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Personal profile

Research interests

My research interests take three paths, all vaguely connected by my interest in algebraic topology:

1. Algebraic K-theory and Wall's D(2)-problem

2. Topological Data Analysis

3. Social network analysis of narrative structures in mythological texts

 

For the first avenue, I have largely been interested in a famous problem in low-dimensional topology, known as Wall's D(2)-problem. In a vague sense, this asks the question of what information is required before we can say that a space is homotopy equivalent to one of dimension 2. In a slightly more formal way, we take a 3-complex with finitely presented fundamental group, and which is cohomologically 2-dimensional. We then ask, is this space homotopy equivalent to a finite complex of dimension 2? To date, we do not know the full answer to this question, although affirmative answers have been found for certain groups, most notably metacyclic groups [1]. This class of groups is my main area of interest. In particular, I have explored their syzygy modules and have attempted to relate this back to the aforementioned problem of Wall.

The second avenue is at the boundary of topology, geometry and data science (cf. R. Ghrist, V. Nanda, A. J. Blumberg). The purpose of topology in a vague sense is to describe shape. In the context of data science, this means my interest is in describing the shape of data. This area is, relatively speaking, still in its infancy with two early key steps coming in the early 2000s when Edelsbrunner et al. introduced the idea of persistent homology [2]. This was then reformulated by Carlsson et al. when persistence barcodes were introduced [3]. Nevertheless, there is already a fairly standard roadmap for the process of describing the shape of data. We take our data and, with little more than some notion of distance between data points, we build a filtered cell complex. In performing this step, we are immediately in the realm of matrix algebra from which we can derive various finite invariants upon which we can perform standard statistical analysis.

The areas of influence of the above is proving to be wide-ranging with applications ranging from politics to biology to physics. Early applications were to image analysis [4] and a study by Benditch et al. looks at the structure of arteries in human brains [5]. An excellent book on the application of topological data analysis (TDA) to genomics and evolution was written by A. J. Blumberg and R. Rabadan [6]. Changing fields, Bhattacharya et al. used a combination of persistent homology and graph search-based algorithms to compute a set of best likely paths-to-goal for an autonomous robot [7]. In physics, Murugan and Robertson illustrated the use of TDA on a data set of fast radio burst observations [8]. Less has been done on using TDA on large climate datasets, however. As such, this is the area in which I am attempting to apply TDA.

Finally, the third avenue, in which I have an interest in applying graph theory and social network analysis to mythological texts. This interest came out of conversations with R. Kenna whose work on western mythology has sparked significant interest since 2011 [9]. My focus thus far has been on Indian mythological epics.

 

[1] Johnson, F.E. A. (2021) "Metacyclic groups and the D(2) problem". World Scientific

[2] Edelsbrunner; Letscher; Zomorodian (2002-11-01). “Topological persistence and simplification”. Discrete & Computational Geometry. 28 (4): 511–533.

[3] Carlsson, Gunnar; Zomorodian, Afra; Collins, Anne; Guibas, Leonidas J. (2005) "Persistence barcodes for shapes". International Journal of Shape Modeling. 11 (2): 149–187

[4] H. Adams and G. Carlsson ( 2009). “On the nonlinear statistics of range image patches”. SIAM J. Imaging Sci., 2(1):110–117.

[5] Bendich, J. Marron, E. Miller, A. Pielcoh, and S. Skwerer (2016). “Persistent homology analysis of brain artery trees”. To appear in Ann. Appl. Stat.

[6] Rabadan, R., & Blumberg, A. (2019). “Topological Data Analysis for Genomics and Evolution: Topology in Biology”. Cambridge: Cambridge University Press. doi:10.1017/9781316671665

[7] S. Bhattacharya, R. Ghrist, and V. Kumar (2015). “Persistent homology for path planning in uncertain environments”. IEEE Trans. on Robotics, 31(3):578–590.

[8] J. Murugan and D. Robertson. “An Introduction to topological data analysis for physicists: From LGM to FRBs”. arXiv:1904.11044

[9] P Mac Carron and R.Kenna (2012). “Universal properties of mythological networks”. EPL 99 28002

Teaching specialisms

My primary area of specialism is in teaching algebra and its applications. This has predominantly included linear algebra and group theory, although I also have lots of experience teaching number theory, ring theory, and homological algebra. I am particularly interested in showcasing ways in which traditionally 'pure' subjects can be used to tackle a range of problems.

I have also taught logic, calculus and, more recently, topics relating to data science such as data mining.

Throughout my teaching, I stress three key aspects: Structure, Intuition, Practice. For Structure, I believe it is key to understand how topics and fields fit together. This enables students to obtain a bird's-eye view of the topic and facilitates heuristic thinking. Next, Intuition helps student remember the variety of definitions and results by understanding the key messages behind them. It also helps students understand what technique needs to be applied to a given problem and when. Finally, Practice aids computation and allows students to free their minds from the nitty gritty details and instead take a step back and view the problem as a whole.

Education/Academic qualification

Education, Associate Fellowship of the Higher Education Academy (AFHEA), Coventry University

1 Jan 20201 Oct 2020

Award Date: 1 Oct 2020

Algebraic Topology and Algebraic K-Theory, PhD, Group algebras of metacyclic type, University College London (UCL)

1 Sept 20131 Mar 2018

Award Date: 1 Mar 2018

Mathematics, MSci, Leray-Serre Spectral Sequences, University College London (UCL)

1 Sept 200931 Aug 2013

Award Date: 1 Sept 2013

External positions

Associate Lecturer, University of Reading

1 Sept 201931 Aug 2023

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