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I'm curious if it is common or quite rare for mathematicians to change their area of expertise, whether it be after their PHD or mid-career, how radical this change might be, whether it's often for reasons of evolving tastes or rather conforming to the type of research done at their institution, and any other thoughts anyone has on how mathematicians stake out the fields they will be producing research in, thanks =].

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The great mathematicians usually seem to be involved in many fields. I can't say much about great mathematicians today, but I looked up Terry Tao a few days ago and his research interests are extremely varied it seems. – Graphth Oct 23 '11 at 2:36
If I remember correctly, Grothendieck didn't get into algebraic geometry until fairly late (it wasn't his thesis). Another example: Quillen studied partial differential differential equations in graduate school and went on to become a leading homotopy theorist. – Akhil Mathew Oct 23 '11 at 4:32
There are a couple of relevant MO posts here and here, and I like Gower's concise answer in the second link (recommended in a comment in the first link). Adam's answer below is in the same vein. – yasmar Oct 23 '11 at 10:55
To supplement Akhil's comment: many people are aware that Grothendieck did not begin in algebraic geometry, many people seem unaware of the fact that he made fundamental contributions to the field he began in--- the theory of topological linear spaces, and specifically Banach spaces (Albrecht Pietsch's History of Banach spaces and linear operators is a good reference, although for Banach space enthusiasts only). There are people in this area who say things like "if I remember correctly, Grothendieck went on to do things besides Banach space theory," and aren't kidding. – leslie townes Oct 23 '11 at 22:26
Wow, excellent comments everyone, very informative, thanks – Zaubertrank Oct 31 '11 at 7:56

I think it's rare for people to make a radical shift all at once, largely because it takes a long time to master a field. However, it is much more common to start in field $X$, get interested in an adjacent field $X'$, then in an adjacent field $X''$, etc. Iterate this enough times, and you end up someplace quite different from where you started.

If you are interested in working in a field which is a little different from your usual one, the best way is to seek out a collaborator in that field. Probably my most productive collaboration started because I got interested in a problem in a field in which I was not an expert and approached someone who was.

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If you define "area of expertise" narrowly, then all mathematicians change areas of expertise. It's extremely rare for people to focus on only one kind of problem. If you look at the publications of anyone who has been publishing for decades, you will see considerable drift in what they work on.

If you define "area of expertise" broadly, I think you will find that most people do not change areas of expertise. Rather than "expertise", it might be more accurate to talk about interest. Most people's broad mathematical interests do not change that much. If someone doesn't care about geometry at age 30, you will probably not find them doing geometry at 50. If someone is steeped in geometry in their 20s, what they do in their 50s often has a geometrical "flavor" even if is not pure geometry. The methods and tools that people use change a lot, sometimes drastically. But the underlying interests tend not to. There are exceptions, of course, but that's why people notice them--- they're exceptional. Nobody is astonished to see an geometer, broadly defined, stay an geometer.

All of the abrupt shifters I personally know switched shortly after grad school. These shifts often occur because one's focus in grad school is determined more by the environment (e.g. an advisor, the faculty at at a single school) than the individual. It takes years to develop mathematical tastes. In the meantime you sort of adopt, or try out, the tastes of the people you are learning from. It's not unusual for a grad student to accumulate enough material for a dissertation in area X, while in realizing in the process that they are more interested in Y. So you see CVs where someone does a thesis and a handful of papers in X, and suddenly it's just Y after that.

You ask if researchers shift to "conform to the type of research done at their institution." I would say a definite no to this. At research universities you do have to conform (at least partially) to get hired in the first place. If you look at, for example, you will see that research universities usually specify (at least partially) a research area in job announcements. So: if you don't conform a little, you don't get hired--- or, more likely, you don't apply in the first place. If you are hired then you have interests in common with at least some faculty, and it's natural that you might work with them. But generally speaking, once you're hired, nobody really cares what your research direction is, provided that you have one (and are otherwise playing an active role in the department). If you only publish sole-authored papers, or only publish with people from universities besides the one that you work for, no reasonable person will hold that against you. There is no pressure to conform in the sense that I think you are asking about.

A final thing, on your choice of words in "how mathematicians stake out the fields they will be producing research in." Generally mathematicians do not choose a field first and then begin producing research (except perhaps at the very beginning, ie grad school). Research problems come first, and over time, depending on what the perspectives and tools one acquires, one finds oneself situated in a given field. For example, it's rare for someone who has not done X in the past to just wake up and say "OK, I'm going to do X now." What's more common is they will be working on a problem that isn't explicitly about X, and in trying to solve it, they eventually realize (or learn form someone else) that some X might help. So they go and study X a bit. Sometimes this leads to a solution to the original problem, and case closed, and they don't think about X ever again. And sometimes it's a dead end. But sometimes it leads to them reading the literature on X, thinking about X more often, maybe eventually contributing to the literature on X. This is the kind of gradual shift of interests Adam Smith talked about in his answer. It doesn't generally arise out of a conscious desire to switch fields or to "stake out" a field. It just happens.

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From my experience, a radical area change is rare, but this does happen. In grad school I knew a numerical analyst who started as a number theorist. Another professor I knew in grad school had wandered from field to field throughout his career. He started in topological groups and ended up in mathematical physics. One my colleagues started in differential equations and then mid-career decided he would switch to difference equations (this isn't as big a move as my previous two examples).

Why do people switch? Sometimes a field "dries up". Results get hard to come by and it's time to move on to greener pastures. Sometimes people get bored with their field and just want to try something new. I think this sort of thing is, for the most part, rare because it takes time to build up the background necessary to get started in a new area.

Edit: An interesting example of a major mathematician who "switched" fields is L. E. J. Brouwer. See:

Brouwer left topology and analysis (you may know "Brouwer's Fixed Point Theorem") because of a major philosophical shift. He didn't believe in that stuff anymore.

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It is not that uncommon for interests to drift, but the commonly cited examples (e.g., Grothendieck, Quillen, Serre, Mazur, Gelfand, Witten) of impact on relatively distant fields are extremely atypical. The fact that famous mathematician M was able to do top work in subject X, and then again in a faraway subject Y (and later in Z, and ...) says more about famous mathematicians than what is possible under more common conditions. Late-career changers of specialty such as Mumford, Smale, Freedman or Langlands demonstrate that there is more freedom with a Fields Medal or without teaching obligations. Somebody of more ordinary abilities and circumstances may be more limited.

In computer science Len Adleman spent a year in a biology lab and learned the techniques that he used to found DNA computing, but his case is similar to that of the Fields winners.

Paul Cohen's work in set theory might be an example but he switched to a field that had not reached maturity, still had its most basic questions left open and was not a densely populated research area at the time. Mumford and Langlands both ventured out of very technical, mature subjects into what was at the time relatively open mathematical territory.

Hales' work on computational approaches to the Kepler conjecture and proof formalization was a switch from his original interests in representation theory.

It's hard to cite lower-profile examples, because they are less visible and not heard of as often. Changes are most common in the first few years after the doctorate.

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For a period of time, Robert Langlands switched from number theory to studying mathematical physics in statistical mechanics.

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