Applications of exact and approximate enumeration

枚举是数学的一个分支，它研究的问题是“有多少..“;也许正是这些问题表面上的简单性和普遍性解释了它们的普遍性。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的对象集合的上下文中应用不同类型的枚举方法-随机游动和自避免游动。从那里，我将适用于这些方法，无论是精确的和近似的，问题，如相变的聚合物，随机打结和连接的曲线和问题的增长和共同增长在一个地区的纯数学-几何群论。我还将承担两个数论项目的计算和枚举的味道-编目椭圆曲线和计数素数在算术级数。

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.**