DescriptionOne of the most natural ways to think of groups are as the set of symmetries of some object. For example the Dihedral group \(D_n\) is the set of symmetries of a regular \(n\)-gon. But what happens if we look at the set of symmetries of an infinite object? And, given a group, can we always find an object that it is the set of symmetries for?This project will introduce you to the world of Geometric Group Theory which sits at the intersection of algebra, topology, formal language theory... and a bunch of other fields. In this field we investigate groups, not by working with them directly, but by looking at the geometric properties of the objects they act on. As Cayley discovered in 1878, one of the most useful of these geometric objects is the Cayley Graph which, given a generation set, completely encodes the group structure. The main focus of the group project will therefore be Cayley Graphs and the results relating to them (collectively known as Cayley's Theorems). We will also look at other graphs related to groups and begin to introduce some of the formal language theory necessary to investigate some of the weird and wonderful infinite groups that geometric group theorists have come across over the years.
Group ProjectThe group project will revolve around learning basic concepts and results in the field of Combinatorial Group Theory. By the end of the group project we will have explored the following topics:
Mode of operation and evidence of learningThis project will revolve around learning through reading with a focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding.Students will demonstrate their understanding of the subject matter by solving relevant problems, exploring and constructing examples, investigating theoretical applications of the material, and clearly communicating it in both written and oral formats. Individual ProjectThe individual project will build on the knowledge we have gained in the group project and will explore additional advanced topics. A few examples of topics you would be able to investigate include (but are not limited to):
Mode of operation and evidence of learningThis project will revolve around learning through reading with a focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding.Students will demonstrate their understanding of the subject matter by solving relevant problems, exploring and constructing examples, investigating theoretical applications of the material, and clearly communicating it in both written and oral formats. Prerequisites and Companion modulesPrerequisites: Algebra II.Having done Discrete Mathematics would also be useful, but is not essential, as we will end up defining graphs slightly differently from that course. I would expect a student doing this project to also be taking Galois Theory, Groups and Geometry III, but it is not a compulsory corequisite. Note for MMath students about potential links to fourth-year modules: We will spend a lot of our time dealing with group actions, which is also one way of defining a representation of a group. This project may therefore be complimentary for any students planning to take Representation Theory VI, even for the simple reason that it will hopefully make you far more comfortable with group theory. References
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