EAPSI: Counting Pointed Dynamical Systems Over Finite Fields

该奖项支持研究，以促进对有限域上动力系统基本约束的理解。这些系统发挥着重要的社会作用；例如，它们是目前世界上使用的椭圆曲线密码术的核心。动力系统由环境空间以及空间中点如何随时间移动的规则组成。从有限域构造的动力系统是基本的数学兴趣对象，无论是在理论上还是在现代密码学的实际应用中，都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，人们常常试图理解表现最好的点：不动的点，它们不随时间移动；更一般的周期性点，它们在有限的时间后最终会回到原来的位置。这个项目是有限域上二次动力系统的计算探索，这是最简单有趣的例子，有效地由单个二次多项式定义。研究人员将在不同的有限域上，检查有多少系统具有给定的周期点结构的计数模式。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。这提供了访问他们在计算数论以及重要的高性能计算资源工作的独特强大的研究人员组。这些点动力系统的计数将在1)有限域上进行，其大小是固定素数的增长幂，2)有限域的素数大小对于不同的素数。由于Lang-Weil估计，第一种方法可以用来检验动力学曲线约简的不可约性，动力学曲线是参数化所考虑的动力系统的重要几何对象。第二种方法近年来在揭示模空间序列中意想不到的几何结构方面取得了丰硕的成果，特别是在具有越来越多标记点的固定属曲线模空间的情况下。研究人员还将应用最近开发的闭点筛法来解决d次多项式动力系统如何接近随机d到1动力系统的问题。该奖项由美国国家科学基金会和澳大利亚科学院共同资助，隶属于东亚和太平洋暑期研究所项目，支持美国研究生进行暑期研究。

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.