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.
Best Answer
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