How my supervisor saved my PhD — the Goldilocks principle for mathematics

Mathematics is all in the asking

I almost flunked my maths PhD.

I had been forewarned that the journey to graduation would be paved with uncertainty; nothing like the smooth, linear path I travelled as an undergraduate. Research is a messy enterprise, and pure mathematics takes no prisoners.

Almost two years in, I was caught in the jumbled mess that is the hallmark of research. To acquire a PhD, one must make an original contribution to their field’s body of knowledge (or similar; I can’t recall the exact wording). However subjective that criteria might seem, I was clearly not meeting it.

The lengths we’ll go to for those two letters (image source)

Trawling

I had spent most of the first year trawling through the literature, familiarising myself with the established body of knowledge in my field of Functional Analysis. Research is a fine line between plagiarism and creativity: you have to take inspiration from existing ideas and develop novel approaches that build on existing knowledge.

The PhD mathematician’s first charge is to identify a question worth pursuing — one that will invite creative solutions, and that can be answered within the 4-year timeframe of a doctorate.

By the first year’s end, I had my question pinned down. The second year would be all-out assault. That was the plan, anyway.

Toiling

Year two was the year of insignificance. I attacked my problem but met with resistance each time. I sensed breakthroughs were close, only to learn that my senses had betrayed me. My mathematical intuition — that holistic quality I had relied on in years past — had evaporated altogether.

When you perpetually fail to solve a maths problem, it can mean one of two things. Either the problem can’t be solved, or it is too damn difficult. With my research question, I had assumed the latter. Perhaps I just needed to take a deeper dive into the journals to uncover new lines of attack. A fleeting insight in the shower confirmed my gravest fears.

Archimedes in reverse

I had always dreamt of my own Archimedean triumph, where inspiration would strike me at the most inopportune moment. One morning, as I showered, that moment arrived. My intuitions finally awoke. But triumph turned to disaster as the mathematical breakthrough that came to me was not the solution to the problem, but the realisation that the problem could not be solved. For the past year, I had been chasing a wild goose; a maths problem with no solution. I had asked the wrong bloody question.

I stumbled into my weekly supervision a picture of dejection. Why waste my supervisor’s time, or even my own, with meek attempts at mathematical discovery? Impostor syndrome had taken firm hold of me; I was ready to quit. And cry.

My doctoral supervisor fits many of the stereotypes that mathematicians carry. Needless to say, I knew better than to lean on him for sympathy or emotional support. I shared the devastating discovery that my research to date was all for naught, my tone marked by desperation and a hint of shame. I honestly believed I would make unwanted history as his first failed graduate student.

Pivot

My supervisor then did what he does best. Paused, reflected, scratched his scalp. He drew a blank piece of paper and scribbled away, murmuring something inaudible. After another pause, he handed over the paper and declared in a neutral tone: ‘this problem might be worth your effort.’

His presentation may have been subdued — a bit awkward, even — but the message was riveting. He was not abandoning me! We talked through the intrigue of this latest problem, which had no resemblence to the one I had unwittingly pursued over the past year. In a single moment, with a single problem, my supervisor had pivoted my research.

Floodgates

It worked, too. My progress was not immediate. I continued to toil a while, but there was always a sense that the breakthrough was on the horizon. This time, my senses came good. I derived a partial solution to my supervisor’s problem, which opened the floodgates to further insights. Within a few months of that initial milestone, I had solved the problem in its entirety. Within a few more, I had extended the problem in several directions, asking new questions for myself. Smart, answerable questions too, the solution to one of which would later be labelled a ‘desideratum’ by my examiner (a mark of honour for mathematicians).

A noticeable shift occurred in my third year: I found myself driving the weekly meetings, educating my supervisor on new insights I had gathered. This transition marked the final stages of my PhD; I sprinted to graduation with renewed spirit and confidence in my mathematical abilities. My identity as a mathematician was secure.

I was under no illusion — I owed the turnaround to my supervisor. He possessed the knowledge and experience to pose the right kind of problem. He had no solution at the time, just a feeling that it fell within the scope of a doctoral research project.

Having devoted a year to an insoluble problem, I had only respect and admiration for my supervisor’s instincts. And gratitude.