By Kareem Carr

Fields medalist Terence Tao is undoubtedly a very successful mathematician. He works primarily in harmonic analysis, partial differential equations, geometric combinatorics, arithmetic combinatorics, analytic number theory, compressed sensing and algebraic combinatorics. I interviewed him over email to gain some insight into how aspiring mathematicians can become successful too.

Q: There is an extensive list of career advice on your page which I encourage everyone to read. What do you think is the most important piece of advice on your website for young mathematicians?

A: It would depend on the mathematician! For instance, I know some who are very hard working, but don’t ask the type of “dumb” questions that would advance their knowledge nearly as much as they should; and then I know other young mathematicians who are exactly the opposite. But perhaps my favorite piece of advice on those pages isn’t even my own, it is the initial quote by Erica Jong: “Advice is what we ask for when we already know the answer but wish we didn’t”.

Ultimately you should follow advice not because someone tells you to, but because it was something that you already knew you should be doing.

Q: As a long time reader of your blog, it’s hard not to notice how productive you are. You seem to get an amazing amount of work done. How many hours a day do you spend doing mathematics as opposed to other things?

A: It varies tremendously from day to day – I look at what I’m scheduled to be doing that day, and the various tasks that I need to get around to (of varying levels of mathematical sophistication), and also examine my own energy levels and motivation, and figure it out from there. I don’t always accomplish what I might initially intend to do, but I usually make progress on something (even if this “something” is just the task of replenishing my own motivation levels). One nice thing about having a blog is that it provides something to do when one wants to do something reasonably sophisticated mathematically (e.g. learn about topic X properly), but doesn’t have the time and energy to really work on an open problem or something. (I usually can’t maintain that level of focused concentration for more than an hour or two at a time anyway.)

But there are definitely some days in which I am too fatigued or caught up in non-mathematical tasks to get much “real” work done. That’s usually a good day to do errands, proofread papers or blog articles, and respond to email đź™‚

Q: You’ve written that you adapt your work schedule to match your energy levels. What time of day do you find you are most productive at mathematics and why do you think that is?

A: Again, it varies from day to day. Certainly if I give a talk or lecture, or even just a deep conversation, then I tend to be quite tired for several hours afterwards and not really able to do any advanced mathematics. And if I’m distracted or worried by something, then I usually can’t concentrate on maths. Conversely, if I, or one of my coauthors, have just found some interesting thread of an argument that I’m itching to pursue further, then I can often block out everything else and work on it. Other than that, though, I can’t really predict what my energy levels will be at any moment other than the immediate present.

Q: Do you think programming is a helpful skill for a mathematician? If so, which languages do you think are the most useful?

A: Certainly it is useful to know at least one language, so that one can do some rudimentary computations whenever necessary. If one has to do a really large-scale computation, then it’s likely that one would have to learn some customised tools and packages and so no specific prior language background would be particularly useful – rather, a general familiarity with how programming languages work would be more valuable. (Though, for certain specialised subfields of mathematics, specific software packages could be of particular use, of course.)

I also think that there are some useful analogies between writing computer programs and writing mathematical papers; in my pages on writing papers, for instance, I discuss how computer programming philosophies such as encapsulation and information hiding can help one structure a paper in a more reader-friendly manner.

Q: You have collaborators in several fields. What is the most important element to having a successful mathematical collaboration, especially between fields?

A: I feel that collaboration is most productive (and enjoyable) when it arises from a genuine friendship, and not just a business deal. In particular, one should not be worrying too much about how to apportion the credit or the workload for a project, and one should always be trying to communicate one’s own thoughts as clearly as possible to the other collaborators. At least one of my collaborators insists on strictly adhering to the “Hardy-Littlewood rules of collaboration”; with most of my other collaborators, we don’t adhere to these rules to the letter, but we certainly follow the spirit of them in most cases.

Q: One of my professors once said that a big part of mathematics is frustration management. How do you manage your frustration?

A: I tend to have several things to work on at any given time; when I get stuck on one of them (which often happens), I write up how far I managed to get, and turn attention to something else. Also, I have a fair number of things to do which are not as difficult as solving an open problem (e.g. blogging about a known piece of mathematics), so that’s usually a good way to evade frustration. When one comes back to the problem a few months or years later, with some more tricks under one’s belt and a fresh perspective, often one can see a way to make progress that one didn’t see before.

Wow, this is great! Thanks, Professor Tao, for giving us this good advice.

Very nice read.