The internal set theory, as proposed by Edward Nelson in 1977, blurs the line between finite and infinite sets in a very simple, effective and controlled way.
This PhD project is aimed at a systematic development of the theory of finite and pseudofinite groups in the language of the internal set theory. This is motivated by problems in a branch of computational group theory, the so-called black box recognition of finite groups. Its typical object is a group generated by several matrices of large size, say, 100 by 100, over a finite field. Individual elements of such a group can be easily manipulated by a computer; however, the size of the whole group is astronomical, and arguments leading to identification of the structure of the group are being de facto carried out in an infinite object. The internal set theory provides tools that allow us to deal with finite objects and numbers that are, in effect, infinite. This is an exciting, unusual, but accessible topic for study.
Read more at http://www.maths.manchester.ac.uk/~avb/pdf/PhD_Topic_Internal_Set_Theory.pdf.
Prerequisites for the project: university level courses in algebra. Some knowledge of mathematical logic is desirable.