Dear Early Career

I am a graduate student in mathematics just starting to specialize to my research area. I am enjoying graduate school, but wonder how can I expect to solve major open problems in my area?

—Motivated

Dear Motivated,

The truth is that you shouldn’t really expect to solve a major open problem at this stage—it happens very rarely. At graduate school you’ll hopefully develop a level of expertise in your research area so that you have detailed intuition about why the primary conjectures in your area are hard and understand what sort of developments would be required to make progress. Essentially, the first goal is to really understand why problems are considered “major.” Along the way, you will hopefully identify (with the help of mentors) interesting problems which are tractable and advance your area, and these might form the basis of your PhD thesis.

Whether successful or not, working on mathematical problems commensurate with your current level of expertise will provide valuable information about what might work and what doesn’t. For instance, you may be able to prove the result under an additional assumption. Even in cases when you can prove exactly the theorem that you were after, you’ll likely be aware of the limitations of the tools that were used in the proof. As you interact with your research community, you’ll hopefully find future research problems that will enable you to test these limitations and absorb more techniques introduced by other mathematicians, thereby expanding your knowledge of the subject. As this process repeats, you’ll find yourself in a progressively better position to close the gaps necessary to solve problems or develop theory. At this more experienced stage one still typically doesn’t expect to solve major open problems, but you might find the process of building your knowledge base and developing the subject in directions you find interesting (along with communicating these developments) to be the most satisfying parts of research in mathematics.

—Early Career editors

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