How to Self-Study in Ph.D math

Question

I was recently admitted to the Top 10 Universities in Math Ph.D. The department is really strong at the Number Theory, I am primarily interested in the Number Theory so it is a good thing. I will be most likely find some faculties that research the Number Theory and wrote a Ph.D thesis on that(provided things go well).

Although, I am also interested in Mathematical Physics. Potentially in the future, I would like to study topics between to Mathematical Physics (for example Mirror symmetries) and Number Theory. Having said that at the University, no one studies Mirror Symmetry, KdV equation, Hitchin System, or Geometric Langlands(there are experts of usual Langlands though)... This means that almost no faculties researches on the topic that I am curious about it.

So my question is in this case should I abandon my curiosity about Physics, and simply focused on Number Theory? Or even there is no facilities are researching on Mathematical Physics, is it worth to keep study it? In this case how to self-study it as a graduate student?

Answer 1

For US study you will be busy enough with one topic. Your first goal is to pass the likely comprehensive exams and find an advisor. This probably requires some advanced coursework. Then you need to dive very deep into your research topic. And, keep an open mind about the specialization until you need to choose a dissertation advisor.

However, focus on what is most important.

But, you don't need to "abandon" other mathematical areas, just delay deep dives. Put them on the back burner for a while until you are in a position to set your own agenda.

Along the way, however, you will need to take breaks for your own mental health. Exercise is good for that. But fairly casual reading (and note taking) of other math ideas can also give you a break as your mind switches "gears" in a new area.

A good practice when you read, either in your specialization or elsewhere, is to keep a notebook of ideas that you might want to explore in the future. Review it occasionally and add current thoughts. Finishing a doctorate with a notebook of possible research is a great place to be.

It is possible, in face to face situations, to have informal conversations with many faculty members (coffee lounge is good for this). If you know they have a specialty that interests you, ask for papers you might explore for this. It is good to have some guidance.

Once you get a tenured position you will be free to wander the highways and byways of math, physics, ...

Good luck. Have fun. But focus.

Answer 2

Potentially in the future, I would like to study topics between to Mathematical Physics

The future can wait a while.

So first priority : get your PhD in any mathematical topic.

Once you have a PhD you can try steer your career in whatever way you want. However until you have a PhD (which for most people is going to be stressful and non-trivial) you need to focus on getting it the simplest way possible with as few other goals interfering as possible.

If you want to keep an eye on the mathematical physics side (whatever you personally mean by that - there are different interpretations) then that's fine. But don't let that become a distraction from the primary focus : the PhD. I would not think of it as self-studying mathematical physics - that's perhaps pushing the boundaries of letting it distract from the goal of a PhD.