"On Life and Math", Chapter 4: Bonn Hiver (Part 1 of 3)

Starting in the winter of my second year of grad school, I spent a year stationed at the Max Planck Institute for Mathematics (MPIM) in Bonn, Germany, due to my advisor being dually appointed there. Eric came too.

Bonn was… cold. The weather was wet; the interactions were dry. It reminded me of suburban America: life felt lifeless. Even many Germans snarkily referred to Bonn as “a great place to travel from”. (It’s well-connected by train to more vibrant destinations — Berlin, Munich, Amsterdam, and Paris.)

The math was cold too. Lectures felt rigid: utterly decontextualized; no attention to beauty; definition, theorem, proof. I missed the flashiness, seductivity, and mystery of the lectures back home. It felt like many German mathematicians treated their craft like any other job: they worked hard, but there wasn’t any magic to it.¹

Shortly after getting situated at the MPIM, Eric and I reached out to some grad students studying homotopy theory at the nearby Uni Bonn, and we arranged to come visit them at their office. I remember arriving at something like 11:15am. After friendly but brief introductions, they kindly requested that we leave them alone and return at 12 noon to join them for lunch. This was quite a culture shock, coming from Berkeley where many grad students would happily chat with visitors for hours on end — even one-time visitors, to say nothing of long-term visitors such as Eric and myself. After finally finding and parking ourselves in an empty classroom and probably getting about 10 minutes of not-actual-work done, Eric and I returned at noon and headed with the others to the Mensa — the large and highly subsidized dining hall typical of Germany universities, where the food generally tastes exactly as good as its sticker price — and proceeded to have a stunningly terse and awkward conversation. I recall it consisting of a sizable quantity of uncomfortable silence, punctuated by people abruptly monologuing on various mathematical ideas or problems at entirely inappropriate levels of specificity and detail.²

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It took me a while to get my mathematical footing. In recognition of my inclination towards homotopy theory, my advisor onboarded me to the homotopy-theoretic side of a problem that he and some collaborators had been working on. I tussled with it for a few weeks, but didn’t make any further headway than they had already made.³ That would turn out to be the last time that I ever had a genuine one-on-one mathematical interaction with my advisor, who for this and other reasons is not listed on my Mathematics Genealogy Project page (see below).

I contribute a rare parentless node to the directed graph recorded in the Mathematics Genealogy Project’s database.

Thankfully, I soon fell in with a number of other mathematicians, among them two homotopy theory postdocs — Justin and Dave — who were particularly knowledgeable and generous with their time and energy. Along with Eric’s continuing guidance, I was able to eke out a sense of mathematical community — though it was far from the level of camaraderie offered by Berkeley’s spirited math grad student body.

It was in this context that I became enchanted by the mathematical object known as Topological Modular Forms, or TMF for short. This gorgeous object was (and remains) arguably the crown jewel of chromatic homotopy theory, and posits richly enticing connections with the exquisite field of number theory. I was completely certain that I wanted to work on TMF. Even the mere name “TMF” took on the mouthfeel of perfection.

TMF being as alluring as it is, I was far from alone in my enchantment. But like so many starry-eyed grad students before me, I had no idea how to go about actually doing something with it.⁴

I’m reminded of the following quote from Dave Eggers’s foreword to Infinite Jest, David Foster Wallace’s 1,079-page masterpiece.

This book is like a spaceship with no recognizable components, no rivets or bolts, no entry points, no way to take it apart. It is very shiny, and it has no discernible flaws.

As with Infinite Jest, so with TMF: a shiny and flawless spaceship with no entry points.

1. Of course, there were a handful of exceptions, i.e. German mathematicians who visibly reveled in the magic of mathematics. And of course, I’m only sharing my own impressions here. I don’t doubt that German mathematicians felt as deeply called to mathematics as I did: they just generally weren’t in the habit of showing it — at least not in a way that I could recognize.

2. There are many different ways of talking math, and at most a narrow range of them are appropriate in any given context.

For instance, if one is chatting at tea time with a visitor from another institution who works in a very different field of math, one should generally keep the discussion light on details and technical jargon. By contrast, if one is giving a lecture in a grad student learning seminar, one should generally strive to clearly and fully explain all technical details.

Relatedly, there’s a whole little social code around “going to the board” (meaning a whiteboard or blackboard) in a casual conversation, an invitation to which generally signals the speaker’s desire to get more explicit and ramp up their level of precision and/or depth. Without a board (which acts as an extended “working memory”), math conversations should generally not involve more than a small cast of characters — at least if the goal is to genuinely communicate ideas, instead of just to speak impressively.

Of course, there are plenty of contexts where it’s not really appropriate to talk math at all! And separately but somehow relatedly, there are also plenty of interactions among mathematicians — far too many, really — where it feels very much like everyone involved wants to talk about something besides math, but for whatever reason nobody is able to successfully pivot the group towards any other conversation topic.

3. Specifically, the project he suggested amounted to running a certain spectral sequence computation involving Spin cobordism. Spectral sequence computations can be notoriously difficult, with large swaths of them remaining unresolved for decades at a stretch — or perhaps for the rest of humanity’s existence, for all we know. So, it wasn’t the least bit surprising that I was unable to make further headway.

4. Just as “ending up at Harvard” is spoken as a faux-humble euphemism that belies an effortful and earnest striving towards status, so might mathematicians speak of “doing something” with a given mathematical concept or object.