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Algorithms in algebra and topology

Periodic Reporting for period 1 - GroupsComputability (Algorithms in algebra and topology)

Reporting period: 2015-10-01 to 2017-09-30

"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, which is a finite set of building blocks, along with a finite set of rules on when we can substitute one collection 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.

Groups, and higher-dimensional geometric shapes, appear in many other areas of study. One such application is in cryptography, where we use groups to develop the underlying framework of the cryptosystems used to keep internet transactions safe and secure; in this context, being unable to compute various questions in group theory makes them useful for cryptographic applications. Another application is in physics, where we use both groups and our knowledge of geometry to understand the `shape' of the universe; in this context, the ability to compute various properties of the geometric objects that we are working with is of great help in understanding them.

The purpose of this project was to investigate other algorithmic questions in group theory, beyond the word problem, to determine if they are computable. We aimed to then apply these results to particular classes of higher-dimensional geometric objects, identifying whether certain problems relating to them are computable or not. One of the key tools in doing this, which was developed during the project, is the use of ""embedding theorems""; these are constructions which allow us to realise groups as sub-structures of other groups in a ""nice"" (= algorithmic) manner. Using this, it is then possible to export undecidable problems from one group to another.

This project also involved the PI carrying out various teaching activities, in order to improve and refine his teaching and communication skills. This included teaching undergraduate and graduate courses, and supervising masters theses and undergraduate summer research projects."
"The PI worked with Dr Rishi Vyas on developing a more general understanding of the Higman embedding theorem. This involved a detailed survey/analysis of a construction by Cohen, to ascertain that it possessed further properties than originally known. This was in the context of how the torsion of the group (counting how many times we need to ""wind"" a group element in order to annihilate it) behaves under this embedding construction. We were able to demonstrate that the embedding construction preserved not only the ""type"" of torsion, but also how the group embedding behaves when we ""remove"" all the torsion elements.
This resulted in a preprint on arxiv.org which has been submitted for publication.

The PI worked with Dr Zachiri McKenzie, on constructing ""universal"" group with a certain type of torsion, containing all groups with that type of torsion. Using the techniques developed earlier in the project, we showed that if the type of torsion was sufficiently controlled (i.e. enumerable by a Turing machine), then such a universal group exists.
This resulted in a preprint on arxiv.org.
The PI and McKenzie also carried out a survey of Slobodskoi's work on the impossibility of computing which statements are true in all finite groups (groups with only finitely many elements). This was extremely important, as the original paper was poorly written, contained errors and omissions, and was very difficult to source.
This resulted in a survey article, which is in preparation.

The PI worked with Michael Hill investigating an explicit construction of a universal group (one containing all groups), following on from the above work with McKenzie. We were able to show that this could be applied to construct a universal group with no torsion elements.
This resulted in a preprint on arxiv.org which has been submitted for publication.

The PI has been investigating the Adian-Rabin embedding theorem in groups. A recent work by Bridson and Wilton gives a modified version of this construction which preserves various properties of finite groups. Their work made use of the above result of Slobodskoi which was surveyed by the PI and McKenzie. The PI has a simplified version of the construction given by Bridson and Wilton, and this is a manuscript in preparation.

The PI delivered several lecture courses during the fellowship. For each of these, he prepared a thorough set of printed notes and set of exercises. He is now in negotiations with Cambridge University Press to convert one of these sets into a graduate text book titled ""Decision problems in group theory"" as part of the London Mathematical Society Student Text series.

The PI formulated, developed and delivered a completely new teaching program on ""Ethics in Mathematics"" for the undergraduate mathematics students in Cambridge. This consisted of a 16 hour seminar series, and associated lecture notes. This work was done entirely ""from scratch""; no resources currently exist for this sort of teaching to mathematicians. The PI is currently in negotiations with Cambridge University Press to produce a book titled ""Ethics for the working mathematician"", to help the mathematics community understand and appreciate the impact and harm that mathematics can have on society."
"This project has helped further our understanding of groups as a mathematical tool for explaining abstract mathematical objects. The relative substructures and embeddings of groups (that is, the way ""groups fit inside other groups"") is now much more well understood, especially in the context of what happens to torsion elements. We now know the structure of an explicit universal group with no torsion, as well as how torsion behaves under the Higman embedding theorem. We have a much-simplified version of the Bridson-Wilton construction, which helps us understand how finite groups behave under certain embeddings.

In terms of the wider societal implications of the project, the key output here has been the formation of a teaching syllabus on ethics in mathematics, and the accompanying resources and text. This is a pioneering step; up until now, no European university has taught ethics to undergraduate mathematicians, and yet this group of professionals is having an immense impact on society through emerging technologies such as the internet. Our modern society is being run by mathematicians, and yet this group has practically no ethical training whatsoever. This idea is now disseminating to other universities in Europe, and will hopefully help mitigate the damage to society that mathematicians may do in the future."
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