Create something impactful or just generalize results?

Question

I am a second year Pure Maths PhD student in Asia. My research interest lies in functional analysis, particularly Banach Space Theory. My aim is to work in academia as professor.

Recently my PhD advisor told me that it is not good for academic career if I just simply generalize some results. For example, if someone has done A+, then I should not just consider doing A++, as this kind of research very hard to continue.

Instead, I should create something on my own and try to investigate it. This will make my research more doable and more lively.

Now I am a bit loss. I know that if I can create something significant then of course it is a good thing and it will impact my future academic career. But I know that it is not easy in research to just create something impactful, especially I left 2 years only for my PhD.

I am not sure whether I should follow my advisor, that is, spend a bulk of my time to think of something completely new , or should I stick with my original plan, that is, try to generalize some results?

Answer 1

The people who can truly follow your advisor's advice are actually very rare. The advice is good, but not necessarily the best way to begin your career. In the long term you want to look at what can be done that is entirely fresh, but in the short term, you need to finish your degree and find employment.

If you want to make a big impact, IMO, you need to be able to work in a highly collaborative environment in which a lot of productive people are willing to share ideas in seminars and such. Depending on your current university, that may already be happening around you. But when it comes to finding employment, seek a place with a larger, rather than a smaller, faculty. Or, if it is small, make sure that they collaborate closely on semi-related ideas.

However, you need to follow your advisor's advice enough so that s/he will approve of and sign your dissertation. Most people at your stage of career are happy enough with A++. But it is even better if in doing that you gain insight into the larger issues of mathematical creativity.