GroupsComputability · Algorithms in algebra and topology

Group theory is the study of symmetry in mathematical objects, such as rotations of geometric shapes.Groups help us understand the underlying structure of mathematical objects by revealing their symmetries.To understand groups we need an efficient way to describe them. Some groups admit a finite presentation;a finite set of building blocks, along with a finite collection of rules on when we can substitute one setof blocks for another. These descriptions are convenient. However, results in algebra and logic showthat such descriptions are not always suitable to work with, as certain problems (e.g., the word problem,of deciding if two distinct collections of blocks represent the same group element) are incomputable; nocomputer can be built to always answer this. We can embed incomputable problems from groups intogeometry, to show that the homeomorphism problem, of recognising if two geometric shapes are equivalentunder smooth deformation, is incomputable in all dimensions above three. Thus we can't computationallyclassify geometric shapes in higher dimensions; we can't identify the unique distinguishing features ofeach shape. The study of generic computability (problems which can be computed most of the time) isa useful area in mathematics. Conversely, showing a problem can't be computed most of the time givesrise to applications in cryptography: generically incomputable problems are an excellent tool in the theorybehind cryptosystems. This proposal will deal with incomputable and generically incomputable problems.We will investigate certain problems in group theory to determine if they are computable, or genericallycomputable, or neither. We will apply these results to particular classess of higher-dimensional geometricobjects, identifying whether certain problems relating to them are computable or not. The project will becarried out at the University of Cambridge, under the supervision of Dr. Henry Wilton.