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.