I love the variety of beautiful places (physical and intellectual) that math takes me. This week math has taken me to the south of France. What a wonderful place to do math, or really anything. Not least because France has recently taken the extraordinarily cheering step of electing a Fields Medalist to Parliament. I was at the Centre International de Recontres Mathematiques (CIRM), a math center near Marseille. The conference: Arithmetic, Geometry, Cryptography, and Coding Theory (AGC^2T), a biannual conference which began in and has been held at CIRM since 1987. I hate to go overboard, but it might have been the nicest conference ever. It wasn’t just the daily hike to the nearby Calanques to swim in the Mediterranean, though that was incredible. The talks were great, which I say with full information because I went to all 35 of them. Even more, though, I just enjoyed talking with these very interesting people. The conference organizers, and really all the participants, deserve a great deal of credit for creating an exceptionally relaxed and positive atmosphere.
In graduate school I spent a lot of time reading Algebraic Curves over a Finite Field, by James Hirschfeld, Gabor Korchmaros, and Fernando Torres. James Hirschfeld was a member of the scientific committee for the conference, so I got to meet him for the first time this week. We started chatting at a coffee break, and I didn’t realize at first that I had spent so much time in the company of his work. He was very personable, and I eventually told him that I had in fact purchased three copies of this book over the years (losing one in a move across the country and another in some unknown way). I asked how he came to start writing books, and he said that in fact he started his first book because his research wasn’t going well. He thought that writing a survey of the field would help him regroup. I’ll quote here something he had earlier written about this: “In 1972, when my research was going poorly, I decided to write a one-volume survey of the field of mathematics that I had been working in, that is, the combinatorics of finite projective spaces. … As the book was supposed to be a survey of the field, I decided to begin the book by compiling as complete a bibliography as possible.” He said that in compiling the bibliography, he read many papers and had a lot of ideas that helped him move forward with his research.
I found this narrative, well, not exactly surprising, because it makes sense—writing a deep survey like this would be incredibly enlightening. I would love to do it, and I do like writing survey/expository pieces. It seems impossible, at this juncture, that I will ever have the time and focused energy to do such a thing, but he had only been out of graduate school for 6 years at that point (the same as me, now). Ahem. Anyway, it was not surprising that writing the book worked, but it was somewhat unexpected and inspiring to me that he freely stated that his research had been going poorly. Nobody ever says their research is going poorly in public! It is so rare! The market, the culture of the field, and our own dedication to positive thinking all encourage us to accentuate the positive, and greatly fear the negative. Since, when things aren’t going well, it becomes very easy to think that we actually aren’t smart enough and we probably shouldn’t be doing this after all. For those of us that are trying to get tenure or the next job, it seems like an act of self-destruction to send the message that we are stuck, or struggling in any significant way. So, I guess it really struck me to hear someone I respected casually mention that their research had been going poorly.
Now, poor is a relative term. We are each working from our own vision of what our research life should be, so one person’s poor could be another person’s ideal. But this anecdote, which Hirschfeld just mentioned in passing, really got me wondering what people do when things are going poorly. In particular, how do the mathematicians that I admire get unstuck or find inspiration in their research lives? I often imagine that the “mathematical experts” of the world don’t have this sort of trouble. On the other hand, I spend some time every semester normalizing struggle and convincing my students that the experts struggle too. Students in algebra and number theory classes that I teach are generally assigned to watch The Proof, the Nova documentary about Andrew Wiles and the proof of Fermat’s Last Theorem (based on a book by Simon Singh). There are several reasons I share this film, but one of the things I always point out is just how long it took to prove this statement, and how many brilliant people got utterly stuck on the problem. The idea of being stuck on this scale, of making mistakes and struggling for years, seems to be a bit of a revelation and a comfort to my often-stuck students.
Ask the Experts: Getting Unstuck
With all this in mind, I decided to ask some really bright, successful mathematicians about being stuck. What I wanted was to ask about times when they had really struggled, and felt that their research was going poorly. But, in the end, the specifics of this seemed too personal to ask people to share in someone else’s blog. I just couldn’t bring myself to put people on the spot about this. What I did manage, though, was to talk to some great people and get some more general advice about what some experts have done when they were stuck or looking for inspiration. And of course, since being good at math means being good at this kind of struggle, they had really good suggestions.
I asked Bjorn Poonen, Clause Shannon Professor of Mathematics at MIT, if he had a good story about being stuck in research. After thinking a bit, he said that he had often been stuck but that there wasn’t really a good story. He said, “At one point I was stuck for a year, and that was pretty much the story—I was stuck.” Bjorn says that when he has no good ideas to move forward on a problem, he finds it helpful to explain the problem to someone else, “not necessarily to get their advice, but just because the process of explaining it forces me to rethink it.”
Irene Bouw, Professor of Mathematics at Ulm University, had an immediate answer to my question of what she does when she is stalled on a problem: “Just leave it for a bit–that is the best thing. After you have left it alone for some time, you have to go back over it from the start and check every single detail.” Irene is a realist about the situation, though, in a way that perhaps those looking back at the comparative wealth of time of graduate school can relate to. “There is no recipe that always works. Also, this method worked better when I was a PhD student and I had more time. I don’t have time to be stuck anymore. Now, maybe I come back to it, and I decide it is not so interesting, and leave it. Of course, this is more possible when you have tenure, and maybe you have more problems than you have time to work on.” She gets to one of the hardest questions for a pre-tenure person—still there is no time, but there is great pressure to produce results. When should you push through on something that is less interesting to you now, for the sake of salvaging a publication, and when you should just cut your losses and move on?
Everett Howe confirms that he, too, has been stuck, and in particular has sometimes felt stalled by a lack of inspiration. He said that he has found that going to a conference and just talking to people about their problems has actually been a good way to find direction. “When a friend or someone that I meet at a conference asks me something that they would like to know for their own research, and I have some ideas on how to answer, it gives me the incentive to work on their question. The personal connection provides the motivation.” I found it really interesting to hear Everett elucidate one of the ways that generally meeting people and building friendships within math, and particularly going to conferences, can be so worthwhile—these things can open up a whole new world of questions to care about, and maybe you actually know how to solve some of them!
Thank you very much to everyone who talked with me at the conference, and especially to James, Bjorn, Irene, and Everett. Your thoughts on research setbacks? Conferences in the south of France? Please share in the comments.
I should mention here that I was thinking about some of the ideas in this post weeks ago. In an earlier comment thread, I asked Joseph Silverman some questions about his math life and how he dealt with being stuck in his research. Just in case someone out there doesn’t read all the comments on every PhD + Epsilon post, I’ll share an excerpt here:
Me: “What do you do when you are stuck, and you may not know anyone who is interested in your problem? Is it strange to cold-call an expert, or to impose on your advisor? If you ask an expert for advice, is it your responsibility to add them to the project? ”
JS: “There are lots of strategies when stuck. Try to work out an example or a special case. Try to find a counterexample (which often leads to a proof). Try to read something that seems relevant to your problem. Talking to your advisor is fine. Put the problem aside and work on something else for a month or two. Cold-emailing an expert probably should be saved as a last resort, but talking to experts at conferences is a good idea. (Then, when you email them, they know who you are.) If there’s a specific fact you want to know, MathOverflow is good, but before posting, write out your question locally,read and re-read, wait a day or two, re-read again, then post. Also first search MO to see if your question has already been asked and answered. It’s amazing how much stuff is there.
“Generally, if anyone offers you advice, you should acknowledge that advice in the acknowledgements of the paper. But it has to be pretty substantial before you ask someone to be a co-author. There’s no hard and fast rule.”
Thanks again to Joe for taking the time to read the blog and answer my many, many questions!
Postscript: A few more thoughts from an expert
I had such a good time talking with James Hirschfeld about his mathematical life that I wanted to share a little more of our conversation here. I asked him a few questions about how things had changed in mathematics, his early career, and how he found problems. For your enjoyment, here is a sampling of things that that I found particularly interesting from his answers:
“Considering my career, I always remember that in my early career I was particularly fortunate: in the mid 1960s, if you had a PhD you could get a job anywhere. There was a great post-war expansion of the university system. So, I know that my experience is extremely different than it is today.
“But when I think back to the start of my research career, my MSc supervisor suggested a very particular problem—working on the double six theorem over the field with four elements. When I moved on to a PhD program, my supervisor didn’t suggest a new problem; he said just continue your previous work, with the double-six over larger finite fields. I did this, and of course it’s very nice to have a specific problem to work on. The fault with this is that there is a larger context to consider. What I should have done is more wider reading. Algebraic geometry has been through so many phases. After the classical stuff, there was Andre Weil doing it more abstractly, then Grothendieck even more abstractly. As a PhD student, it’s good to have a particular problem and get somewhere with it, but it’s important to acquire this broader knowledge, rather than just doing some calculation.
“And how do mathematical problems arise, anyway? Of course, there are a very few people who are doing something truly original. Otherwise there are two ways. You can generalize something or connect to similar problems. But equally well, we get problems from other subjects. Physics or social sciences or biology. And there is an interconnectedness to look for within the fields: In my area of finite geometry, for example, it was the 1980s before geometers and coding theorists realized that they were working on the same problems.”