Applications of exact and approximate enumeration

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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.**