Once I was told that in mathematics, after one gets a PhD, it is very hard to change one's field of specialization (within maths). Is this true? What are the reasons? How common are counterexamples?
In your answers please also specify what country are you talking about, if you believe it matters.
I think it's quite misleading to hold up examples of very talented mathematicians or mathematicians from the distant past. The answer depends heavily on the field you want to switch into, hold closely related it is to the field you wrote your dissertation in, and how talented and hard-working you are.
In the most classical, long-established subfields of mathematics, there is a large amount of background one needs to learn to be able to do significant, original research. To take the worst example I know, a graduate student who wants to work in the general area of the Langlands program needs about 18 months to 2 years of dedicated study after their second year graduate courses to get to the point where they can tackle some problem of interest to the research community, and this is with an advisor to guide their study and steer them away from pitfalls that would result from having an incomplete knowledge of the field. (In particular, this means the graduate student likely still has some blind spots in their knowledge that would greatly slow down their research if they didn't have an advisor.)
A graduate student doesn't only have the advantage of an advisor; he or she also has a good deal more time. Most postdocs and almost all professors have more teaching responsibilities than graduate students, and professors also have service responsibilities which increase as one gets older. Furthermore, one also has to do enough research to write somewhere between one and two reasonably significant papers a year (depending on subfield) in order to be competitive for jobs that allow time for research and eventually to earn tenure in such a job. If one wants to switch into a new field, then one presumably has to do this research in their old field while learning the new field.
If someone in representation theory or algebraic geometry or other parts of number theory wants to switch to working on Langlands, then he or she needs to learn the material in one second year graduate course (because they already studied the other two or three that a complete beginner needs) and another 18 months of specialized study. It's true that some patterns of thought will be familiar, even if the specific ideas are different, so one is going to learn somewhat faster when one learns their second field, but moving into a new field still requires at least a year of dedicated study unless someone is an unusually fast learner or extraordinarily hard working. Most people don't have the time and energy to fit an extra year of work above their other duties within any reasonable timeframe. Fifty years ago, one could have given up a couple years of paper-writing to accomplish the switch, but someone trying that today would never get another job that allows significant time for research in today's far more competitive job market.
Most areas of research don't require as much background as Langlands, but unless one is moving into an essentially brand new field requiring minimal background, switching fields requires a substantial amount of time that one simply rarely has after obtaining a PhD.
Lack of an advisor can be an issue, but it is less likely to be one than lack of time. Many fields do have a significant amount of "folklore" that is well-known to experts but not clearly stated in print anywhere. These are usually ideas that are too advanced and specialized to appear in a graduate textbook, but at the same time too easy to be the subject of a research paper. At some point this folklore is used to establish more significant results in a paper, but since "everyone" knows it, it might not be explained very clearly or be easily found by someone who needs it for some other purpose. However, most fields have experts who are quite friendly and willing to explain the necessary folklore to new entrants to the field, and, at worst, one does things in a clumsy way in his or her first papers in a new field and has some folklore pointed out by a referee. If one appears talented and capable, then it is not so hard to get some help. Researchers in countries not connected to the international mathematical community (such as in Africa) generally face much more significant problems with having access to experts than people trying to switch fields.
Is it possible? Yes, but it's very hard.
Plenty of people change fields, and it is probably easier in math than in other disciplines, since we don't need any equipment. And for most people, the transition is very natural rather than forced.
Assuming that you were interested in the topic of your PhD thesis, when your interest changes gradually, so will the problems that you're interested in. But since many problems in modern mathematics actually require some training to even understand, the problems you're interested in are probably going to be in topics adjacent to your PhD thesis. And in this way, you can shift to various problems.
If you think that you're interested in a topic that is completely unrelated to your PhD thesis, be wary, because it should be really hard to judge whether a problem is interesting or not. It's very unusual to be interested in a problem without knowing any of the current techniques of the field, and you should remember that many of the cranks also approach problems this way. This being said, if you're able to find a mentor in this new field who is willing to work with you on a problem, then you might be able to overcome this difficulty.
There are many doctorates who have done their PhD in one specialization and moved their work to another. Receiving a PhD in a domain of specialization doesn't mean that they are confined to it. Mathematics is no different in this aspect.
Some of the Math professors I know did change fields and worked quite successfully. Mathematicians who work on widely applicable fields, such as statistics, computational mathematical modelling, linear and non-linear optimization methods, are considered to be versatile. They change fields and can even move across to nearly any other fields in STEM to apply their expertise.
If you are specific about counterexamples, here are a few:
So, how realistic is it to switch fields in Mathematics?
It is very realistic. Most of the famous mathematicians (as in the list above) have done it and and there are evidence stating so many typical mathematicians current generation continue to do so successfully.
[EDIT] Just saw the comment on changing within maths. So yes, new discoveries are often made by linking apparently foreing, or weakly connected, areas. Think for instance about the proof of the Fermat conjecture by Andrew Wiles. I'l add other examples later.
The specialization you invest on in a PhD IS NOT you. A (well-chosen) PhD is a unique chance in life to focus on a topic for 3+ years, and what you learn can be deployed in many other domains.
On a general side, mathematics has become a tool for selection. And doing great in maths does not mean you cannot do something else. Mathematics stories abound about great mathematicians who hesitated between two paths before a PhD, for instance with Greek or Latin, when those where selection tools (a century ago). Mathematics is a (if not THE) common language of science.
On the local side, it also depends which the side of mathematics you are on, and your power of conviction with respect to the new field you want to play on. Optimization, functional analysis, combinatorics, logics, arithmetics, even topology can lead to a number of topics: data analysis, bioinformatics, economics.
I am a not a fully trained mathematician with electrical engineering background. My personal path was on pure arithmetics (Diophantine equations) and applied data compression. I switched to harmonic analysis (wavelets). I now apply what I learned (I mean the methodology) in data processing: chemistry, biotechnologies, engine management, even real-time simulation. Most of what I use comes from my shallow math background
Mathematical reasoning is a great tool.
I usually tell people something which is almost opposite (usually in the context of having an advisor who is inexpert in what the student wants to research): you can change your research area after you get your PhD, and this is fairly common.
Here are some points to keep in mind:
it's very common to work on other types of problems after you finish your PhD
often problems you work on (and this is true of most good problems) are related to other areas, so you may be naturally led to other areas from your present work. also most areas are closely connected to many other areas.
some fields are easier to get into than others (both in an absolute sense, and also depending where you are now)
you haven't specified what you consider to be "fields" (they could be defined quite broadly or narrowly)
many problems lie at the intersection of many fields, so possibly you could do something very similar and be considered in a different field. a lot of people in area say they do representation theory, some say they do number theory, and some aren't sure.
at certain stages of your career, you have less time (say in a given semester) to spend learning new mathematics. but also at certain stages of your career you have less pressure to produce theorems.
moving to different places/talking to different people can lead naturally to working on different problems
it's possible to dabble in different fields, or work in multiple fields at once
In my experience, most mathematicians enjoy learning new things and will end up working on different things throughout their career. Certainly the majority do not make a huge field shift (e.g., PDEs to geometric representation theory), but I think many (possibly a majority) of us shift quite a bit from our thesis to the point where it feels like we're doing a different kind of mathematics. Often this is gradual, but could be sudden, and we may float back and forth between different areas. I think most of the faculty at my department (at least among those whose research I'm fairly familiar with) have at least done some work out of their "comfort zones" (e.g., algebraic number theory to analytic number theory, Riemannian geometry to combinatorial probability, etc.)
In summary, I think it's not unrealistic to change fields, particularly if that's what you want to do. Personally, I have been a number theorist since grad school, but in the 12 years since my PhD have enjoyed (and continue to) working on other things in combinatorics, harmonic analysis, representation theory, hyperbolic geometry, finite group theory, etc as well--I just choose to stay a number theorist because I want to. Just be aware that (1) some areas may be hard for you to "break into," particularly without a mentor/collaborator, and (2) it may take some time to learn the appropriate things to transition.
An opening note, I am an undergraduate student, working towards becoming a research mathematician, so perhaps I am not qualified and experience enough to answer this question, but here goes
I do believe most research mathematicians have multiple areas interests among a few particular fields. Furthermore there often aren't well-defined boundaries between different fields of mathematics, if there were, well mathematics wouldn't be where it is today.
Obtaining a PhD is more an entrance to the world of mathematical research I would say than defining the type of mathematician you want to be. Though more often than not the topics PhD students would pick for their thesis would be closely related to the areas of mathematics they find interesting.
It doesn't mean that because you obtained a PhD in say Algebraic Topology, you are destined to be a Topologist for the rest of your life.
An example I can give, is that to study Algebraic Geometry (let alone write a thesis on it), one needs to have a good understanding of Analysis, Abstract Algebra, Commutative Algebra, Algebraic Topology and a bit of Differential Geometry. So I don't see why a mathematician who obtained a PhD in Algebraic Geometry couldn't also do research in Algebraic Topology, Differential Geometry or even in Algebraic Number Theory.
Different fields of mathematics are not as separated as you may think, apart from research in some of the foundational stuff I would say (Logic, Category Theory etc.)
