The importance of foresight in a research problem

Image courtesy of OpenClipart-Vectors/27376 via Pixabay

Many academics, not just mathematicians, will describe doing research as [intellectually] exploring uncharted territory. However, research becomes much easier if you know the related results and resources in your field, which only comes with time and experience (as I have seen firsthand through my PhD journey to date).

My old boss and mentor Po-Shen Loh once told me, “for your research, always stay one step ahead and try to anticipate your advisors’ questions.” It has taken me a while to figure out how, exactly, to act on this advice, and it has come down to a few main points.

1. At least scan a large number of papers, only reading carefully the ones that appear to be especially useful or interesting
2. When applying an existing framework to a new problem, study the framework carefully and entirely to identify any possible complications beforehand (Example: for my research, many of these complications have involved trying to extend results known for scalar-valued functions to vector-valued functions)
3. Always have additional questions to explore on hand, even if you don’t know how to answer them

The exact way to proceed likely depends on your advisor’s mentorship style, and also how frequently you meet with him/her. Point #1 is largely about efficiency; while one…