How many topics outside of his or her specialization is an average mathematician familiar with?
For example, does an average group theorist know enough of partial differential equations to pass a test in a graduate-level PDE course?
Also, what are the "must-know" topics for any aspiring mathematician? Why?
As a graduate student, should I focus more on breadth (choosing a wide range of classes that are relatively pair-wise unrelated, e.g., group theory and PDEs) or depth (e.g., measure theory and functional analysis)?
Your question is philosophical rather than mathematical.
A colleague of mine told me the following metaphor / illustration once when I was a bachelor student and he did his PhD. And since now some years have passed I can relate.
It is hard to write it. Think about drawing a huge circle in the air, zooming in, and then drawing a huge circle again.
This is all knowledge:
All knowledge contains a lot, and math is only a tiny part in it - marked with the cross:
[---------------------------------------x----] | Zooming in: [xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx]
Mathematical research is divided into many topics. Algebra, number theory, and many others, but also numerical mathematics. That is this tiny part here:
[xxxxxxxxxxxxxxxxxxxoxxxxxxxxxxxxxxxxxxxxxxxx] | Zooming in: [oooooooooooooooooooooooooooooooooooooooooooo]
Numerical Math is divided into several topics as well, like ODE numerics, optimisation etc. And one of them is FEM-Theory for PDEs.
And that is the part of knowledge, where I feel comfortable saying "I know a bit more than most other people in the world".
Now after some years, I would extend that illustration one more step: My knowledge in that part rather looks like
[ ρ ρρ ρ ρ ρ ρ ρ]
I still only know "a bit" about it, most of it I don't know, and most of what I had learned is already forgotten.
(Actually FEM-Theory is still a huge topic, that contains e.g. different kinds of PDEs [elliptic, parabolic, hyperbolic, other]. So you could do the "zooming" several times more.)
Another small wisdom is: Someone who finished school thinks he knows everything. Once he gained his masters degree, he knows that he knows nothing. And after the PhD he knows that everyone around him knows nothing as well.
Asking about your focus: IMO use the first few years to explore topics in math to find out what you like. Then go deeper - if you found what you like.
Are there "must know" topics? There are basics that you learn in the first few terms. Without them it is hard to "speak" and "do" math. You will learn the tools that you need to dig deeper. After that feel free to enjoy math :)
If your research focus is for example on PDE numerics (as mine is) but you also like pure math - go ahead and take a lecture. Will it help you? Maybe, maybe not. But for sure you had fun gaining knowledge, and that is what counts.
Don't think too much about what lectures to attend. Everything will turn out all right. I think most mathematicians will agree with that statement.
The answer to your question is easy:
No, an average mathematician specialized in, say, algebraic geometry could not pass without preparation a graduate level exam on partial differential equations.
Wait, it's worse than that: he couldn't even pass an undergraduate level exam on partial differential equations.
Wait, it's even worse: he couldn't pass an exam in algebraic geometry on a different specialized topic from his own. For example an elementary exam on the classification of singularities if he is specialised in Hilbert schemes.
Conversely I would be very surprised if a notorious analyst who recently got a Fields medal could solve the exercises in, say, Chapter 5 of Fulton's Algebraic Curves, the standard introduction to undergraduate algebraic geometry.
1) What I wrote is easy to confirm in private but impossible to prove in public:
I can't very well write that in a recent conversation XXX, a respected probabilist, abundantly proved that he had no idea what the fundamental group of the circle is.
2) If author YYY wrote an article on partial differential equations using techniques from amenable group, this doesn't imply that other specialists in his field know any group theory.
It doesn't even prove that YYY knew much about group theory: he may have realised that group theory was involved in his research and interviewed a group theorist who would have told him about amenable groups.
3) On the bright side some very exceptional mathematicians seem to know a lot about nearly every subject in mathematics: Atiyah, Deligne, Serre, Tao come to mind.
My sad conjecture is that their number is a function tending to zero as time passes.
And although I couldn't ace an analysis exam, I'm aware what this means for an $\mathbb N$-valued function...
My two cents: unless you have a magical brain, or are some sort of epoch-making genius, you're probably going to find that you can only hold only so much mathematics in your mind at any given time. So, for practical reasons—both with respect to writing a dissertation, and with respect to making a career for one's self—you should probably stick to one or two closely related areas, so that you might have sufficient expertise to make yourself useful to a research institution or to whatever it is that you wish to do with your future.
That being said, I've found that elbow grease and skill in mathematics are often woefully uncorrelated with one another. Rather, skill is often dependent more on how much mathematics one has seen. To that end, I would say, though you should definitely pick a subject area or two to call your own, you should strive keep an open mind and maintain an active interest in as wide a variety of mathematical disciplines as possible.
I often find that reading (even if only casually) about forms of mathematics unrelated to my research areas provides a wealth of new ideas and insights. The more patterns and phenomena you are acquainted with, the better the chance that you'll notice something of interest intruding upon your work, and that might give you some intuition you might not have otherwise had. At the very least, it will help you know what topics or sources (or collaborators...) to look up when you stumble across something outside of your area of greatest expertise.
Edit: One more thing. Linear algebra. To paraphrase Benedict Gross, there's no such thing as knowing too much Linear algebra. It's freakin' everywhere.
There is, of course, terrific ambiguity in the question. But, with any interpretation, the answer would be generally, "no, most practitioners of some part of X do not remember all of X... because they do not need to".
Thus, if only because most even-very-smart people's memories fade with time, there'll only be a slight residue of the standard-basic things in the mind of mathematicians who're working on one particular sort of thing for some years. Apart from teaching calculus, there's scant need to remember much else. Yes, from the viewpoint of scholarship, this is potentially distressing, but, in fact, in nearly all professional mathematics situations, there's scant motivation/reward for genuine scholarship. It somehow does not fit into salary-increase formulas, tenure, or much else. (Not that I myself care whether I try to understand things "for pay", or not...)
True, most graduate programs in the U.S. in mathematics do attempt to engender some minimal competency/appreciation for a big part of basic mathematics, but after "passing qualifiers" it seems that the vast majority of people do not find much interest in further pursuing broad scholarship, either in principle or for possible direct benefits.
Also, I take issue with the (what I think is) simplistic picture that "specialization" is like "zooming in with a microscope", and so on. Sure, this is a defensible world-view, and subject-wise world-view, and, sure, by one's actions one can make it be an accurate description... but I think it is not accurate of the reality. Specifically, I do not see the genuine ideas as being nearly so "localized" as a "physical zoom-microscope" would be relevant-to. That is, the idea that "math" can in any reasonable way be depicted as a physical thing, entailing all the local-ness that that implies, I think is wildly inaccurate. Again, yes, we can make it be accurate, if nothing else by ignorance or ignorant-fiat. But...
The question of how many mathematics topics an average mathematician knows, heavily depends on two definitions:
Of course it also depends on other definitions (like mathematician) but to a lesser extent.
Let us define levels of topics in the as follows, loosly based on wikipedia:
Now, based on personal experience and an image of the average mathematician, I can answer how much such a mathematician would know about this, for each level:
Note that if you move beyond level 4, you get so specific that you may not find complete graduate courses on such a topic. Hence my conclusion:
Based on personal experience, I expect an average mathematician to have decent knowledge of between 5% and 15% of topics on the graduate course level
I spent several years on a project to read the first 1-2 chapters of at least one math book on each shelf of the university library. It was an attempt to gain an unbiased survey of mathematics. It was good for me, but it was a luxury: the forced march through a PhD program and into academia offers little time for such behavior. Yet it's important: all of the very best, most famous mathematicians are clearly employing cross-disciplinary tools in their work. And, for me, personally, it was a kind of level-up: suddenly, everything is easier.
Specializing in one field is kind of like lifting weights with just your right arm, ignoring the core, back and legs: it leaves you surprisingly weak and incapable. When you have to master many different styles of abstraction, you get better at abstraction, in general, even in your chosen specialty. This, to me, was the big unexpected surprise.
For the more quantitative question asked here: could I "pass a test in graduate level XYZ course?" for a 1st-year, 1st semester course, maybe, probably. Sort-of. Exams tend to pose questions using phrasing and notation that are closely aligned with the class textbook, and this notation can vary strongly from one textbook to another. So for that, prep would be needed.The point is that such prep becomes easier.
There is a great dearth of textbooks at graduate-level in mathematics written with the aim of teaching the "true subject" as quickly as possible. "Smooth Manifolds by Sinha" is one such book. If many books of such type become available, then scholarship in mathematics wouldn't be a thing of laugh.
Certainly not. For example, the great mathematician Grothendieck was insufficiently well acquainted with arithmetic to recognize the integer $57$ as a non-prime. The many accounts of this story can be accessed by an internet search for the key terms; say, look for grothendieck prime 57.