Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize

My take is that life is ultimately about taking the chance to do interesting things when they crop up.

I disagree, but it might be because I interpret the spirit of the question differently. I would not call an activity involving some collaboration between people at different universities (including email), "solitary." It may not be very socially engaging, but it's categorically not a solitary endeavor. People are working together.

To speak to your examples directly: as you're aware, they're extreme outliers. They're not just extreme because of the level of achievement; they're also nearly heroic in scope for a single person to work on without collaboration. It just doesn't happen much at that level of sophistication and complexity.

More to the point, if you peel back those examples you see quite a lot of non-solitary work. For example, Perelman refused the Fields medal with a declaration that the substantial prior work was just as important for the solution as his own work. He's a pretty big champion of the important of the community aspect of mathematics. Likewise Wiles actually needed collaboration to fix a critical flaw in his original proof of Fermat's Last Theorem.

And that really captures the heart of what I'm getting at, which is what mathematics is "about" by practice and intention. Most mathematicians engage with the research community by attending conferences and collaborating with their peers, even if that collaboration is largely asynchronous. They keep their fingers on the pulse of the academic landscape by talking with other people. In the cases where someone produces noteworthy research on their own, it still exists in a social framework.

A mathematical proof is intrinsically a social endeavor. To prove a theorem is to convince someone else that it must follow from a series of axioms, definitions and other theorems. Sometimes you can do the work on your own, but it won't matter if it's disconnected from the research community, and the easiest way to stay connected to the research community is through collaboration.

If no one else in the world agrees with your proof, your proof is meaningless (as an example, see the controversy around Mochizuki's Inter-universal Teichmüller theory). This is what peer review is for. Moreover if you construct a new mathematical theory, it's only meaningful insofar as it builds upon existing work developed by other researchers. The choice of axioms and definitions must be interesting and significant within the context of an existing body of work.

This is one of the reasons you so rarely see nontrivial research produced by isolation. It's (approximately) a myth, and instances of it happening receive outsized attention precisely because they're extraordinarily rare and unconventional. In reality, it's usually not a good sign for a researcher to work on something in solitude. In most cases it coincides with the stagnation of a mathematician's career, not the glorious unveiling of a solution to a decades-old open problem or a grand new theory.

I can see why the idea that mathematics is a solitary endeavor (or suited to solitary work) is an attractive one, because the most visible parts of it (writing papers, solving problems) can be done alone. But it's usually very inadvisable to go it alone in learning or practicing mathematics.

I typed in a comment rejected as too long, clicked on the wrong icon, and lost it.

My view of the prospects is, in a word, poor.

Others have described the prospects for an academic math career. I would add, should include a lot of politics, friends, alliances, etc.

Outside of academics, long the main source of jobs in math was in US national security, with some of the major companies getting money for national security or just lots of smaller opportunities within 100 miles of the Washington Monument.

For Wall Street, I still have the letter back from Fisher Black at Goldman Sachs saying that he saw no opportunities for optimization on Wall Street. I sent lots of resume copies, went on a few interviews, had not yet heard of James Simons, and got nothing.

I've nearly never seen a business slot wanting a Ph.D. in math.

My experience in large organizations is that they want more people very much like the people they already have. What I saw doing math in business, e.g., FedEx, IBM, a staff slot at Georgetown U., is that for any good results there is a LOT of bitter resentment above, below, and from all sides. Math just isn't one of the job titles or descriptions.

My view is that mostly a job as an employee is too unstable to be financially responsible. For an exception, get a unionized slot in a monopoly, e.g., an electric utility. Generally it is necessary to start, own, and run a successful business.

My view is that computer science has run out of gas, e.g., can no longer extrapolate from what D. Knuth did and wrote about, and doesn't have good enough problems or tools for a good future. My view is that the best future of computer science is to be essentially a branch of applied math, complete with theorems and proofs, but not nearly enough of the computer science profs have sufficient math backgrounds.

My view is that practical computing is rapidly becoming routine, Web pages, database transactions, server farm and network system management.

My view is that automation is doing so well that we are in line for a huge amount of unemployment. E.g., I saw a lot of people on farms of 50 to 200 acres go very poor as only large operations with fewer people per acre and much more capital equipment did well. E.g., I put together a pizza recipe, a pizza for 1 in 10 minutes. The crust is the star of the show, and it has 9 cents of flour.

Some people are making money hiring illegal immigrants as common labor, having them sleep on the floor, a dozen or so to a room, and paying them in cash. After a few years of that, back home in Central America with the cash, they can live much better than before. US citizens can't play that game. That game is from currency exchange rates, and I'm still looking for the causes of such a valuable dollar.

My view is that computing and applied math are super tough ways to start a business but about the best there is. So, my startup, a Web site, has some applied math I derived, complete with theorems and proofs, based on some advanced pure math prerequisites; that math is the crucial enabling core of the startup, but users will not be aware of anything mathematical. The math is a secret sauce advantage, without the math, with results difficult to duplicate or equal. I don't know anyone else who has done such a thing. I don't know any one at all close who had done anything at all valuable with math outside of US national security.

In simple, first cut terms, I see no careers in math.

I've had my resume on Indeed for months with no responses at all. A few years ago I sent 1000+ resume copies with no meaningful responses. As far as I can tell, anyone over 40 years old with a good background in pure and applied math and computing is absolutely, positively, totally unemployable at anything, including minimum wage.

The industrialized countries are already so poor that the birth rates are so low they are rapidly going extinct.

My view is that, due to automation doing so well, so many people will be left with nothing to do that we are in for massive unemployment.