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Math At the end of my Physics PhD, Switch to Mathematics?

  1. Jun 22, 2017 #1
    Hello! I am writing this to get some ideas for what to do with my future. This will be a long post 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 want a career in science and 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:

    1. To learn about the Atiyah–Singer index theorem.
    2. To learn more about the Riemann Hypothesis.
    3. Probably learning about Inter-universal Teichmüller theory, although if I choose this I will surely need many years :)
    4. Algebraic topology.
    5. 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 :)
     
  2. jcsd
  3. Jun 22, 2017 #2

    Charles Link

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    @batboio I find your perspective interesting. It seems that theoretical physics and/or mathematics can be very much an adventure if you land on the right path, but there are paths where you wind up spinning your wheels. I am presently retired, but I can say the same was the case from time to time in my career=some experimental efforts and some coursework offered little or no return, while others were very much an adventure. My recommendation is very much what you are already doing=stick with it and be patient=eventually you'll stumble across something that provides the adventure and excitement of new findings that you are looking for.
     
  4. Jun 22, 2017 #3

    Chronos

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    Some people are inclined to gravitate from physics to engineering whereas you appear to be leaning in the opposite direction. Engineering is merely applied physics and physics is little more than applied mathematics. Many of the most spectacular successes in physics [e.g., GR, QFT] originated from underlying mathematical principles and virtually all of them demand mathematical validation before they gain widespread acceptance. The distinction between a theoretical physicist and mathematician is often very blurry. My suggestion would be to focus on the branches of math that apply to the physical principles you find most interesting. The next great breakthrough in physics is more likely lurking on a white board than under the icy gaze of some elaborate instrument.
     
  5. Jun 23, 2017 #4
    Switching now sounds like a recipe for never finishing. I would strongly recommend that you press on, finish what you have started. A switch later, or a search for something more in between would be much easier after you finish.
     
  6. Jun 24, 2017 #5
    Thank you all for the replies. The more opinions the better :) I read now that I may have not written this clearly enough. I don't plan to drop my PhD. I will most definitely finish it and then look for something different :)
     
  7. Jun 24, 2017 #6
    Engineering is absolutely not applied physics. Design principles abstract away from the underlying physics. Moreover the objective is not an accurate answer, it's an answer that is good enough. Some disciplines of engineering are more mathematical (e.g. control systems) than physical.

    Even worse, physics has an experimental and a theoretical end. The former is obviously not "little more than applied mathematics", and less obviously the latter is definitely not applied mathematics. Physical principles are not mathematics. They are principles with mathematical formulations. The distinction between mathematician and theoretical physicist is abundantly clear to anybody with an awareness of physics that actually connects with experiments. Similarly to how an engineer does not necessarily demand the most accurate model, a physicist does not demand the most thoroughly consistent mathematical rigor. This is historically well precedented (e.g. path integrals, calculus, neither of which required "mathematical validation before they [gained widespread acceptance]").

    I see little evidence that beginning with mathematical principles leads to breakthroughs in physics. The order is typically reversed. Einstein did not learn about differential geometry before he deduced general relativity. Gel-mann did not learn about Lie groups before applying them to nuclear physics. There has been much speculation that we have entered a new era where the order ought to reverse, and we should look first to mathematics. However, this has produced nothing that has been experimentally validated, aside from topological methods in condensed matter, which, sadly, are a greatly inferior achievement (so far, anyway, unless topological insulators or superconductors receive quality experiments) to anything similar in high energy.

    Indeed, all the evidence suggests that empirical observations play a fundamental role in triggering new developments, which are then built upon with mathematical analysis. The only thing I'll say in Chronos' defense is that everything I've said so far is just a principle or heuristic. It is possible that we have in fact reversed the traditional order of things. While most of physics consists of principles with mathematical formulations, many have argued that quantum mechanics violates this interpretation. So, perhaps there is some evidence for that end of things, but I think it's very hasty to hold such a point of view.
     
  8. Jun 25, 2017 #7
     

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  9. Jun 25, 2017 #8
    He looks very bemused about your apparent ignorance of the concept of emergence.
     
  10. Jul 21, 2017 #9
    Maybe look into Mathematical Physics as a bridge.
     
  11. Jul 22, 2017 #10

    radium

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    I think your comment about the progress of theoretical physics only applies to a limited subfield of theoretical physics. There have been a lot of experimental and theoretical advances made in condensed matter physics in the recent years. Many people in the less phenomenological areas high energy theory have become very interested in condensed matter problems for this reason. Have you considered this area?
     
  12. Jul 23, 2017 #11

    mathwonk

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    If you wish to finish your thesis, I would suggest focusing there, and trying to generate some enthusiasm for the topic. In my experience (admittedly only within mathematics), a thesis is not something that just pops out after a year of routine calculation, but is by far the most difficult and intensive part of the graduate experience. If you actually complete an acceptable thesis project, I believe it likely you will have found some enjoyable topics to pursue. I have heard of a few cases of famous mathematicians who were pursuing thesis topics they found unintertesting, obtained professional positions that freed them from finishing, and immediately took up more interesting and challenging projects, (still within mathematics), but they are somewhat rare.
     
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