Research Statement Stephen Binns

My primary research interest is in computability and complexity theory. More specifically, I have an active research program in the area of effective dimension theory and randomness. Computability theory was first developed in the 1930s and ’40s by Alan Turing who has come to be seen, along with Gödel and Tarski, as one one of the founders of modern mathematical logic. The subject developed over the 20th century and has branched into many different sub disciplines including the study of resource-bounded complexity (in which the famous P6=NP problem is formulated), randomness, and descriptive complexity. It is in these last two fields that I am currently concentrating my research energies. Descriptive complexity begins with the concept of Kolmogorov complexity. This defines the complexity of a finite mathematical object to be the length of the shortest computer program that will output the object. It is meant to capture the intuitive notion that a simple finite object is more succinctly described than a complex one. This idea is quickly extended to infinite objects (characterised as infinite binary sequences) by effective packing dimension, which is defined to be