# Luke Collins' Website

My name is Luke Collins, I am currently a visiting mathematics lecturer at the University of Malta, as well as a Heilbronn doctoral student at UCL, under the joint supervision of Alexey Pokrovskiy and Shoham Letzter.

I graduated from the University of Malta with a first class double honours in Mathematics and Computer Science, and completed a Master's degree in Mathematics at the University of Warwick, graduating with distinction. My main research interests lie in graph theory, combinatorics, and number theory. Recently, I re-established the Malta Mathematical Society, and run it with the help of a group of dedicated undergraduates from the University of Malta.

Outside of academia, I was a researcher at Simply VC, a company specialising in Blockchain development and infrastructure; and I also cofounded the mobile app company LearnD. In my spare time, I enjoy listening to classical music, playing the piano, doing (ethical) hacking and practising Brazilian jiu-jitsu. I like reading about history, politics, theology and philosophy.

I don't really use social media—if you want to get in touch, email me at *luke (at) collins (dot) mt*. I am a member and supporter of the Free Software Foundation, a movement which advocates for users' rights to study the source code of programs executing on their machine. I make use of Arch Linux, and always run free (libre) whenever possible.

## Research

My areas of interest in mathematical research are combinatorics, graph theory and number theory.

### Publications

My Erdős number is ⩽ 3 because of the chain: L. Collins (me) → I. Sciriha → C. Godsil → P. Erdős.

- Luke Collins & Irene Sciriha. The Walks and CDCs of Graphs with the Same Main Eigenspace. Discussiones Mathematiæ Graph Theory (in press),
`doi:10.7151/dmgt.2386`

. - Luke Collins. The Ternary Goldbach Conjecture. Master's Dissertation in Mathematics (2020), University of Warwick, UK.
- Luke Collins. On Walks and Canonical Double Coverings of Graphs with the Same Main Eigenspace. Undergraduate Dissertation in Mathematics (2019), University of Malta, Malta.
- Luke Collins. Synthesising Safety Runtime Enforcement Monitors for μHML. Undergraduate Dissertation in Computer Science (2019), University of Malta, Malta.
- Luke Collins & Irene Sciriha. Two-graphs and NSSDs: An Algebraic Approach. Discrete Applied Mathematics (2018), 266, pp. 92–102,
`doi:10.1016/j.dam.2018.05.003`

.

### Walks and CDCs

Recently I've been doing quite a bit of research related to walks and an operation called the *canonical double cover* (CDC) of a graph. Read more about this research here.

## Teaching

I give some courses at the University of Malta. Some of the resources I've prepared for the courses can be found in this section.

### Notes and Exercises

Note that PDFs which are not linked yet, or have version number < 1.0, are still work-in-progress.

- Courses I'm teaching this year at the University of Malta: Other courses I have taught in the past: CHE1215, CHE1217 and TET1013.
- LaTeX Course I delivered for the Doctoral School
- Master's level algorithms course I delivered for IDEA Academy:

- Course Description | PDF Notes (extract)

- Undergraduate and Postgraduate Notes
- Pure Mathematics at A-level and Intermediate
- MATSEC formulæ exam booklet: A4 version | A5 printable version
- Complete list of O-level plane geometry theorems

### Tuition

I occasionally give private tuition for anything related to mathematics and computer science. If you would like me to tutor you, email me at *luke (at) collins (dot) mt.*

### Miscellaneous

- Here is a tool for visualising the effect of 2×2 matrices: matrix visualiser.
- Here is a tool to draw automata diagrams: automaton designer.
- Here is a tool for generating truth tables: truth table generator.

## Other Projects

### Programming Projects

I have worked on various coding projects over the years, including:- a multiplayer snakes game written in C,
- a multiplayer tile game written in Java,
- a compiler for a custom Turing complete language written in C++,
- a logic parser and normaliser written in Haskell.