Thin Counting in Moduli Spaces

模空间中的稀疏计数

• 批准号: 1701357
• 负责人: Michael Magee
• 金额: $ 14.53万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2017-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701357&HistoricalAwards=false
• 关键词: Thin; Counting; Moduli; Spaces

One of the goals of number theory is to count prime numbers in sequences of integers. In the last ten to fifteen years there has been a drive to understand a more general question: Find primes in integer sequences associated to infinite groups of symmetries of finite-dimensional spaces. Formulation of this general question followed the realization that many fundamental integer sequences are connected to groups of symmetries, for example the entries of Pythagorean triples, curvatures appearing in Apollonian circle packings, and the denominators of continued fractions formed using only a finite alphabet. In many important cases, the relevant group of symmetries is 'thin,' meaning that it is sparser than expected, and hence is not subject to classical techniques. As a result, new technology that draws on many different areas of mathematics has been developed to treat questions about thin groups and their associated integer sequences. The research in this project will study primes in integer sequences that arise in the study of moduli spaces. A moduli space is a mathematical object that encodes all geometric structures of a given type on an object, up to natural identifications. Moduli spaces often have associated groups of symmetries. These are built from integers, and so invite many intriguing number theoretic questions. In fact, these symmetry groups are sometimes thin, giving surprising new applications of the theory of thin groups. The PI will study two types of counting problems. The first concerns the number fields associated to periodic orbits of the Teichmueller flow on moduli spaces of abelian differentials. The PI has recently generalized Selberg's 3/16 Theorem to higher genus moduli spaces. This result will be used to obtain a congruence count for periodic orbits of the Teichmueller flow. In addition, Selberg's Theorem will be extended to affine invariant submanifolds of moduli space. The second type of problem concerns the Markoff-Hurwitz affine variety and its automorphism group. The PI will study the prime factors of the coordinates of integer points on this variety by obtaining congruence lattice point counting estimates and using sieve methods. This requires understanding the orbits of the automorphism group on the points of the variety over finite fields, and extensions of this question to square-free moduli. Questions about the statistics of the series of permutations obtained by the actions of a fixed global automorphism on the points on the variety over various finite fields will be studied. These questions are related to surprising numerical results obtained in collaboration with undergraduate students.

数论的目标之一是计算整数序列中的素数。在过去的10到15年里，人们一直在努力理解一个更普遍的问题：在与有限维空间的无限对称群相关联的整数序列中找到素数。这个一般性问题的表述是在认识到许多基本整数序列与对称群相联系之后，例如毕达哥拉斯三元组的条目，阿波罗圆填充中出现的曲率，以及仅使用有限字母表形成的连分数的增子。在许多重要的情况下，相关的对称群是“稀疏的”，这意味着它比预期的要稀疏，因此不受经典技术的影响。因此，利用许多不同的数学领域的新技术已经被开发出来，以处理有关瘦群及其相关整数序列的问题。在这个项目中的研究将研究在模空间的研究中出现的整数序列中的素数。模空间是一个数学对象，它编码对象上给定类型的所有几何结构，直到自然标识。模空间通常有相关的对称群。这些都是由整数构建的，因此会引发许多有趣的数论问题。事实上，这些对称群有时是薄的，这给薄群理论带来了令人惊讶的新应用。PI将研究两种类型的计数问题。第一个问题涉及到的Teichmueller流的模空间的阿贝尔微分的周期轨道的数域。PI最近将Selberg的3/16定理推广到更高亏格的模空间。这个结果将被用来获得一个同余计数的周期轨道的Teichmueller流。此外，还将Selberg定理推广到模空间的仿射不变子流形。第二类问题涉及Markoff-Hurwitz仿射簇及其自同构群。PI将通过获得同余格点计数估计和使用筛选方法来研究该品种上整数点坐标的素因子。这需要理解有限域上簇的点上的自同构群的轨道，并将这个问题扩展到无平方模。将研究由固定的全局自同构在各种有限域上的簇上的点上的作用所得到的一系列置换的统计问题。这些问题与与本科生合作获得的令人惊讶的数值结果有关。