Research in Tandem (Part 3)

by radimentary

Today I want to end this discussion about research collaboration with my most useful tip for grad students: build an explicit model of how collaborators work, especially your PhD advisor.

One of your primary goals in graduate school is to set aside 20% of your brain for simulating your advisor, who is typically the best mathematician you are in close contact with. Learn and imitate their reflexes, their tastes, their decision trees. Spend substantial chunks of time during research meetings being curious about minds and modelling how other mathematicians operate. How did they come up with this? What do they know that I don’t? Why did they try this approach first?

Even if this is the only thing you manage to do in grad school, you end up as a low-resolution clone of your advisor – which is not ideal but nevertheless a better-than-average outcome.

Here are seven points of inquiry to jumpstart your quest to model another mathematician.

Research Direction

  1. What problems do they work on?
  2. How do they choose these problems?
  3. How do they weight the important of a problem versus aesthetic interest in it, versus the actual likelihood of actually solving the problem?


  1. Who do they work with most frequently?
  2. What qualities do they praise about their closest collaborators? How is labor usually divided in their collaborations?
  3. By what criteria do they evaluate other mathematicians?

Your relationship

  1. What exactly do they want from you?
  2. Conversely, what exactly do you have to offer them?
  3. Most mathematicians are somewhat motivated by genuine care for young people, but there are pragmatic considerations beyond that. Can you help realize their mathematical vision? Do you carry out humble work that makes their life easier? Are you stimulating and enjoyable to be around?

Patterns of thought

  1. What patterns do you notice in their thinking over time?
  2. What are their common first refrains when working on a problem?
  3. Which pictures, techniques and lemmas do they rely on time and time again to orient themselves?


  1. What are their glaring weaknesses?
  2. From where you’re standing, are these weaknesses gaps that you can fill, or dump stats that you should deprioritize as well?
  3. Do they ever advise you “do as I say, not as I do”? How seriously should you take such advice?


  1. How did they get started in math?
  2. Getting into orbit requires different strategies from staying in space; what did they do at the start of their own career?
  3. What mathematicians did they themselves admire and learn the most from?

Work-life balance

  1. What is their working life like?
  2. How much time do they spend on teaching, traveling, and administrative nonsense?
  3. Would you actually want to work a day in their shoes? If not, what would you adjust to make it ideal for you?