Applications of exact and approximate enumeration

枚举法是数学的一个分支，它研究形式为“多少……”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的普遍存在。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问题上。本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-随机漫游和自我回避漫游。从那时起，我将应用这些方法，无论是精确的还是近似的，来解决聚合物中的相变、随机打结和曲线连接等问题，以及纯数学领域--几何群论中的增长和共生问题。我还将承担两个具有计算和计数特色的数论项目--对椭圆曲线进行分类和在算术级数中计算素数。**

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.**