Anyhow, as an algebraic geometer, the place I learned about Uhlenbeck's work was in compactification of moduli of curves. Basically, say I want to look at all the polynomial curves that can live in a certain 'environment'. (One 'application' is string theory -- what particle interactions can occur given a certain set of energy constraints? The particle interactions are curves traced out over time by particles/Riemann surfaces traced out over time by the little loops that represent closed strings in string theory; the energy constraints constrain the shapes the particle interaction can take.) If you want to look at limits of families of these interactions, you're looking at some odd behavior. An example is the equation xy = t^2 -- this gives you nice hyperbolas for values of t not equal to zero, but as t -> 0, you get something singular, two crossed lines. Or x^2 - y^2 - z^2 = t^2: you have a nice smooth hyperboloid when t neq 0, and a double cone when t=0. However, these examples are just showing you the idea of how singularities appear in families of smooth polynomials. Uhlenbeck's older work really worked with the system of constraints that I mentioned -- figuring out what can appear under those constraints is a considerably more complicated problem!

This is a very imprecise discussion above and I'm totally blurring together Uhlenbeck's bubbling and compactness theorems from gauge theory -- but I never followed her work in a systematic way but instead was plunged into seeing its aftereffects from another related field that the work affected.

Last quote really resonates: “Along the way I have made great friends and worked with a number of creative and interesting people. I have been saved from boredom, dourness, and self-absorption. One cannot ask for more.”

Quite a wide career.

This sentence struck me as slightly odd if only because Erdõs was renowned for his social approach to mathematics[1], and my layperson's understanding of mathematics departments being somewhat collaborative within their sub-fields.

I know there are some notable examples of other solo breakthrough endeavours, such as Shinichi Mochizuki[2] or Yitang Zhang[3], and but those examples seemed to be exceptions to the rule.

Perhaps I'm channeling too much Terence Tao[4], or maybe the lone wolf researcher only applies to literal geniuses who work exclusively in academia.

Is mathematics research still a primarily solitary activity?

[1] https://en.wikipedia.org/wiki/Paul_Erd%C5%91s

[2] https://www.quantamagazine.org/titans-of-mathematics-clash-o...

[3] https://www.cnet.com/news/yitang-zhang-a-prime-number-proof-...

[4] https://terrytao.wordpress.com/career-advice/does-one-have-t...

Other solo works include Wiles's proof of Fermat (which admittedly later had to be fixed with the help of his student Richard Taylor), Perelman's proof of Poincare, and much of Scholze's foundational work on perfectoid spaces. Besides Poincare, three of the other millennium problems were posed by a single author (Riemann, Hodge, and Cook).

If you look at the recent arxiv listings for differential geometry say, there are many solo authored papers: https://arxiv.org/list/math.DG/recent

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.

> Dr. Erdos traveled from meeting to meeting, carrying a half-empty suitcase and staying with mathematicians wherever he went. His colleagues took care of him, lending him money, feeding him, buying him clothes and even doing his taxes. In return, he showered them with ideas and challenges -- with problems to be solved and brilliant ways of attacking them. (https://www.nytimes.com/1996/09/24/us/paul-erdos-83-a-wayfar...)

Even in the rarefied club of oddball math personalities, he stood out.

But he was an oddball too, for sure.

Indeed a very interesting mathematical family. It will be hard for future members not to revert to the mean :)

also i would add Grigori Perelman to your list of mathematicians working primarily by themselves.

No one ever explained to me how a career in academic math research would work: I never suspected that just digging into questions that were fun and curious and maybe someday somewhat useful, making discoveries, and writing papers could be a career that could support a family.

In particular, I still don't understand why the US has so much respect for research universities: My guess was that college was for teaching the students. But in a research university, nearly all the effort by the professors is their research, and the teaching is a sideline. E.g., might have a really good researcher in far out topics in analysis teaching junior level linear algebra. So, the money to the university, for people directly paying full tuition often a LOT of money, is only a little for that teaching but much more for the research. An analogy for restaurants would be that each hamburger had to come from some three star Michelin place and cost $200.

E.g., when I was a B-school prof, I was really discouraged to see that what the B-school was teaching had next to nothing to do with business. Instead, the school was interested in research, e.g., the question P vs NP. If medicine were taught that way, then no one would go to a hospital no matter how badly they hurt. B-school tried to be academic research, not professional training.

Besides, for me as a college student, I had to conclude that after junior year level material, the profs really needed to understand what they were teaching much better than they did. For that, they just needed to do a much better job learning and presenting what was already in the library and without more research. E.g., I never had a math or physics prof who had a good grasp of the theory and applications of Stokes theorem, classic potential theory, multipole coordinates, or both the theory AND applications of Fourier theory. None of my undergrad physics profs knew either general relativity or quantum mechanics at all well. I wanted to learn that stuff, not for research but for applications.

Math research solitary? It always has been for me. For learning the standard stuff, sure, can have people to talk to. But if they are in the same course and it is competitive, then maybe they don't want to talk?

The reason my math research was solitary is mostly just because the new work I was doing was so focused that no one else around knew much about it or had much interest in it. For a lot of math research, being focused is close to necessary and, then, the solitary aspect gets to be close to automatic.

Moreover it appears that there can be an effect from outside: Some people are more socially skilled, polished, talented, interested, motivated, etc. than others. In some fields, such social skills are important for success. But in math, if you can learn it, mostly on your own, then you can teach it -- good social skills can help but can get by with not much. This is even more the case for math research: Pick a problem, do some research, get some results, type in a paper, send it to an appropriate journal, maybe all with essentially no contact from anyone else.

So, since in math can get by with less than the best social skills, people without great social skills can find math, be successful, and stay, and, then, the result is that comparatively math is not a very social field.

But, a lot of people can learn to be social: Gee, a lot of grade school girls (sorry to bring gender into this) have just astounding social skills: E.g., it seems clear enough that long ago Hollywood discovered that among child actors the girls were much better -- they paid attention what was going on socially, reacted to others, had lots of facial expressions on tap, were good at glancing, averting, head tossing, pose striking, cases of body language, expressiveness in their voices, etc. Well, what a grade school girl commonly knows others can pick up, too! Okay performance in the lessons is not seriously difficult.

Reading some of what Uhlenbeck said about herself, she was doing well in the E. Fromm recommended social activity "Giving knowledge of oneself" and that can help a person be social because they are letting others know who they are. Or little so interests people as other people; if some other person communicates no better than a stone wall, then there's not much for others to be interested in; Fromm's giving knowledge can help with that.

For a nutshell description, my view of mathematicians is that they are less manipulative than most other people. Well, in some activities, can seem to get progress by manipulation, but in math, can't prove a theorem by a fake, manipulative proof!

> So, the money to the university, for people directly paying full tuition often a LOT of money, is only a little for that teaching but much more for the research.

It's the other way around. The faculty are paid to do research from grants they bring in, they are evaluated based on their research, and the university requires them to teach on the side.

> E.g., I never had a math or physics prof who had a good grasp of the theory and applications of Stokes theorem, classic potential theory, multipole coordinates, or both the theory AND applications of Fourier theory. None of my undergrad physics profs knew either general relativity or quantum mechanics at all well.

How odd. I went to a state university, and I'd say that pretty much all my physics professors had a good grasp of all of this besides general relativity, and several understood all of this at a really deep level and used it in applications.

Not true for most departments outside of engineering and some of the sciences, and not true for most math departments. Most faculty members in other departments get little to no grant funding. Math departments always have a small number of faculty members chasing grants, but the rest of them do not. Yet they still are evaluated based on research.

In my experience, the level of back stabbing in math departments is a lot lower precisely because of the lack of pressure to acquire grant money.

So, in effect, yes: If Congress wants to fund research, then they should fund research institutes.

I just don't like student tuition or state tax money funding STEM field research.

There are research institutions: JPL, Brookhaven, Oak Ridge, Los Alamos, SLAC, Fermi Lab, NIH in Bethesda, ....

I am interested in pursuing a PhD in Pure Mathematics. What are the career prospects for it? Based on what I read academic prospects don't seem too thrilling. It appears you may have to do multiple post-docs at low pay, and still not be able to be hired at a good school. What is the path to becoming a professor and making a livable salary?

If that is the case what are the industry prospects? Would whatever software company want to hire you, if your pure math specialization isn't directly applicable to what they're focused on? In turn, it seems better to learn how to code. But will your pure math specialization (as in not applicable to anything anyone is doing) + coding abilities set you apart?

I would rather study mathematics, pure mathematics and not be unemployable. But I don't really know what the job prospects are like. Could you speak to that, say, from a realistic perspective?

If your goal is to maximize lifetime earnings, a PhD is maybe not the right choice, but if you want to spend years studying something you love while financially breaking even then go for it. I definitely don't regret it, even though I would be worth a lot more if I had gone into tech straight out of college.

Btw, finance is also a common destination for pure math PhD's looking to leave academia.

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

It's a risky choice, unless you're very good. You should go in with your eyes open, knowing that you'll be lucky to get an academic job afterwards, especially if you're in a 'pure' field without many applications. For industrial jobs, you'll probably have to re-train through self-study. After a maths phd, doing so may come easily, but it will also take effort.

You'll spend three to seven years (dependent on your location and ability) working on an extremely specialised topic. There is no guarantee that you'll finish: circumstances beyond your control like your relationship with your advisor can derail your progress regardless of your ability. There is also a 'point of no return' after a year or two where it becomes very painful to cut your losses and walk away with nothing.

Doing a maths PhD could open some interesting doors. It will improve your employability, but it's an inefficient way to do so. The only good reason to do one is that you really like your research topic and want to spend years immersing yourself in it.

The key to my getting my Ph.D. was research, research, and research. The first I did was on a little question that came up in a course. In two weeks I got a surprisingly good answer that was clearly publishable. Later I did publish, no problem, in JOTA. That work gave me a halo in the department and made the rest easier.

For my dissertation, I had identified the problem and made good, intuitive progress on it before I entered grad school. In my first summer I turned the intuitive stuff into math, theorems and proofs, wrote an 80 page manuscript, and that was the research for my dissertation. So, I found my own problem, found my own solution, did the work with no direction, and submitted the final work. So, in no meaningful sense did I have any dissertation advisors. So, I didn't have to depend on an advisor.

Myself I'm not particularly worried about maximising income or whatever. I like what I'm studying and researching, and I'm paid enough to live a comfortable life at least for now. I could honestly do this for the rest of my life (which is what an academia career is, I suppose), but if I ever get fed up with it I also know I have loads of doors open everywhere, which is certainly a comforting thought.

Best of luck!

I'd say TCS and pure math are about equivalent in terms of the career opportunties they offer.

Something I forgot to mention: under no circumstances should you self-fund a PhD. Make sure you have a stipend to keep food on the table. Self-funding magnifies the risks I've described tenfold.

This is a realistic perspective of what grad school is like in the average-to-bad case. I could identify with parts of it:

http://nautil.us/issue/56/perspective/what-does-any-of-this-...

Second this. I was told before starting "If you have to pay to do a PhD you're doing it wrong." This may not apply to fields like Literature, but it most certainly applies to Math and CS.

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.

Country wealth and birth rates are not positively correlated, and due to immigration, countries can have positive population growth even as their birth rate is below replacement.

> 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.

To conclude from this, despite everything that has happened in the interim, that math is unemployable is an, uhm, mistake. As I recall, President Obama was lamenting the brain drain to Wall Street speaking precisely of the number of math PhDs lured there from more fruitful pursuits. Not to mention, as I recall there is an equation bearing Mr. Black's name, the accurate/efficient solution of which has had some importance in finance.

Yes, Black-Scholes etc. have been important, were for a while via "covered call options", or as in the E. Thorpe, Beat the Street -- essentially the same as what Black-Scholes did the math for with a Brownian motion assumption, etc.

I got an interview at Morgan-Stanley, gave a little lecture on my recent work in anomaly detection, indicated that I wanted to work in developing applied math, algorithms, and software for automated trading, but was ignored. They just wanted people to hack code for back office record keeping. The money making was elsewhere in the company and was based on more traditional ideas, e.g., investment banking.

For automated trading, listen to a James Simons remark: Someone walks in and is just convinced that they should short GM and go long the German Mark or some such. Well, that's more traditional stock picking or hedge fund portfolio management, and Simons made clear that his group, no way, would ever do such a thing. Instead, each buy/sell decision was from some data, ideas, analysis, and software already thoroughly tested; when that software said buy/sell, there wasn't much doubt. Well, that's what Simons did. Don't expect many other people on Wall Street ever to have done such a thing -- it's a lot easier to have a gut feel to short GM and go long the Mark.

Lawyers have learned: A working lawyer reports only to another lawyer. Well, math guys need such a rule, but outside of academics and some national security shops, there are nearly no math guys to report to. So, my desire to do applied math, software, etc. for automated trading was just ignored. Once I was invited to lunch in lower Manhattan for a guy claiming to be recruiting for Goldman-Sachs .... There just is no respect for a mathematician on Wall Street, no matter what Obama said.

Other than James Simons, not clear that math made much money for anyone on Wall Street.

This is very odd. How can a physics professor not know undergraduate physics material?

My guess is that there are a lot of experimental physicists who are not at all clear on Fourier series/transforms, the wave equation, good proofs of Stokes theorem, the actual definition of Hilbert space, Bell's inequality, or the differential geometry for general relativity. Still fewer know the exterior algebra of differential forms.

Where is his name spelled like this?