I have been in pursuit of answers to difficult questions throughout the course of this summer research project. When not conducting research at the university, I could often be found at the local climbing gym getting in some much needed exercise with friends. Possibly the most difficult question faced all summer was this pearler from one mate: “So what have you actually been doing as a mathematics researcher?” Great question. My first instinct was to think of all the interesting details surrounding my project, but of course these could not be condensed into a digestible answer for someone who decided in high school that maths wasn’t for them. I’d like to say I had this well-targeted answer for them the first time:

Mathematics research is like projecting a difficult boulder problem.

This will require a brief detour to explain some terminology. Bouldering is a distilled form of rock climbing. It can be practiced indoors with plastic holds or outdoors on natural rock features. As far as equipment goes it requires no ropes or harnesses, only climbing shoes and safety mats. Each climb, or problem, has a solution consisting of a sequence of moves starting at the bottom and finishing at the top. The climber must use their technique, strength, flexibility, balance and problem solving skills in order to complete a particular problem. Boulder problems are given a grade according to their difficulty. Grades range from V0, achievable for new climbers, up to V17 for a few of the world’s best climbers. Clearly not all boulder problems are as easily completed as others, and dedicated climbers will often invest significant effort working on, or projecting, a difficult climb.

Now, how is mathematics research like projecting a difficult boulder problem?

Typically the initial approach is to stand back and observe the problem and formulate some preliminary ideas for how the problem is best approached.  You might draw upon past experience and identify similarities to problems previously encountered. You might make note of areas where you expect to encounter difficulties requiring creative solutions. With this big picture in mind, the next step is to get hands-on with the problem. The first attempt might confirm your initial ideas or bring to your attention important things you missed on first glance. It is then through testing new approaches or new sequences in successive attempts that a greater understanding of the problem and its intricacies is gained. It can occur that the problem proves too much. You may get stuck trying the same things to no avail, or realise you lack the skills necessary to master this particular problem. You may simply need to leave the problem temporarily to acquire those skills by working studiously. Perhaps consulting someone more experienced or working with another on the problem may prove fruitful. When the necessary breakthrough is made, the problem can finally be completed from start to finish. It is incredibly satisfying, and to those who appreciate the process begs the question, “What next?”

Though this summer research period is at its end, my project stands incomplete. Through honours I intend to keep chipping away for that breakthrough. I give my summer research project a grade of V6.

Luke Yerbury
The University of Newcastle