EAPSI: Counting Pointed Dynamical Systems Over Finite Fields

EAPSI：计算有限域上的指向动力系统

• 批准号: 1714003
• 负责人: Joseph Gunther
• 金额: $ 0.54万
• 依托单位: Gunther Joseph
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714003&HistoricalAwards=false
• 关键词: EAPSI; Counting; Pointed; Dynamical; Systems

This award supports research to advance understanding of fundamental constraints on dynamical systems over finite fields. Such systems play an important societal role; for example, they are central to the elliptic curve cryptography currently in use around the world. A dynamical system consists of an ambient space, along with a rule for how points in that space move over time. Dynamical systems constructed from finite fields are objects of fundamental mathematical interest, both theoretically and for practical applications to modern cryptography, which involves iterating a process that is hopefully hard to reverse. In studying a dynamical system, one often tries to understand the points that are the most well-behaved: fixed points, which do not move over time, and more generally periodic points, which are eventually brought back to their original locations after a finite amount of time. This project is a computational exploration of quadratic dynamical systems over finite fields, which are the simplest interesting examples, efficiently defined by a single quadratic polynomial. The researcher will examine patterns in the counts, over different finite fields, of how many systems having a given structure of periodic points there are. The project will be conducted at University of New South Wales in Sydney, Australia, under the mentorship of Professor John Roberts. This provides access to their uniquely strong group of researchers working in computational number theory along with important high-performance computating resources.These counts of pointed dynamical systems will be conducted over 1) finite fields whose sizes are growing powers of a fixed prime number, and 2) finite fields of prime size for different primes. The first approach can be used, thanks to the Lang-Weil estimates, to examine the irreducibility of reductions of dynatomical curves, important geometric objects which parameterized the dynamical systems under consideration. The second approach has been fruitful in recent years in uncovering unexpected geometric structure in sequences of moduli spaces, in particular in the case of the moduli space of curves of fixed genus with an increasing number of marked points. The researcher will also apply recently developed closed-point sieve methods to questions of how closely degree d polynomial dynamical systems can approximate random d-to-1 dynamical systems.This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the Australian Academy of Science.

该奖项支持研究，以促进对有限域动力系统基本约束的理解。这种系统发挥着重要的社会作用;例如，它们是目前世界各地使用的椭圆曲线密码学的核心。一个动力学系统由一个周围的空间组成，沿着的是一个关于空间中的点如何随时间移动的规则。从有限域构造的动力系统是基本数学兴趣的对象，无论是理论上还是现代密码学的实际应用，其中涉及迭代一个希望难以逆转的过程。在研究一个动力系统时，人们经常试图理解那些行为最好的点：不动点，它们不随时间移动，更一般的是周期点，它们在有限的时间后最终会回到它们的原始位置。这个项目是有限域上的二次动力系统的计算探索，这是最简单的有趣的例子，有效地定义了一个单一的二次多项式。研究人员将检查模式的计数，在不同的有限领域，有多少系统具有给定的结构的周期点。该项目将在澳大利亚悉尼的新南威尔士大学进行，由John Roberts教授指导。这提供了访问他们独特的强大的研究人员在计算数论工作沿着重要的高性能计算资源。这些计数的指向动力系统将进行1）有限域的大小是一个固定的素数的增长的权力，和2）有限域的素数大小为不同的素数。 第一种方法可以使用，由于朗-韦伊估计，检查不可约的减少dynatomical曲线，重要的几何对象参数化的动力系统正在考虑。第二种方法是卓有成效的，近年来在发现意想不到的几何结构序列的模空间，特别是在模空间的情况下，曲线的固定亏格与越来越多的标记点。研究人员还将应用最近开发的闭点筛方法来解决d次多项式动力系统如何接近随机d-to-1动力系统的问题。该奖项是东亚和太平洋夏季研究所计划下的一个美国研究生的夏季研究，由NSF和澳大利亚科学院共同资助。