Throughout my seven years as a math grad student, there was a constant struggle for attention between “what I was supposed to be working on” and “what my heart was into.” This theme played out in different ways over those seven years.
When I first arrived in Princeton in 1978, I discovered that I was the only one in my entering class who hadn’t already taken graduate-level courses as an undergrad. Worse still, the Math Department, catering to its typical clientele, offered no introductory graduate courses. None at all. Our mandate was to learn—or, for my classmates, review—the basic stuff on our own, and then dive into the research-level courses that the various professors offered.
I was adrift. What should I have been doing? Studying even harder about Noetherian rings, the Radon-Nikodym theorem, etc. etc., all at a level of abstraction that I found neither useful nor interesting. What was my heart into? Learning some concrete geometry and topology, along with some general relativity. You can guess what I spent most of my time doing.
As you might imagine, my qualifying exams did not go well. On my first attempt, they let me off the hook on the basic topics with a humiliating but welcome remark that “you can learn this stuff when you teach it.” But for the advanced topics, they wanted me to try again the next year. They weren’t impressed then either, but they decided that I should start working on a thesis anyhow.
By then, I’d signed on with Bill Thurston as my advisor. I don’t remember what problems I was supposed to be working on, but I made no progress with them. What was my heart into? By that time, my fellow grad students, colored chalk in hand, had introduced me to the beautiful world of low-dimensional topology, and I’d fallen in love with it. I’d also noticed that all books on the subject involved a lot of graduate-level algebra and analysis that really wasn’t needed to understand and appreciate the core concepts of geometric topology. So, I decided to ignore my thesis problems and instead took it upon myself to write the “missing book,” something that would welcome everyone—from high school students on up—into the magic world of multi-connected spaces.
That project took several years, with the book, The Shape of Space, finally completed in my fifth year as a grad student. My grad student stipend ended that year, so during what would have been my sixth year, I taught at Stockton State College and saved up enough money to return to Princeton and write a thesis in the seventh year. I don’t remember what I was supposed to be working on, but while Bill Thurston was away for the winter holidays, I decided to grant myself an indulgence to investigate some especially simple 3-manifolds and write a little computer program to find hyperbolic structures for some of them. I was lucky enough to stumble onto the smallest closed hyperbolic 3-manifold. During my first meeting with Bill after the holidays, I figured I’d briefly mention this little discovery. Bill responded with some ideas for how I might move forward with these investigations and how I might extend my little computer program. He said not a word about my old boring thesis topic, so by the time the hour was over, it was clear I had his blessing to do a thesis with my software for putting hyperbolic structures on 3-manifolds. “What my heart was into” had finally become “what I was supposed to be doing”!
Throughout six-and-a-half of my seven grad school years, I’d always felt that I would have been better off with a different advisor at a different school, with a firmer hand to guide me. But in retrospect, Thurston’s approach was exactly the right one: he let me find my own path. By the end of those seven years, I had some research software (SnapPea) and a book (The Shape of Space). In my heart, the book was my true dissertation. Even though it did nothing to satisfy Princeton’s Ph.D. requirements, it set the tone for my whole professional life, both in content (geometry, topology, cosmology) and style (exposition for the general public).
As for my advice to current students, I dearly wish I could say “follow your heart” and leave it at that. But, of course, it’s not quite that simple. One needs a paying job. So, my advice is: follow your heart, and find a way to turn your passion into your livelihood. I wish you good courage and the best of luck.
Jeff Weeks is a free-lance geometer, topologist and occasional cosmologist. His favorite successful projects are the third edition of his book The Shape of Space and the Geometry Games software. His favorite unsuccessful project was collaborating with cosmologists to deduce the topology of the universe from the cosmic microwave background. Support over the years has come from the National Science Foundation, a MacArthur Fellowship, lectures, and occasional work for science museums. In his spare time, he enjoys biking (spring, summer, autumn) and cross-country skiing (winter) with his wife, and learning foreign languages.
Mikael Vejdemo-Johansson, Assistant Professor of Data Science at CUNY College of Staten Island, started out in computational homological algebra before moving into topological data analysis (TDA) a decade ago. After a MSc at Stockholm University and a Dr rer nat at Friedrich-Schiller-University Jena, he went through 8 years of postdocs at Stanford, St Andrews and KTH before settling in New York at CUNY. Beside TDA and running the blog 
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