Thin Counting in Moduli Spaces

数论的目标之一是计算整数序列中的素数。在过去的十到十五年里，人们一直在努力理解一个更一般的问题：在与有限维空间的无限群对称有关的整数序列中寻找素数。这一一般问题的表述是在认识到许多基本整数序列与对称组相连之后，例如毕达哥拉斯三元组的项、出现在阿波罗圆填充中的曲率以及仅使用有限字母表形成的连分式的分母。在许多重要的情况下，相关的对称组是“薄的”，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，利用许多不同数学领域的新技术已经被开发出来，以处理关于稀疏群及其相关的整数序列的问题。这个项目的研究将研究模空间研究中出现的整数序列中的素数。模空间是一种数学对象，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。它们是从整数建立的，因此引发了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。第一个问题是关于交换微分模空间上Teichmueller流的周期轨道的数场。PI最近将Selberg的3/16定理推广到更高亏格的模空间。这一结果将被用来获得Teichmueller流的周期轨道的同余计数。此外，Selberg定理将推广到模空间的仿射不变子流形上。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法来研究该变种上整点坐标的素因数。这需要了解有限域上自同构群在变化点上的轨道，并将这个问题推广到无平方模。我们将研究各种有限域上由一个固定的整体自同构作用所得到的置换序列的统计量问题。这些问题与与本科生合作获得的令人惊讶的数字结果有关。

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.