of mathematics. Its integer solutions have a tree structure, and investigation of the arithmetic properties of these Markoff numbers (the first of which is associated with the golden ratio, in a sense the most badly approximable irrational number) leads to important questions in graph theory. This research project investigates the connectivity of graphs related to the Markoff tree. In addition to its importance for number theory, this investigation has deep connections and applications to, among other topics, the product replacement algorithm (the most prevalent but still poorly understood tool in computational group theory) and expander graphs in computer science.Investigation of the arithmetic properties of Markoff numbers leads to the question of whether the graphs obtained by the modular reduction of the Markoff tree are connected (strong approximation). Superstrong approximation is the assertion that these graphs are in fact highly connected, that is to say form a family of expanders. The past decade saw a remarkable explosion of activity in the area of superstrong approximation for thin groups (Zariski dense subgroups of infinite index). Much of the research was driven by development of the affine sieve. In the case of thin groups with Levi factor of its Zariski closure semisimple, the strong and superstrong approximation and their applications in affine sieve are by now well-understood. On the other hand, the tori pose particularly difficult problems, both in terms of sparsity of elements in an orbit and their Diophantine properties as well as in terms of strong approximation, which in this case amounts to Artin's primitive root conjecture. This research project will investigate strong and superstrong approximation in a setting that is intermediate in level of difficulty between that of tori and that of thin linear groups, namely, that of nonlinear actions on a surface defined by the Markoff equation as well as in the context of other surfaces of Markoff type. Superstrong approximation results for thin linear groups will play an important role in this investigation, as will techniques and methods related to progress on Lang's conjecture and to the classification of algebraic Painlevé VI equations.

无理数的存在自古以来就吸引着人类;用有理数逼近它们在理论和应用上都非常重要。马尔可夫丢番图方程出现在马尔可夫的基本工作（在1879年）的特点，这些无理数是严重逼近。这个方程出现在几个数学领域。它的整数解有一个树形结构，对这些马尔可夫数的算术性质的研究（第一个马尔可夫数与黄金比例有关，在某种意义上是最难逼近的无理数）导致了图论中的重要问题。 本研究计画探讨与Markoff树相关的图的连通性。 除了对数论的重要性外，这项研究还与其他主题有着深刻的联系和应用，产品替换算法Markoff数是计算群论中最流行但仍不为人所知的工具，它是计算机科学中的扩展图。对Markoff数的算术性质的研究引出了一个问题，即通过Markoff树的模约简得到的图是否连通（强近似）。超强逼近是断言这些图实际上是高度连通的，也就是说形成一个扩张族。在过去的十年里，瘦群（无限指数的Zariski稠密子群）的超强逼近领域的活动出现了显着的爆发。大部分的研究都是由仿射筛的发展推动的。在Zebriki闭包半单的Levi因子为零的薄群的情形下，强逼近和超强逼近及其在仿射筛法中的应用已得到充分的理解。另一方面，环面提出特别困难的问题，无论是在稀疏的元素在轨道和丢番图性质，以及在强逼近，这在这种情况下相当于阿廷的原根猜想。本研究项目将调查强和超强逼近的设置是中间的难度水平之间的环面和薄线性组，即，非线性行动的表面上定义的马尔可夫方程以及在上下文中的其他表面的马尔可夫类型。薄线性群的超强逼近结果将在这项调查中发挥重要作用，因为将技术和方法有关的进展朗的猜想和分类的代数Painlevé VI方程。