Algorithms in algebra and topology

ended

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 set of blocks for another. These descriptions are convenient. However, results in algebra and logic show that 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; no computer can be built to always answer this. We can embed incomputable problems from groups into geometry, to show that the homeomorphism problem, of recognising if two geometric shapes are equivalent under smooth deformation, is incomputable in all dimensions above three. Thus we can't computationally classify geometric shapes in higher dimensions; we can't identify the unique distinguishing features of each shape. The study of generic computability (problems which can be computed most of the time) is a useful area in mathematics. Conversely, showing a problem can't be computed most of the time gives rise to applications in cryptography: generically incomputable problems are an excellent tool in the theory behind 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 generically computable, or neither. We will apply these results to particular classess of higher-dimensional geometric objects, identifying whether certain problems relating to them are computable or not. The project will be carried out at the University of Cambridge, under the supervision of Dr. Henry Wilton.