Research Reflections, Rotations and Translations - A Mathematician’s Thoughts on Research

Don’t we already know everything about maths? What does maths research actually entail? What’s 129,204 divided by 3,492? These are some of the questions I found myself asking before I began my research project. In this blog post I’m going to talk about some of my experiences in finding the answers.

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Before I embarked on my research adventure, I expected to spend my days writing pages and pages of conjecture and proof, exploring and expanding the field of rigidity theory at lightening speed. To me, this seemed like a logical guess; during lectures we are always stating new theorems and learning their proofs, so it would make sense that I would be spending my time proving lots of theorems of my own. This, however, was not the case.

Initially, I had severely underestimated the importance of really understanding the existing material on rigidity theory. I thought that I would be able to read a few papers, understand some basic examples and then begin working on the open problems. In reality, I had to watch several lectures from an introductory graduate course before I could even begin to comprehend any of the existing material! After that, I quickly learnt that it was not enough to simply read and passively follow along with the proofs; instead, I had to scrutinise the papers and understand the content to the point that I could apply my newfound knowledge to unseen examples and to reproduce existing proofs. It was this familiarity and experience that was the key to moving on to the second phase of my research.

Over the past three weeks, now that I have a sound understanding, I have been creating a Python package that allows us to computationally model, manipulate and visualise the frameworks we study in rigidity theory (see below) – to date this is my most significant contribution to the field. Hopefully, with this package, I will be able to run exhaustive searches on a special type of framework that will ultimately culminate in my own proof of a currently open conjecture!

My shift in perspective on what it actually means to research maths highlighted an important lesson to me; despite studying rigidity theory, it is imperative to be flexible and able to adapt my expectations. It is through the self-leadership skills I have developed during my scholarship that I am able to confidently embrace these changes.

I would like to take this opportunity to thank my supervisor, Dr. Louis Theran, for supporting and guiding me through this challenging research experience, and Lord Laidlaw and the wider Laidlaw Foundation community for affording me this incredible opportunity. Not only have I gained an in-depth insight into life as a researcher, I have been able to meet and engage with myriad wonderful scholars from all across the globe whilst learning valuable lessons that I am eager to apply to all of my future leadership endeavours! This truly is an incredible experience, and I can’t wait to further my leadership skills next year.

Thanks for reading!

:)

(P.S. It’s 37. Did it bother you enough to work it out too?)

Three dimensional screw periodic framework
A screw periodic framework generated using my Python package.
Source: Joseph Edwards (https://github.com/theran/laidlaw-origami

Joseph Edwards

Student, University of St Andrews

I am currently a third year student studying for an MMath Mathematics at the University of St Andrews. During the summer of 2021, I will be undertaking a six week research project into Algebraic Constraint Geometry and Graph Rigidity.

Alongside maths, I also really enjoy playing around with computers! At the moment, I am learning how to use HTML and Ruby so that I can host and maintain my own website. I also hope to take a course on machine learning soon.

Outside of academia, I am a big fan of the outdoors. For the past two years I have been learning to kayak and have really enjoyed the opportunity to explore Scotland whilst doing so.

Comments

Go to the profile of Inkindi Mutoni Sabine
4 months ago

Hey, Edwards, thanks for sharing your research reflections:) Didn't it happen to you that when you realized that you first had to understand the existing materials brought you back and maybe slowed down your research? I mean I also had the same experience when I started my Laidlaw research: I thought that I was going to follow a straight path, read new materials and then implement my project. But, I noticed that it was very necessary to first understand the materials I was given in class before adding more and this somehow slowed down the process and implementation of my project (which made me think that I am not doing the right thing)... How did you just manage to first understand the existing materials and believe that it's what you should first do?

Go to the profile of Joseph Edwards
4 months ago

Yes, it definitely took a lot longer than I expected before I felt like I was truly doing research of my own! For me, the most helpful thing was having regular meetings with my supervisor. He directed me to lots of useful sources and, because of how often we spoke, assured me that I was staying on track. 

Go to the profile of Inkindi Mutoni Sabine
4 months ago

Yeah, I understand. That's so helpful.