The Undergraduate Research Support Scheme (URSS)Link opens in a new window enables Warwick undergraduate students to carry out an interdisciplinary summer research or public engagement project. The scheme is open to all undergraduate students from any year or course, (with the exception of exchange students). In the summer of 2023, the Warwick Mathematics Institute hosted 26 projects, covering a wide range of topics in pure and applied mathematics. Below, Paul Lezeau (URSS 2022) describes his experience.
I was fortunate enough to work on two URSS projects - one in the summer of my second year and one in the summer of my third year. The first was about formal mathematics and algebraic number theory. The aim was to formalise the proof (i.e., write a computer proof using the proof assistant Lean Theorem Prover) of the Dedekind-Kummer theorem, an important result in algebraic number theory. The second was on machine learning and aimed to use methods from differential geometry to understand how neural networks behave when their inputs are perturbed slightly, something that has recently gained quite a bit of attention in the artificial intelligence community. (See sidebar on adversarial examples.)
Each project lasted a bit more than two and a half months, and was roughly structured as follows:
1. Learn the mathematics for the project
2. Work on the main objective of the project
3. Write up the work, for example as a report, presentation, or online repository
One of the most interesting – and at first rather surprising - aspects of doing this type of project work is that it was a lot less linear than expected, and the three parts listed above were far from being clean cut. For instance, the learning and working on the main research objective parts tended to blend slightly, since I often had to go back and learn some new concepts to understand a paper I was reading or to investigate a new approach for solving a problem. This also meant learning new maths for the projects felt very different to doing coursework, as it was a lot less of a linear process: I often ended up starting reading papers without having learnt all the prerequisites, slowly making my way towards the details, and learning what I needed as I went.
Another nice aspect of these projects was that they allowed me to have a go at some other research related activities, such as preparing and delivering talks at undergraduate conferences, writing research papers and reports, or applying for funding. This was a great opportunity to learn the basics for these in a low-pressure environment, something which has definitely come in handy more than once in my studies since!
Finally, one of the things I enjoyed most about my URSS projects was that there were many opportunities interact with researchers and other students. Typically, I would meet with my supervisor once a week, in person or online, to talk about my current progress, explain what I had been doing and discuss where to go from there. These chats were very helpful for getting to grips with my project, but also for learning a different perspective – from an experienced mathematician – on the ideas I was working on. During the first project, I also ended up interacting quite a bit online with other mathematicians from all around the world who were working in the same area (the Lean Theorem Prover community is very welcoming to students and beginners, and anyone can join its Zulip chat, which is the main means of communication for its members). During the second project I often met up with other URSS students to work together – although we were doing very different types of mathematics, chatting about our projects turned out to be a lot of fun and sometimes quite helpful!
All these things meant the URSS projects turned out to be fun, interesting, and rewarding experiences, which I would strongly recommend to anyone who is interested in having a go at research!
Paul Lezeau & WMI Magazine staff
Published: 18 January 2023
Adversarial examples are inputs specifically designed to fool machine learning algorithms into giving a wrong answer.
Fooling handwriting recognition
The above images were fed into a machine learning algorithm trained to understand handwritten digits. It answered correctly for the original images. For the modified versions, not only did it answer incorrectly, it reported a very high degree of confidence in the wrong answers!
Adversarial examples are an important topic of research in adversarial machine learning, an area which seeks to study models in an adversarial setting, i.e., one where the model might be the target of some potentially malicious attacks. This is particularly relevant, for instance, in cybersecurity.
Want to learn more? Started by reading:
Interested in generating your own adversarial examples? It's relatively easy! A good place to start is with these tutorials: