Changing PhD area of math?

Question

Some background: I'm currently about to finish my first PhD year (in a 4 year program) in complex geometry and I'm really struggling with it because I have to learn a lot of new mathematics essential for the area and which are completely new to me. My background is not completely in mathematics. I did my bachelor in physics and computer science, and after that I obtained a masters in pure math, where I got to learn the foundational material (analysis, topology, number theory, etc.)

As a consequence I've contemplated the following options:

1. The first gambit: Continue with it and give it my best try. I'm confident I can learn all the material I'm required to, the problem is my learning speed is not fast enough.

2. The second gambit: Change to a more friendly area: I'd like to think of this as an area that requires less foundational material (more vertical perhaps?). An area which may be better suited for me according to my background AND for which my university has a group for it (this limits the options). One of the options would be applied math.

3. The solution I fear most: Quit the PhD. I really don't have the slightest idea what would be of me in this case since I despise the idea of working in a 9-5 job, it's just not my thing.

In fact, all this got triggered by my advisor showing concern about my progress a few months ago, and as of now, they told me to learn certain topics, yet I find them quite hermetic given my foundational gaps, so I often get very stressed while studying this foundational material since I should be also studying what I was told to.

At the moment I'm not personally ready to take options 2 or 3, but these ideas are always in my mind.

What this post is for: I'd like to know what other people in similar situations to mine have done and how it worked for them. Maybe I'm overlooking other better courses of action. Maybe it's normal to feel this way or have some of this problems, or not. Broadly speaking I'm looking for advice from a more experienced person.

Answer 1

There is one piece of advice that seems particularly important to give: do whatever is needed to enjoy working on your PhD. You won't go very far if stress and pressure is all that gets you going, and the probability of getting an academic job after even a very good PhD is too thin to go into PhD without enjoying the PhD years by themselves.

Of course you will have to work hard whatever path you take, and not all the work will be pleasant. But if you are not enjoying your current work at least from time to time, you should definitely change something. Having insufficient foundations is a pretty good explanation for not enjoying oneself, so going to a subject where this problem will disappear feels reasonable to me.

Answer 2

I really feel for you. I had a very similar experience in math graduate school. I won't go in to the details, but let's say I ended up in a research group whose general area I was interested in, but the particular research they did was not as interesting to me and required a rather different background. I struggled to fill in the gaps in my background, but it was difficult not least because I wasn't all that interested in it, and I always felt only half welcome in the group. I ended up basically taking approach 2 from your list, working on a project at the interface between my current group and another group in the department. I wouldn't say the latter group "required less foundational material", but the stuff I wanted to work on had not been that heavily studied before, so it was in some sense easier because I was the first to look at it seriously. Not surprisingly, the results were not particularly breakthrough, however, but I did complete my PhD defense ( a year longer than usual ).

I have since left academics, in no small part due to my experiences in the social and societal aspects of academic math life. In fact, realizing this about the fourth year of my PhD program, I started looking outside of academia for job opportunities, took a couple of classes in more applied fields, and contacted a recruiter. If you decide to go this route, I very strongly recommend contacting someone to help you with the job search. In my experience, the disdain for leaving academics and entering industry amongst pure mathematicians, while mostly tongue-in-cheek, nevertheless means they are clueless about the prospects and the process. Again, one person's experience, but I think it can't hurt to reach out.

I might have been able to struggle harder through the foundational material, but I felt I would have ended up with a thesis in a subject that was not my passion, and it seemed to me at the time at least that once you've gone that far into a specific research field, your post-doctoral position(s) will be in that field, and it will be even harder to switch; I didn't want to spend my career in the same field as my research group.

I don't mean to dissuade you from trying to stay in academics, but I will say that I'm very happy with where I've ended up. I would also caution you, the last year of my graduate school was very difficult and I had to find my own way both in terms of research and job prospects; my advisor could not have cared less about my job search once I was no longer looking in academics.

Answer 3

My experience was a little like yours. I studied algebra as a pure math grad student. Up to a certain point, I actually enjoyed the work even. But, when it came time to really get down to writing a dissertation, select an advisor, etc., I just found that I did not enjoy the subject that much.

I switched to doing an applied math PhD and have found it personally to be much better. I enjoy the subject a lot more, feel more inspired in the field of research, and am ultimately a lot happier.

I sort of "lost" a year or two of time by making this switch, but it was better than spending the next 30 years in a field I did not enjoy.

Grit can only take you so far in a PhD.

Answer 4

You are taking a journey a road less travelled by many educated individuals. There had to be a compelling reason for you to choose y

To quote the first articles last paragraph:

Successful PhD students thrive in a highly intellectual environment, are willing to work very hard with only a possible payoff, love their field of study, and don't mind forgoing impressive paychecks. If this sounds like you, forge ahead!

http://www.princetonreview.com/grad-school-advice/why-you-shouldnt-pursue-phd

Remember that your pursuance of new knowledge will expand your perspective. It is not just solving a new or lingering problem, your cognitive notions will be tested and truths will be challenged and verified. The rigor is real and the demand complex. Your computer science background is your cure for the analytics. Physics roots theoretical understanding to tangible outcomes.

Quoting someone who directly engaged your current field of study:

Because you love math[s].

http://www.math.ucsd.edu/~justin/phdadvice.html

Option two sounds most to my liking because I am a Computer Science and Applied Math fan. Its practical to me.

For you, it sounds like you are going to finish with choice 1 and really should get back to the books to exercise your brain on the routine of the extra courses not seen or remembered from your past (like number theory). You have a lucky foundation that favors your current studies. Have a good journey.