Groups are algebraic structures which arise naturally in contexts where there is symmetry of some sort. Historically groups have been an important tool in understanding many ares of mathematics, and in particular geometry. Geometric group theory reverses this relationship, using the tools of geometry and topology to study groups. Often seen as quite a young area of maths, coming to the fore in the late 1970's and 80's with the work of people like Gromov, it has its roots in the work of Max Dehn at the start of the 20th century.
At the moment I am particularly interested in three areas of geometric group theory, Coxeter groups, Nielsen equivalence, and accessibility of groups.
I first came across Coxeter groups in my masters dissertation, which was supervised by Dr. Pavel Tumarkin at Durham University. These are groups which are generated by reflections and so have particularly strong connections with geometry, indeed they turn up all over mathematics. I have worked in particular on the connection between the geometry of Coxeter groups and the smoothness of certain varieties which live in flag manifolds, in a project supervised by Prof. Konni Rietsch at King's College London.
Generating sets of a group G of size n are in one-to-one correspondence with the surjective homomorphisms from the free group on n generators to G. Two sets of generators for G are called Nielsen equivalent if they differ by an automorphism of the free group. It has been shown that all generating sets of surface groups of the same size are Nielsen equivalent.
ACCESSIBILITY OF GROUPS
Bass-Serre theory tells us how to write a group as an amalgamated product or HNN extension by studying how the group acts on simplicial trees. A natural question to ask is whether there is an upper bound on the number of times you can split a group in either of these ways. If there is the group is said to be accessible. Accessibility is the first step towards building JSJ-decompositions of groups. I started thinking about this topic in a project supervised by Dr. Lars Louder at UCL.
Here is a list of some academic talks I have given in the past.
CUTTING UP SURFACES AND COMMUTATORS IN FREE GROUPS
Warwick DAGGER seminar 13th Jan 2020
Bristol junior geometry seminar 5th Nov 2019
UCL/KCL junior geometry seminar 10th Oct 2019
When is an element g in a group G a product of commutators [a,b]? What is the minimal number of commutators n such that g=[a_1,b_1]...[a_n,b_n]? What are all possible solutions (a_1,b_1,...,a_m,b_m) to the equation
g=[a_1,b_1]...[a_m,b_m]? Very difficult, yet important, questions - but ones whose answers seem to lie in the darkest recesses of combinatorial group theory. Not so! At least for G a free group, these questions can all be answered elegantly and beautifully by cutting up and colouring surfaces. In this talk I shall present solutions to these problems with an emphasis on drawing nice pictures.
THE TOPOLOGY OF KNOTS AND SURFACES
UCL undergraduate colloquium 13th Nov 2019
Topology, or the mathematical study of the shape of things, is one of the most important areas of mathematics, underpinning, aiding, or influencing a diverse range of modern mathematics including geometry, number theory, combinatorics and group theory. It even has applications to quantum mechanics, molecular biology, and computer science. This talk will be a whistle stop tour of some 1 and 2 dimensional topology, including knots and links, Seifert surfaces, polygonal complexes and Euler characteristic, and finishing off with the classification of compact surfaces. - Oh, and the best part? There will be lots of pictures!
CAN FORGETFUL COMPUTERS DO GROUP THEORY?
UCL postgraduate seminar 25th Apr 2019
In 1931 Gödel proved that there were problems in mathematics which can never be solved...fiddlesticks! To add insult to injury in 1936 Church and Turing independently showed that even among those problems which are solvable, there are some which take an infinite amount of time to solve algorithmically...double-fiddlesticks!! Since this time, questions of computability (decision problems) have entered all corners of mathematics, some of the most well-known being posed by Dehn: the word, conjugacy, and isomorphism problems in group theory. I shall introduce a model for a particularly stupid type of computer (a Turing machine with no memory) called a finite state automaton, and discuss whether such computers can do group theory. If/when they can, I'll discuss how they can help solve some of Dehn's decision problems.
AN INTRODUCTION TO GEOMETRIC GROUP THEORY WITH APPLICATIONS TO ACCESSIBILITY
UCL/KCL junior geometry seminar 8th Nov 2018
Geometric group theory aims to study groups using tools from geometry and topology, more precisely to study finitely generated groups by looking at the way they act on geometric and topological spaces. In this talk I shall introduce some of the aims of geometric group theory, and construct a well-known space associated to a finitely presented group. I will talk about algebraic splittings of groups in the sense of van Kampen's Theorem, and finish by sketching the proof of Dunwoody's theorem that finitely presented groups admit only a finite number of such splittings over finite subgroups.
THE DUALITY BETWEEN THE COMBINATORICS AND THE GEOMETRY OF COXETER GROUPS
UCL/KCL junior geometry seminar 2nd Nov 2017
Coxeter groups, or groups generated by reflections which act discretely, arise naturally as the symmetry groups of regular polytopes and periodic tilings. They are typically studied from one of two viewpoints. From the point of view of geometry they give rise to nice simplicial structures which are very important in the study of Lie groups and Lie algebras. On the other hand, thought of as combinatorial groups, they admit a simple constructive solution to the word problem, as well as a solution to the conjugacy problem. I will introduce the basics of both of these approaches and show how the algebraic structure of the group is encoded into the geometry, and illustrate how this can dramatically simplify proofs of combinatorial results, as well as giving us a geometric solution to the word problem.