Applications of exact and approximate enumeration

精确枚举和近似枚举的应用

• 批准号: RGPIN-2017-03751
• 负责人: Rechnitzer, Andrew
• 金额: $ 1.75万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635556
• 关键词: Applications; exact; approximate; enumeration

Enumeration is the branch of mathematics that examines questions of the form "How many..."; it is perhaps the apparent simplicity and yet universal nature of these questions that explains their ubiquity. They arise in many different areas of mathematics and in problems at the interface between mathematics and physics, chemistry and computer science. The aim of this program is to apply different types of enumeration methods in the context of particular sets of objects - random walks and self-avoiding walks. From there, I will apply those methods, both exact and approximate, to problems such as phase transitions in polymers, random knotting and linking of curves and to questions of growth and cogrowth in an area of pure mathematics - geometric group theory. I will also undertake two number theory projects of with a computational and enumerative flavour - cataloguing elliptic curves and counting primes in arithmetic progressions.

枚举是数学的一个分支，用于研究“有多少……”这样的问题；也许正是这些问题表面上的简单而又普遍的本质，解释了它们无处不在的原因。它们出现在数学的许多不同领域以及数学与物理、化学和计算机科学之间的接口问题中。这个程序的目的是在特定对象集的上下文中应用不同类型的枚举方法——随机行走和自我避免行走。从那里，我将应用这些方法，无论是精确的还是近似的，来解决诸如聚合物的相变、随机打结和曲线连接等问题，以及纯数学领域——几何群论中的生长和共生问题。我也将承担两个数论项目，具有计算和枚举的味道-编目椭圆曲线和计数素数在等差数列。