I am writing this to get some ideas for what to do with my future. This will be a long post I think so I should start by introducing myself…
I am currently in the last year of my PhD studies and I will be starting my thesis soon (at least I hope so). My work is on classification theorems in General Relativity. With that said, I feel burned-out. At least that’s what I think it is. It’s not that I am really overworked (I am pretty sure I don’t work as much as I should), but I don’t find my topic interesting anymore. Even worse, I am kinda disappointed in theoretical physics as a whole. I follow daily the gr-qc section of the arxiv and I rarely find something interesting. I feel that theoretical physics is going nowhere right now meaning that countless articles are written with just particular calculations or theories that have no experimental basis (take a random quantum gravitation article). This is not even good math, as mathematics should aim to generalise rather than calculate particular examples. I don’t mean to offend anyone so I apologise if this is the case. Those are just my thoughts on the matter (admittedly coming from a guy who is not even a PhD yet). It’s just that I think physics right now needs the next big experimental result that will throw everyone off (I was really excited for some time when there was some talk of a new particle detected at CERN. This was last year, wasn’t it?).
BUT I still find mathematics immensely interesting. At some point before I began my PhD I thought about switching to pure mathematics. I decided against it with the argument that I could do this later after I got my PhD since in my country we have not yet fully adopted the publish or perish strategy and there are still some permanent positions which are easy to get into. The pay is not that great, but it gives you relative freedom to pursue your interests. Well I am now at this stage, but this means that I will probably work alone for some time and I will be really grateful for some opinions. Some of my ideas for what to do are:
- To learn about the Atiyah–Singer index theorem.
- To learn more about the Riemann Hypothesis.
- Probably learning about Inter-universal Teichmüller theory, although if I choose this I will surely need many years :)
- Algebraic topology.
- Category theory (I have basic knowledge here).
As you can see all of those involve learning new things, but not actively working. I think I can dedicate at most 2 years for this. I guess I am mostly afraid that I will have to start working in a new area at some point and this time without an advisor. The fact that I am reading Hilbert’s biography right now doesn’t help one bit… I know the perspective of history makes science more romantic than it actually is. Still, Hilbert was a great man.
Any advice and/or comments are greatly appreciated :)
P. S. I apologise if this is not the correct site for this kind of question. I thought also about posting this on mathoverflow, but decided this is a better place.
My background is quite similar to yours, but I already switched to math and I did it much earlier. I made my Bachelor in physics, my Master in mathematics, and now I am in the middle of my PhD.
I wrote my Bachelor thesis on a topic in theoretical physics. Of course, not on a level as deep as you might have worked in this field. It was that time that theoretical physics was no longer appealing to me, because (at least where I studied) it was not even close to being mathematical rigorous. One easily lost the grip on what is allowed when dealing with a bunch of formulas, because one never studied the highly subtle inner workings of the transformations and rules one was using. Everything was just understood as good as necessary, as loose as possible. One of the most hated things as a physics student was the highly dubious $\delta$-function. Zero everywhere except in the origin, but enclosing an area of $1$. Once you have accepted this you learn how to differentiate it and things start to get even messier. It is like believing
$$1-1+1-1+1-1+1-\cdots=\frac12$$
because someone showed you some strange way to rearrange everything until it works.
But now on your desire to switch. I think having a background in physics (maybe also in computer science/programming) is highly valuable when starting mathematics. Why? Because then one knows why all of this is done. Many mathematicians solve problems for the problems sake. Then invent a new problem and go on. Other mathematicians just play with symbols on paper, accept definitions unquestioned etc. instead of trying to visualize what is going on. This will make them progress much faster, but they also will leave behind all the reason for why this is done. In general it is always valuable to have several viewpoints on a topic: the formal rigorous view of a mathematician, the mechanical and close-to-reality view of a physicist, the algorithmic and limited-resource-aware view of a computer scientist. Also, in my experience, an understanding in some applied field will really help you explaining the stuff going on in your head. You can draw pictures, animate a process etc. instead of just writing down the formula and saying "You see? Here we got it!".
However, switching to math you should be prepared to meet a lot more rigor. There will be a need to proof anything $-$ no matter how obvious. The special and pathological cases are important, even if they where not the thing you had in mind intially. You cannot argue them away. I know nothing about your mathematical education so far, but usually a physicist receives a comparatively good one. Depending on your current understanding it might be necessary to study some undergraduate topics that were skipped over in physics, e.g. measure theory, discrete mathematics, etc. (I guess one can add here much more depending on preferences).
I must admit that it was the best for me to be a student of math at least for some time, instead of immediately starting with a thesis. All I can recommend you is to read a really rigorous math book to see how things go on in there. A historical book might not be the best start here.