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I have now some problems about my research Career, I would like to tell my stories. I am a Chinese guy, but now a Ph.D. candidate in Germany, in the field of so called 'Geometric Analysis', but I do not feel happy when I work in such a field. Somebody said that a wrong choice of study field or supervisor means several years' Frustration, it describes my situation very well. I choose this geometric analysis only because I want to study geometry with some analytical methods, but gradually I found that the most researcher in this field only have very narrow knowledge about PDE and Riemannian geometry, I think this field is lack of real beautiful idea, but full of papers, I am not judging that this is not good, but this is not my taste.

Compare to the normal researchers in this field, I have relative comprehensive mathematical knowledge, I am not only familiar with second order elliptic PDE and differential geometry (Riemannian geometry, geometry of fiber bundles, Chern-Weil theory) but also with algebraic topology (homology, cohomology, characteristic class and spectral sequence), complex geometry(the whole book of Demailly'Complex differential and analytic geometry' and some complex Hodge theory in the first book of C.Voisin). Several years ago, I have also studied abstract algebraic geometry and Index theorem in courses, but now I am not familiar with such kind of material. In China I have already known, for my future, I should not only have one math-tool. At that time, a professor in algebraic geometry has suggested me that as a young guy, algebraic geometry may be a better choice for the future and invited me to be his student, but I hesitated and refused, that was maybe a stupid decision. During my research in Germany, I was always very depressive because what I was working were only some trivial and unnature PDE estimates.

Now my career is a little hopeless, somebody suggested that I could try to contact some experts who work in Symplectic geometry with analytic methods, like members of Hofer's school in Germany. Yes, I actually want to study some deep topics in (complex-) algebraic geometry or symplectic geometry, but the problem is, I will soon have a Ph.D. degree, how can I change my research area after my graduation? As a normal young guy the experts in another field do not know me at all, maybe it's very hard to get a postdoc position from them, so should I do another Ph.D. in math? Is that worthy? I am frustrated and asking for help, maybe some guys have similar experience. If I cannot research what I like, I must try to find a Job in Industry.

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    $\begingroup$ Many people do change their field after getting their PhD. It depends on how quickly you think you can get up to speed on something new. $\endgroup$
    – Will Sawin
    Commented May 28, 2018 at 15:29
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    $\begingroup$ It is absolutely normal to change the research topic after you get a degree (and even before that, if your thesis is essentially ready.) People know that, including the experts you are worried about. I know several examples where people got two PhD's on different subjects, but they were done in two different not completely unified academic systems (Russian/Western). $\endgroup$
    – Fedor Petrov
    Commented May 28, 2018 at 16:37
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    $\begingroup$ I call myself a geometric analyst and I have changed my research area several times after my PhD. It happened naturally: I got interested in a question and I started investigating. There are big questions that many people are chasing and there are $\endgroup$
    – Liviu Nicolaescu
    Commented May 28, 2018 at 17:33
  • $\begingroup$ Related questions: mathoverflow.net/questions/69937/… and mathoverflow.net/questions/12684/switching-research-fields $\endgroup$
    – Timothy Chow
    Commented May 28, 2018 at 18:36
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    $\begingroup$ It could also be that your expectations were a bit unrealistic, and it's not so much the specific area you happen to be in now. Your description (I'm paraphrasing what you actually wrote) "lots of pedestrian papers, few brilliant ideas" describes the vast majority of the research (I'm tempted to write "research") in just about any area. $\endgroup$
    – Christian Remling
    Commented May 28, 2018 at 18:59

3 Answers 3

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Finish this degree, then switch to whatever interests you. Many (most?) mathematicians change fields at some point in their research careers.

Bob Solovay told me once that the most important research was the first new thing you did after you got your degree. His thesis was A Functorial Form of the Differentiable Riemann–Roch theorem. He finished it in a hurry so he could move on to mathematical logic, where he's famous.

https://en.wikipedia.org/wiki/Robert_M._Solovay

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  • $\begingroup$ And Solovay also made a major contribution to computational number theory, the Solovay-Strassen Primality Test. I believe he told me once it came from a single conversation, but I may misremember or he may have exaggerated. $\endgroup$
    – Andrew Lazarus
    Commented May 29, 2018 at 18:40
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    $\begingroup$ ...and the Solovay-Kitaev theorem in quantum computation. $\endgroup$
    – Adam P. Goucher
    Commented Jun 19, 2020 at 0:30
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With enough motivation, you can learn new areas and go there. I started my first two papers in complex analysis, related to the Schrodinger equation. I am now doing algebraic combinatorics related to representation theory, some quasi-symmetric functions and enumerative aspects of combinatorics. On the side, I have also worked a bit on polytopes, and an unfinished project which are related to invariant measures and Julia fractals.

There is no point of taking a second PhD - having a PhD means you should be mature enough to read research articles and study mathematics by yourself without having to take classes. You are also (I hope) familiar with the ethical aspects of research and the submission/review process. You should also know what constitutes a well-written paper and you are now familiar with LaTeX and mathematical software and how to present mathematics in form of posters and talks.

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    $\begingroup$ @ZhiqiangSun I second that. Just get your degree in whatever you are doing now, continue to work with something you are familiar with and slowly extend into other fields you are interested in (read, try to solve problems, talk to people, attend conferences, etc.). It'll take some time, of course, but you'll be there in a few years. As for in lack of really beautiful idea, but full of papers, that can be said more often than you think, and it is always not the field, but the general culture. In any field there are tons of beautiful open questions; nobody just knows how to approach them. $\endgroup$
    – fedja
    Commented May 28, 2018 at 16:29
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I understand the OP is frustrated with tight job market and those are valid concerns, but his/her description of the geometric analysis as a shallow subject is ridiculous. Parts of geometric analysis certainly attract top people and have seen remarkable recent progress. How a new Ph.D. could fail to notice these happenings is a mystery.

@ZhiqiangSun: Now that you have a Ph.D. you are free to tackle "nontrivial and natural problems". There are plenty of those in geometric analysis, and as to whether the subject involves any "beautiful ideas", it is my opinion that it surely does. If some papers seem shallow, ignore them. Based on your stated background you might enjoy working on degenerations of Kaehler metrics, which involves a healthy mix of geometric PDE (eg Kaehler-Ricci flow) and algebraic/complex geometry. You may wish to start by reading recent works of Simon Donaldson and Gang Tian.

It is common for mathematicians to change their research area several times during their career. Doing so after the PhD is quite possible, but it could be risky careerwise. It may be best to proceed slowly and expand to adjacent areas. Moving between two subfield of geometric analysis is not really a big change and many people do so.

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    $\begingroup$ Thanks a lot, that was my research topic during master study in China and I enjoyed it, that's why I have learned so much complex geometry. But the problem is, these kinds of topics concerning canonical metrics on Kähler manifolds are almost monopolized by the schools of Donaldson and Tian, it's not wise to do that. The mehods of elliptic equations are also widely used in symplectic topology and spectral geometry, so I am not judging that geometric analysis is a shallow area but some mixtures of geometry , topology and analysis should be exactly my taste. $\endgroup$
    – Zhiqiang Sun
    Commented May 28, 2018 at 22:05
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    $\begingroup$ I do not know about "monopolized" but even if it is true you could join one of the mentioned groups, or do other related things such as Kaehler-Ricci flow. Generally, a good strategy is to get involved in what you find exciting because this is where you have the best chance of contributing. It is hard to do math you don't enjoy. $\endgroup$
    – Igor Belegradek
    Commented May 28, 2018 at 22:39
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    $\begingroup$ @ZhiqiangSun: I wish to add that while it helps to have a guide (i.e., mentor, coauthor, or advisor) when entering a new area, there are plenty of examples of people who start doing good work outside established groups. Whether this can work in your situation remains to be seen, of course. $\endgroup$
    – Igor Belegradek
    Commented May 28, 2018 at 23:11
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    $\begingroup$ @ZhiqiangSun * these kinds of topics ... are almost monopolized by the schools of Donaldson and Tian, it's not wise to do that*. I wonder if I should say that Harmonic Analysis is monopolized by Stein's school, so it was very unwise for me to do my PhD in that area and to continue to play with my homemade contraptions there for the next 25 years.There is no ownership on mathematical problems, much less on fields of research, though there are people who will make a fuss if you "don't emphasize enough it was their question" (an actual quote from a letter of a (Chinese) mathematician to me). $\endgroup$
    – fedja
    Commented May 29, 2018 at 6:30
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    $\begingroup$ I think your metaphor is not so fair, you should compare harmonic analysis to geometric analysis, i think harmonic analysis is a bigger world than geometric analysis, and the study of canonical metrics is only a small subset from geometric analysis, so this specified and technical subarea is ful of top mathematicans from three groups(the schools of Donaldson,Tian, Chen), so as a young beginner, one should avoid direct competition with them, but try to know them and join them or just follow their papers and keep doing some small results $\endgroup$
    – Zhiqiang Sun
    Commented May 29, 2018 at 14:32

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