Maths

Group cohomology is a mathematical field that sits at the intersection of algebraic topology, commutative algebra, representation theory, group theory, invariant theory, and homological algebra. It has its origins in the late 1800s, and started as a proper own field in the 1930s/1940s. 

To get a feeling for what I do, read below for an intuitive and non-technical introduction. In more technical terms I am working on the mod-p group cohomology of 2x2 special linear and general linear groups with entries in the integers mod p^n, n>1. The cohomology of matrix groups with entries in fields is well studied, however GL_2(Z/p^nZ) and SL_2(Z/p^nZ) have entries in finite rings, and commonly used methods for finding the cohomology are not applicable. I am currently writing up my PhD thesis which includes the computation of the cohomology of Sylow-3-subgroups of SL_2(Z/3^nZ) for all n. Furthermore I have general result for all primes and both SL_2 and GL_2 related to the Lyndon-Hochschild-Serre spectral sequences associated to suitible group extensions of their Sylow-p-subgroups..  Both results will be published shortly after thesis submission.

Talks I have given since starting my PhD and settling into the beautiful area of group cohomology

02/2024 Cohomology of Matrix Groups Through the Ages.  HIMA Postdoc Seminar, Manchester 

10/2023 On the mod p cohomology of certain matrix groups.  Linfoot Number Theory Seminar, Bristol

09/2023 Towards the cohomology of  SL_2(Z/p^nZ).  Algebra Seminar, Aberdeen 

07/2023 On the Cohomology of Congruence Subgroups. A festival to remember Victor Snaith, Bristol

05/2023 Towards the cohomology of SL_2(Z/p^nZ). Junior Number Theory Seminar, Warwick

05/2023 The mod-3 cohomology of a Sylow-3-subgroup of SL_2(Z/3^nZ). Manchester Algebra Seminar

04/2023 Working with Spectral Sequences.  Junior Algebra Seminar, Manchester

04/2023 On the Cohomology of Congruence Subgroups. Women in Number Theory and Geometry, Chesterfield

12/2022 On the Cohomology of Certain Matrix Groups. General Maths Seminar, Royal Holloway University London

11/2022 Through the Looking Glass: Homology, Cohomology, and Duality. Galois Lecture, MathSoc, Univeristy of Manchester

03/2022 An example of the Lyndon-Hochschild-Serre specral sequence. Junior Algebra Seminar, University of Oxford 

11/2021 A Historic Approach to Group Cohomology. Pure Postgrad Seminar, Univeristy of Manchester

11/2021 Towards the cohomology of GL_2(Z/3^nZ). Junior London Algebra Seminar, London (online)

09/2021 Towards the cohomology of GL_2(Z/3^nZ). Young Algebraist’s Conference, EPFL Lausanne, Switzerland

10/2020 An Intro to Profinite Groups and their Cohomology. Pure Postgrad Seminar, University of Manchester (online)

01/2020 Homological Algebra and its Applications to Groups. Pure Postgrad Seminar, University of Manchester

12/2019 The Language of Pure Mathematics and some of its Tools. PDS in Pub, University of Manchester 

Some talks I gave as part of assessed coursework at the University of Goettingen

The experimental search of the Higgs Boson. Seminar for particle physics

Clifford Algebra and Spin Group. Advanced seminar for differential geometry

Tropical Manifolds. Advanced seminar in discrete differential geometry

Tropical Curves. Advanced seminar in discrete differential geometry

Chaotic behavior in continuous and discrete dynamics. Seminar on complex dynamics