Computability Theory and its Applications

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Up to the mid 19th century or so, mathematics was mostly algorithmic. Proofs of existence were usually done by giving an actual construction of the object. As mathematics began to become more abstract, people began to study the notions of algorithms and computability. The first stumbling block was on the definition of what it meant to be algorithmic or computable. When faced with an algorithm, people would agree that it was one, but how would one show that something could not be computed, or solved by an algorithm? By the 1930s, work of Turing and others culminated in an acceptable notion of computability.Since the 1930s, computability theory has developed far beyond just checking what is computable and what is not. We have notions of relative computability, where we say A is computed from B if there is some program that can compute A if it is allowed to refer (finitely often) to an infinite tape coding B.In this project, I propose to study various areas of mathematics, and use computability theory to examine the complexity of the proofs. This can help answer questions such as how difficult is it to prove one thing compared to another? Is the original proof the most efficient? How difficult is it to tell that similar objects are actually similar? By studying these properties, we can discover interesting phenomena about the area of mathematics being studied, often leading to further insight.