Is Pure Maths Hard

[Arnold Gilgen]

Inotice that many pure math PhD students, doing things like algebraic geometry or representation theory, drop out and enter industry and become software developers. The ones who finish their pure math PhDs also drop out and do things such as blogging, tutoring, etc. These students are pretty "top of the heap" too, attending the top pure math PhD programs like at UC Berkeley and UCLA.

Then I notice that many applied math PhD students typically finish, and also get post-doc offers. These PhD students are typically doing numerical methods for the solutions of partial differential equations.

In numerical PDEs, it seems that just a little progress leads to a PhD degree, e.g. solving numerically a PDE in one space dimension. (I know firsthand of someone getting their applied math PhD from an Ivy League program for solving a PDE in 1D.)

I am almost in the exact situation you describe in your question. I finished my PhD in Algebra (coding theory, lot's of linear algebra and representation theory) and am now working as a software developer. As a fun fact, this wasn't always planned. When I started my PhD, I was still dreaming about getting a PostDoc and ending up as a tenured prof at a prestigious university.

Of my group of students at a pure math department, with a few world renowned old timers, I would not say algebra was considered the harder path. On the contrary geometry and analysis students, such as those studying general relativity or geometric analysis, such as stochastic differential equations over manifolds, were generally seen as having more brass. Ironically geometry and PDEs are closer to what's studied in "applied math" but this was a pure math department and numerical analysis was not studied.

But I think your point of view applies to the difference between pure math and applied math in general. Of some 15-20 of my grad school friends, most completed their PhDs and eventually worked for Google or in cyber-security or for IBM, such as myself, after perhaps teaching math for 3 or 4 years. I think 5 have settled down as math professors.

The simple answer to your question is just that pure math is very hard. I cannot speak to applied math training, I was not much exposed to it, but from what I understand proving an unknown statement or theorem is not necessarily required, whereas it generally is for a pure math thesis. It is occasionally said in academics that obtaining a pure math degree is the biggest challenge among PhDs. As the decades pass the bar is set higher and higher because math is a progress field and it is being completed. A similar sentiment is often expressed regarding physics and other hard sciences.

A theoretical math PhD is (obviously) extremely difficult and most of the topics in it tend not be directly beneficial towards future employment in anything but more work in abstract theoretical math. While many people think this is what they want to do at the beginning of the degree, the reality of the work is often far worse or different than what they are prepared for.

Now this is hardly an uncommon situation in many of the degrees out there but there's a catch for people in degrees like pure math that isn't true (as an example) for someone getting their PhD in history or literature: the skillset of a pure math PhD student is extremely lucrative. When you can either continue to bang your head against the wall on incredibly difficult and esoteric topics, or get an offer the next day for a 100k + paying job in tech or banking, I don't think it's much of a surprise a lot choose the latter. Often times they have recruiters seeking them out telling and them as much.

The difference for applied PhD's is that the end product is even more lucrative/valued and opens doors that someone with a masters degree isn't qualified for. If you complete a PhD in something like cryptography or applied AI/ML, you will have career opportunities in Google, Microsoft, blah blah blah, as they are specifically seeking out people who are experts in those topics. Whereas with pure math PhD's they will still seek you out, but only because people with those degrees also have a skillset they value, not because they care about your thesis on >.

First, it is not accurate to use "algebra" to refer to broad swaths of mathematics, any more than it is accurate to refer to "pure math", in fact. One immediate objection is already partly related to the issue of the question, namely, these are labels very often used by outsiders to refer to "not what they themselves do", and/or reflections of their own lack of awareness of mathematics outside their immediate competence. (It is not a moral failing to not be a universal scholar, but it starts to become some sort of problem when one loses sight of the skewing that one's own ignorance may engender.)

More significantly, I really don't think that there is any coherent "subject" that is "algebra", for example. And in any case number theory is not a subset of "algebra", nor is algebraic geometry, nor "representation theory" (despite the weird and technically inaccurate wikipedia entry for it), nor... logic? set theory?

Is there "pure math"? Well, depends what you mean, obviously. A very common usage of the phrase is by people who have a limited understanding of mathematics, and use the phrase to refer to things they don't understand, and don't see the point of. Also, some people who are "pure mathematicians" give themselves this label as some sort of idealistic thing. BUT in reality the best mathematics tends to be relevant to lots of things. Yes, sometimes people get involved in "deep background", and due to non-trivial difficulties do not manage, in the lifetimes, to return to resolve the original questions that sent them off on their inquiries.

My 40 years of observation of people working on PhD's in math at good places does not indicate that "pure math" people have a harder time either finishing or getting jobs... except for the subset of those people for whom the label does truly refer to something they've discovered they don't care about.

And, again, the potential relevance of any part of mathematics to things outside of mathematics proper is huge. Category theory to computer science!?! Stochastic differential equations to finance? Elliptic curves to cryptography!?! More elementarily: quaternions and 3D games (not to mention aerospace?!?)

To recap: the framing of the question is unsurprisingly naive, but, also, inaccurate in the current reality. Not so surprising that beginners are not aware. Also, skewed language not only reflects misunderstanding or ignorance of reality, but can limit one's vocabulary so as to create difficulties in having coherent discussions about reality.

(It may be worth noting that in the U.S. some of the "research experiences for undergrads" are (perhaps necessarily) so artificial as to be pretty silly. Maybe fun, fun to be with other enthusiastic kids, but often quite misleading about what genuine contemporary mathematics is, and what "research" in it would mean. But/and it often seems to happen that the people slide into an apparent enthusiasm for an alleged part of mathematics that they give some naive name... and, often, become disillusioned when the part of actual mathematics that has that name is not at all what they thought they'd bargained for.)

Yes, it's not easy to complete a PhD, and it's not easy to get a job, and in all cases one can easily feel that one has no hope to be any sort of heroic contributor. But may that last bit is asking too much for most of us, in any case.

Seriously, for completion of the degree, itself, it's not that any kind of math is any easier than any other, unless by mischance standards are lowered. And it is not easy to gauge the latter. That is, often, novelty is a valuable thing... even if it doesn't pay off... and serious novelty is truly harder to achieve in topics that have been around for 200 years.

Some less constructive "criticisms" of my earlier remarks seem to hinge on the allegedly obvious esotericn-ness and irrelevance of sheaves... or something... etc. Ok, I have to disagree with this, and claim that such math is old news, and has proven its value in understanding basic things. The possible fact that novices do not understand/appreciate its value... while a significant fact... only makes discussion more difficult.

My own general subject of interest, number theory (and its applications), nicely shows the vacuity of attempted partitioning of "math" into "analysis, algebra, ..." (though, yes, not so many years ago my own university's math dept had a supremely idiotic categorization as the basis for a hiring plan, etc.)

E.g., one of my recent PhD students (Kim K-L) solved a differential equation in automorphic forms that expresses four-loop (if I remember correctly) graviton interations in string theory. Simplistic classification is...?

What I do observe is true, again, is that novices' naive ideas about "what subjects are" leads them often to naive decisions, causing them to lose enthusiasm when the reality catches up... Perhaps a significant difference between "applied math" (maybe the questioner means "modeling"???) and (then???) "proving theorems" is that the bait-and-switch on theorem-proving might be perceived as far worse than in engineering-oriented "applied math".

The amazing thing about (good) math is that it is not only relevant but decisive in so many human endeavors. Whatever one's specialty, if one is _good_at_it_, one will have both specific and abstract mathematical chops, but/and demonstrated resilience to certain sorts of scientific/intellectual adversity. Crazily-enough, not all STEM disciplines teach that.

Because I'm OK at Applied Math but failed Pure Math two times, I asked my instructors and a brilliant friend studying the undergraduate Math Tripos at the University of Cambridge for advice. They all said that there's no secret or shortcut, that you must know your material, practice a lot, and keep trying. But I've been doing all this and still failed two times. So I Googled to see if even Cambridge math undergraduates fail PM. What should I learn from the quotes? Is Applied Math easier? Why or why not? I bolded the text.

3a8082e126