My Research
PhD project:  The homology of periodic cell complexes
Invited talks:
- 23rd August 2022, London TDA Seminar, Queen Mary University of London
- 5th December 2022, EPFL Applied Topology Seminar, EPFL (Slides)
- 23rd May 2023, London TDA Seminar, Queen Mary University of London
Contributed talks:
- 20-24th June 2022, ATMCS10, Oxford University
Preprints:
- Quantifying the homology of periodic cell complexes, A.O., V. Robins, 2022
A d-periodic cell complex is a cell complex which permits a free action by a free abelian group of rank d. These model very large, homogeneous data sets. Think, for instance, of a graph modelling a crystal - which has translational symmetries. The problem is that we can't study these infinite spaces on computers, and instead we must store and study a finite representation which do not respect its overall topology. These representations lose some topological features of the infinite periodic complex and also create new topological features. For instance, the maximal quotient graph with respect to translation of disconnected 3-periodic graph in the image has a single connected component and three single edge 1-cycles.
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My research uses combinatorical and algebraic techniques to attempt to reconstruct the homology of a periodic cell complex from a finite representative (with minimal additional data). Most recently, I have been working with my supervisor and Dr Amit Patel to achieve this with objects known as "bisheaves".
I am also interested in the question of how to determine when two finite spaces represent the same periodic space, but have made less progress on this front (but see here for a cool recent preprint by another group working on this problem).
Undergraduate projects
My undergraduate degree at ANU allowed me to complete a number of research projects during and in place of coursework
Theoretical Astrophysics: Emission lines from electron excitations in cyanide (HCN) molecules have been shown to have a strong correlation with star formation rates. I was able to do a project with Christoph Federrath and Mark Krumholz analysing numerical simulations to determine an empirical relationship between star formation rates and HCN emission. This was the first calibration of its kind worldwide and lead to a paper which you can view here.
Algebraic Geometry: In a class on computational algebraic geometry taught by Martin Helmer, I was able to do a fun report on how one can use techniques from algebraic geometry to solve graph colouring problems and sudoku puzzles (and vice versa). I created some working Macaulay2 code which uses this to solve sudoku puzzles (and their 2x2 equivalent shidoku) in an admittedly unfeasibly long time.
Theoretical Physics: I was able to do a research project with David Williams which involved analytical and numerical computations of how coplanar and coaxial molecular gears interact via a Lennard-Jones potential.