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I'm looking for examples of four kinds of things:

  1. Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying examinations) but that most mathematicians aren't really expected to know/remember: Some things that come to my mind are Sylow's theorems and their applications (for mathematicians outside of group theory and geometric group theory) and point set topology (except perhaps logicians and some algebraic geometers). If there are other examples of this kind of stuff, why is it taught in undergraduate courses? I can think of three explanations: (a) it is useful to learn (either the content or the techniques) at least once, even if people forget; (b) it is so important for people who go into that area of mathematics that it's worth subjecting everyone else to it; (c) inertia.
  2. Material that is not taught or covered in undergraduate courses and/or in most first-year graduate work, but that professional mathematicians across multiple specialties are supposed to be comfortable with. Things that might fit the bill (but I'm not sure) are various techniques in combinatorics and elementary number theory, and ideas from category theory. But I'm not really sure.
  3. On a related note to (1), mathematical skills that undergraduates get good at while studying the courses but that most of them forget even if they become mathematicians. Examples include all the tricks and techniques for integration, Sylow's theorem tricks.
  4. In contrast to (3), skills that people get better at in general as they do more and more mathematics. This probably includes things like a better understanding of quotients, asymptotic behavior, universal properties, product spaces, multiple layers of abstraction (like a norm on a space of operators on a space of linear functionals on a space of functions on a topological space, or one of those typical things in category theory).

All the things above are guesses and I'm curious to hear what items others have in mind and whether people think there exists any notable divide or difference of the kind I've suggested above between what undergraduates learn/get good at and what mathematicians are expected to be good at.

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I'm not sure I buy your examples in (1). Both the Sylow theorems and large chunks of what I learned in point-set topology strike me as genuinely important and widely applicable throughout mathematics. After all, the Sylow theorems are good for more than solving problems like "classify groups of order foo". And the contents of a good first course in topology include, along with Sorgenfrey planes and long lines and Hilbert cubes etc, plenty of stuff people other than "logicians and some algebraic geometers" use: e.g. connectedness, compactness, product and qt topologies,... – Sam Lichtenstein May 18 '10 at 2:16
I'm a HUGE fan of point set topology and you'd get a war from me if any university I was on staff at or getting my PHD at decided it was meaningless and dropped it from the cirricula altogether. I DO have some reservations about the archaic way it's TAUGHT,though. But that's another post......... – The Mathemagician May 18 '10 at 2:55
I'm not so sure that "This area of math isn't useful at all for the area I'm going to do research in" is a good philosophy, even if you're older and have already made a good decision about what you want to focus in. The reason is: many great mathematical achievements have come about by combining ideas from diverse areas of math, and the more you know about more areas of math, the more likely you are to use intuition or even a method from another area of math in your own research, or even discover a deep connection between two areas of math. But maybe this isn't the best philosophy. Thoughts? – David Corwin Jul 6 '10 at 19:24
Looking over all the examples given, I'm not convinced that we really have any good examples for category (1): stuff that is generally covered in undergrad math courses but isn't all that important, and I think people have come up with quite a bit of stuff in category (2) -- tensor products, for example. What this probably means is that there's not enough time in a typical undergraduate math curriculum to cover everything that's of wide importance. – Peter Shor Jul 6 '10 at 21:01
Dear Davidac897, the Langlands program was created by Langlands. He had much more than a working knowledge of both number theory and representation theory. More precisely, he came at things from representation theory, in which he developed the general theory of Eisenstein series (a problem in representation theory and analysis, which also has arithmetic content), but he was aware of the fundamental open problem in algebraic number theory (the creation of a non-abelian class field theory). He then found a hint, in his study of Eisenstein series, that there could be an application of ... – Emerton Jul 7 '10 at 22:59

12 Answers 12

One category of mathematical result that belongs to 1 is statements that you need to know are true and that have complicated proofs. Obviously some such proofs are worth knowing because they will help you find other, similar proofs. But not all of them fall into that category. For example, almost all mathematicians can get by just knowing that it is possible to construct a complete ordered field. And perhaps a more important example: many mathematicians use Lebesgue measure, but all most mathematicians need to know is a few basic facts about it, and not the full details of the construction and proof that it works. Another result I remember my undergraduate lecturer more or less explicitly apologizing for was the simplicial approximation theorem, which I remember disliking intensely.

Why do we teach results like this? One reason is that when we teach we are not just equipping people with the tools they need for research, but also demonstrating that we can build up the edifice of mathematics from just a few basic axioms. One can argue about whether we really do this, but I think we tend to do enough to convince any reasonable person that it can in principle be done. If we were to start leaving lots of gaps (there's this thing called Lebesgue measure ... it has the following properties ... it can be shown that these properties are consistent but the proof is tedious and I'll omit it) then this valuable aspect of a mathematics course would be in danger of being lost.

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This is a very thoughtful and honest response. In particular, it is a strange feature of "intermediate level mathematics" (roughly advanced undergrad through the first year or two of grad school) that almost every result that we state in a course also gets proved. I think I agree with this pedagogy but not necessarily for the standard reasons provided for it: i.e., that otherwise we are not being "rigorous". (In truth rigor is something to be looked for in the community of all mathematicians, not in a particular undergrad classroom.)... – Pete L. Clark Jan 2 '11 at 12:40
...Maybe a more honest explanation is that we need to instill in students a suitable respect for mathematical proof. Anyway, this creates situations that gowers mentions above, where in undergrad courses lecturers feel the need to give complete -- albeit pedestrian and ultimately unenlightening -- proofs of important theorems to "stay honest". – Pete L. Clark Jan 2 '11 at 12:47
The construction of Lebesgue measure is a great example. To prove this, and only this, in a course, is little help to anyone. Better would be to discuss general criteria for extending and constructing measures (Caratheodory and all that). Better still would be to construct Haar measure: there's just more content there. Of course, the class in which one sees the construction of Haar measure should probably not be the course in which one is first introduced to Lebesgue measure. – Pete L. Clark Jan 2 '11 at 12:47
This is a good point. I would also add that the emphasis on meticulously proving every fact appearing on the blackboard has a harmful side effect, namely, too little emphasis is put on motivation, intuitive grasp of the methods and the "big picture" of the subject. Most students will surely forget most of the details of the complicated proofs, however, they are much more likely to remember the ideas and the reason 'why' particular issues are important or worth studying. – Michal Kotowski Jan 2 '11 at 14:18
... I see this often during seminars, where unfortunately many students are more or less copying their notes or the content of a book onto the blackboard, careful to include all the formal steps of the proof, but offering little understanding of what is really going on. They are of course emulating the worse aspects of their undergraduate lectures. – Michal Kotowski Jan 2 '11 at 14:19

Seems to me the goal of undergraduate math is to provide experience in concept formation (finding meaning in abstract definitions), a wide variety of examples of structures, relationships, and approaches to proof, and generally develop mathematical thinking. The actual content is less important than suitability of the topic for elementary development.

Remember that traditional undergraduate math is not a professional degree in that it is not intended to provide information needed for specific jobs. Not even graduate study. The expectation is that beginning graduate students should be able to pick basic stuff up quickly, not that they already know it. Non-academic employers of people with math degrees have the same expectation.

Questions about whether specific content is really needed downstream miss the point.

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I agree with this up to a point. It is certainly true that graduate study in mathematics does not build directly upon a fixed syllabus of undergraduate material. I would say that an undergrauate who has had substantial exposure to concepts of analysis and algebra, at least a little bit of topology, and has begun to learn how to think abstractly and reason clearly is well prepared for a PhD program in mathematics.... – Pete L. Clark May 25 '10 at 21:39
However, as someone who has worked in graduate admissions at my good-but-not-fantastic department, I can tell you that we see many apparently reasonably talented applicants with much less preparation than this. Students who have not had a "real" undergraduate analysis class are going to pick up things like measure theory much less quickly, on average, than those who have. So I would hate to see this sentiment used as an excuse to further water down many already weak undergraduate curricula. – Pete L. Clark May 25 '10 at 21:39
@Pete L.Clark: What do you mean when you say: Substantial exposure to concepts of analysis and algebra, at least a little bit of topology. How much do you think an undergraduate should know in terms of syllabus. – S.C. May 9 '11 at 8:16

I think that one can roughly divide course material into four categories:

A) Cool and useful

B) Cool, but not clear if one will "need" it later

C) Painful to learn, but important later

D) Painful to learn and rarely if ever used, i.e. a waste of time.

This classification varies from one person to another. For example, for me differential forms are in class A and the Sylow theorems are in class D, but for someone more algebraically inclined the reverse might be true. (Incidentally I didn't learn differential forms in a course (although I could have) but rather studied them on my own, and in general these seem to be tragically underemphasized in some undergraduate curricula.)

In my own undergraduate education I think I had more of class B than anything else, but even if I didn't need the specific content later, I enjoyed myself and gained "mathematical maturity" and exposure to different kinds of mathematics.

I think that studying material in class C is not so good pedagogically. For me, if I try to learn something before I need it, then by the time I need it I have forgotten it all. In general, I think that courses sometimes have too much of a bottom-up approach, i.e. building up foundations before one knows what they are for. For example, going a bit beyond the undergraduate level, I have seen graduate students who are familiar with intricate technicalities in the foundations of algebraic geometry, but who struggle to come up with examples and don't know the most basic facts about algebraic curves.

In conclusion, I think that in teaching and learning one should not focus so much on what is "needed" but rather on what is "interesting".

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As I try to explain in this post ( I think it's important to think about your comparative advantage relative to your future self. As you go through grad school it becomes much easier to learn mathematics, hence things in C you're better off learning later when its easier and thus less painful! So for now you might as well concentrate on things that are more interesting or taught by a great teacher. – Noah Snyder Jul 8 '10 at 15:49
Schemes certainly fall in class C... – expmat Jan 11 '12 at 17:12

Re 2: I think there is a certain amount of "inbetweener" material in algebra that is too advanced for, say, a college linear algebra class, but is already taken for granted in graduate algebra. After getting my undergraduate degree at Harvard, I'm pretty sure I didn't know what a tensor product was and I definitely didn't know what an exact sequence was. When I started grad school at the same place a year later, I felt a bit of a hillbilly for not knowing these definitions.

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That may really depend on the country. I remember seeing tensor product in at least four different courses on my third year as an undergraduate. – Andrea Ferretti May 18 '10 at 1:56
I didn't think to include tensor products because one of my undergraduate professors (who may or may not have also commented on this very answer) made absolutely certain that I learned about them. But I think it is a reasonably common experience for students not to learn about them. During my first year in graduate school we spent half a semester on tensor algebras in the algebra sequence, but this past year's first year class did not once hear the word "tensor". So apparently it can even vary within an institution from year to year. – Paul Siegel May 18 '10 at 3:42
D&F go into tensors over noncomm. rings without taking some time to appreciate the comm. situation, which is much more developed at a basic level. Rotman writes in his grad algebra book that he introduces tensor prods over noncomm. rings right away because if he did the comm. case and then noncomm. later he'd have to repeat himself, which wouldn't be good. Eh? Once you get tensor products for modules over comm. rings you can then appreciate the twists that happen over noncomm. rings, just like learning comm. ring theory a bit before bothering with the noncomm. case. – KConrad May 18 '10 at 21:13
In my junior year in college, I took Cliff Taubes's graduate course. He tried his best to be intelligible to us undergraduates, but at some point he couldn't restrain himself and wrote 0 --> V --> W --> V/W --> 0 on the blackboard. I'll never forget the look of panic and bewilderment on every undergraduate face as soon as he wrote those zeroes... – Michael Thaddeus Jun 8 '10 at 14:03
(Noah is referring to a comment that I have just deleted.) – S. Carnahan Aug 11 '10 at 14:12

One theme I found repeatedly occurring in education is that teachers delight themselves in not telling anyone directly about the fundamentally important things. For example,

  • The idea of unification is not taught explicitly - Instead, students are given a table of Laplace transforms and enough exercises that the process is anonymously substituted into their heads.

  • Taking the quotients of a set by an equivalence relation - Having never seen how to construct $\mathbb{Z}$ from $\mathbb{N}$ or $\mathbb{Q}$ from $\mathbb{Z}$, the ritual of constructing $\mathbb{R}$ (if it is mentioned at all) appears completely alien and is forgotten immediately. The same student will likely forget (if they are able to understand in the first place) the first isomorphism theorem.

  • Logical language and the deduction rules for proving statements are not mentioned - One is expected to aquire this language as you do with Listen and Repeat cassette tapes. If it is not clear exactly what a proof is, creating one is a great deal more intimidating and difficult! Teaching basic methods of computing science like structural induction on data types should remedy this.

  • Differential forms are mentioned explicitly but we treat the fickle beasts with great caution - If these unreal quantities are allowed to freely mix with numbers and variables, why must we be constantly told that dividing them is "purely formal"? Despite that various foundations of analysis have been made rigorous beginners to this subject do not benefit. Instead they are troubled by it and develop an allergic reaction to $\epsilon$.

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What do you mean, in the specific case of Laplace transforms, by saying that "The idea of unification is not taught explicitly"? Do you mean that students are given only tables, and not the definition? This is not true at least for any book that I have used (e.g., Boyce and DiPrima). – L Spice Dec 15 '13 at 13:37

I think that there are several reasons:

  1. Mathematics require development. Just like the switch from two oranges + three oranges, to a more abstract idea of 2+3 requires time, so is the ability to think clearly about abstract spaces, infinities, functions and relations. It takes time and experience.

  2. Mathematics is like a muscle. You need to work out a lot to have a big muscular brain that can bench-press theorems, later on to be able and maybe invent some of your own (i.e. research)

  3. When most people come to the university they don't know what math is like really. Before my freshman year I wanted to study pretty much every course that would be given. After a while I realized that analysis is not my cup of tea, while set theory is. And although algebra seems nice at first, eventually it goes beyond the scope of my liking. Today as I'm taking my first steps into learning on my own and starting my M.Sc. I can tell that I'm going to focus on set theory, I didn't know that when I first started my journey in math. How did I learn that? By tasting each topic and choosing my favourite flavour.

  4. The previous point brings me to this one: my dad was in the academy for many years (though in history) and he told me before I started my degree that the first degree is horizontal. You learn a little bit about most things, on the second degree you start focusing on some topic, and in your Ph.D. you study the tiny iota of something. But you need a wide base for that.

  5. One lecture in number theory we had some other professor to fill in for the regular teacher of the course. He was talking about how there are infinitely many primes of some form, and he said that there is an iff theorem about it, but we'll only prove the simpler direction. At one point during the proof one of the students (the majority of the class was computer science majors, not math majors like myself) raised his hand and asked if we can't use the other direction of the theorem. The professor answered that "This is mathematics. You don't use things that you haven't proved." and in a way he's right. Especially when you're only taking your first steps. It's important not to skip too far.

  6. Last (but not least) I'd like to partially repeat several of the other points that I made. When I was a freshman, I came up with some idea and shown it to my linear algebra teacher. He was very impressed because I came up with it completely by myself (that was $||\cdot ||_\sup$ norm on $\mathbb{R}^n$ and definitions for metric, etc). He directed me to several topics that I might read about and learn more: topology, functional analysis, and a few more. But he strongly repeated that you make stepping stones in your way to the knowledge, and when you make them too far apart you'll eventually fall down. And he was right - I did fall down several times because I did that.

So we're being taught all that basic mathematics because we ought to know at least some of it, meddling with it helps to develop a sense of intuition about math and of course the abstract thinking process. Moreover, it's good to let the children play with theorems that were ground to dust and cleaned out of possibly mistakes, rather than new cutting edge concepts that might have problems and unclean environments that need extra-care.

And of course, even now when I finish my first degree I can't remember over half of the things I studied a lot for. But I remember the intuition and I have the tools to rediscover the knowledge when I need it.

Hopefully - never :)

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One reason for including certain material in the education of "all mathematicians" ... my example is complex analysis. You may end up teaching at a small college, where you teach all undergraduate math courses. Including complex analysis, since engineering and physics want their students learn it. And it is desirable that the instructor know more than just what is in the textbook.

Many times I have heard complaints of this kind... "My research is in graph theory so I will never need to know complex analysis, why do I have to waste time learning it?"

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Wow, people actually make that complaint? But it's one of the real gems of undergraduate math! And has such down-to-earth applications. Incredible... – BCnrd May 18 '10 at 13:15
@ BC That complaint says a LOT about the sociology of the academic culture-namely,that teaching isn't important to most up-and-coming mathematicains. And sadly,that's because most of them learn early that if you show interest in teaching to most hard-core,prominent mathematicans whom you're trying desperately to impress when you're working towards your PHD or working your first postdoc appointment, they'll snicker and consider you last on any list of "serious" newcomers.It's quite sad,actually. – The Mathemagician May 18 '10 at 18:48
@BC One thing I think many of us can agree on: With THAT attitude,this student won't be a very broad or creative thinker in mathematics,so he better be DAMN good at either pure graph theory or devising applications in his vaunted research. – The Mathemagician May 19 '10 at 0:47
@Andrew: I give undergrads benefit of the doubt on taking a more mature viewpoint later. I am surprised by your dim views about the role of teaching in hiring. I've been involved with hiring for tenure track, postdocs, etc. at some top departments, and never heard anything but positive things for people who successfully combine good teaching with good research. That is, although quality research is the first criterion, I've only ever heard positive reactions when such people do excellent teaching (no snickering, etc.), and negative reactions when the teaching stinks. – BCnrd May 20 '10 at 3:12
Complex analysis is so important for so many different areas of science that I would rather see it taught than, say, exact sequences, even if exact sequences are in general more important for future research mathematicians. Should the undergraduate curriculum really focus solely on creating future research mathematicians? – Peter Shor Jul 8 '10 at 14:25

If you wish to become a solid mathematician, there are certain topics with which you will want to have familiarity, even if you do not intend to delve deeply into them. For instance, most research mathematicians will have had a decent training in point-set topology and will have learnt at least a few of the major theorems and techniques in the area (Urysohn's lemma, Tychonoff's theorem, Urysohn's metrization theorem, partitions of unity etc.).

Ultimately, the extent to which you remember these results and techniques does not determine your quality as a research mathematician. However, open sets and their theory are ubiquituous in nearly every branch of pure mathematics, and having a feel for certain topological concepts is certainly desirable. (Perhaps not essential, however, depending on which branch of mathematics you pursue.)

Again, this is not to say that someone can be dismissed for not knowing point-set topology: there are plenty of ways one can do meaningful research without having a training in point-set topology on the magnitude of Munkres' Topology: A First Course or Kelley's General Topology, for example, and there are at least a few professional mathematicians who demonstrate this.

As for Sylow theory, the topic, of course, falls into the area of finite group theory. (I hasten to add, however, that Sylow theory does have applications to the theory of locally finite (but possibly infinite) groups.) While finite group theory is an exciting subject full of rich structure and powerful theorems, I suspect that there are not too many branches of pure mathematics where the methods of this theory are instrumental to doing meaningful research. One notable exception (to some extent) would be algebraic number theory. For instance, the principal ideal theorem of class field theory can be proven using the techniques of transfer (in group theory). Also, it would be fair to add that most algebraists have a solid training in finite group theory, even if their research interests lie in other aspects of algebra.

Succinctly, I think that it is fair to say that being comfortable with the various techniques of group theory and topology, whether or not you pursue either of these subjects, can be helpful in many areas of mathematics. The subject matter in its exact form may not repeat itself in other areas, but the techniques, ideas and intuitions may do so.

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I think the gap(s) outlined in the original post of this thread are inevitable since there is a limit to how many courses an undergraduate student can cram into his available time before graduating.

Pertinent material for this discussion is the book "All the Mathematics You Missed But Need to Know for Graduate School" by Thomas Garrity. The goal of this book is ambitious, and so naturally it has its shortcomings. It has a number of errors, and one might dispute Garrity's particular selection of topics. But overall I like the book, and would recommend it to any undergraduate who is considering a professional career as a mathematician.

The main body of the book is a series of short chapters, each a very brief introduction to a particular area of mathematics. This can be a useful read even for the mature mathematician who suspects that their mathematical experience may be a bit parochial.

But one of the most useful parts of the book for me, which I read while working on my master's degree, and which I wish I had been exposed to as an undergraduate, is the introductory material that presents some of the broad patterns that are common across all branches of mathematics, and which form an outline for each of the following chapters. For example, he notes that every branch of mathematics is the study of some particular set of mathematical objects. This study includes questions such as how to tell when two objects of some class are essentially the same (isomorphic); when one is a sub-object of another; how new objects can be constructed from old ones; a notion of maps or morphisms between objects of a class that preserve the essential properties that the objects are supposed to capture; the notion of quotients; etc.

As "muad" noted above, some teachers plan on students learning these principles more or less by induction, from having many concrete examples presented to them, and never mentioning them explicitly. While that may be an ideal way to learn the principles, there is no guarantee that a given student will learn them, regardless of the number of examples given. And I have met some mathematicians who, as far as I can tell, aren't aware of them. As an undergraduate, I came away with the sense that the various branches of mathematics were rather disjointed, with no real common patterns. Blame it on poor instruction, or more likely, my being dense. I think I would have benefitted a lot from someone pointing out these patterns to me in an explicit way.

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Another book with roughly the same goal and plan is Eric Schechter's Handbook of Analysis and its Foundations. As the title suggests, this focusses on analysis, but the foundations covered also include quite a bit of algebra (not to mention a very accessible overview of first-order logic and axiomatic set theory). – Toby Bartels Apr 4 '11 at 3:16

As to "Why is it taught in undergraduate courses?"

I think that many constructions needed in advanced mathematics use concepts from multiple different fields. For example, if you want to prove anything about the adeles (in algebraic number theory), you need to have a good background in both algebra and point-set topology.

Sylow's theorem, to cite your example, is interesting in and of itself, but I think that one purpose in teaching it is to give some meat to students who have recently learned the definition of a group. It is definitely nontrivial, one can appreciate and prove it without having to know mathematics outside group theory, and it gives students the chance to apply techniques they have learned (counting orbits and stabilizers and such) without asking them to learn new abstractions.

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-The Sylow theorems are a great example. I think they are taught so ubiquitously because they provide perhaps the most elementary example of the broad pattern in mathematics of determining the structure an interesting object (finite groups) according to an invariant (the order). They also illustrate the power of letting groups act on things. Still, I have never encountered an actual use of the Sylow theorems in real life.

-Jordan / Rational canonical forms. This is a fairly ubiquitous topic in undergraduate / 1st year graduate linear algebra, but it doesn't seem to come up all that often (I can think of a few applications in Lie theory and differential equations). And even when it does come up, you usually just need that the diagonal + nilpotent decomposition exists - you never need to actually do calculations.

-Riemann integration. Many undergraduates take a course based on, for example. Spivak's "Calculus" and/or Rudin's "Principles of Mathematical Analysis" in which Riemann sums are developed in great detail. I went through all that and I barely remember how it works, because now I have my good friend the Lebesgue integral. Of course the Lebesgue integral is more sophisticated and difficult to learn, while undergraduates should have SOME theory of integration available.

-Basic number theory. I remember computing stuff like 7^92 mod 11 and solving x^65 = -1 mod 5, but for the life of me I don't remember half of that stuff. I guess that sort of thing provides a good invitation to more difficult mathematics, and it is probably foundational for actual number theorists.

-I wouldn't really put point set topology on the list. As you say many logicians an algebraic geometers (even number theorists) encounter certain ideas in point set topology, and such notions are also very important in a different way to analysts (e.g. weak topologies, Frechet topologies).


-Homological algebra. This is probably changing these days, but many students aren't seeing basic homological algebra until their first year graduate algebraic topology course.

-Representation theory. There are plenty of very nice, basic theorems that for some reason don't make it into the undergraduate or even graduate curriculum. I'm into analysis and geometry, and even for me I found this to be a big gap in my education.

-Lie theory. I learned it as an undergraduate, but I think most students don't.

-Metric geometry / convex geometry. There are lots of useful ideas here for analysts and even number theorists that aren't commonly taught.

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Re: Jordan/Rational canonical forms: Maybe I'm in the minority, but I very much disagree that they don't come up all that often. I think they're one of the most important things I learned as an undergrad, and quite important in Lie theory (along with their generalizations). For example, they each come up when studying orbits of the nilpotent cone of a Lie algebra and the Kostant section of the adjoint quotient map. I do agree, however, that you often only need to know of the existence of these canonical forms, but I don't see how that's a strike against them. – Mike Skirvin May 18 '10 at 2:18
Maybe it does come up more often than I claim, though what I wrote is consistent with my personal experience (perhaps limited in many ways). Still, when this stuff is taught in undergrad algebra the focus is often on problems like "find the JCF of the following matrix" or "classify all possible JCF's of matrices with property X" rather than on conceptualizing things in a way that is relevant later (say, via the structure theory for modules over a PID). – Paul Siegel May 18 '10 at 3:51
Nate, the Lebesgue integral makes sense on functions with values in "Banach spaces, for instance". It's the Bochner integral. Look in Lang's Real and Functional Analysis. I found that viewpoint much more refreshing than what is done almost everywhere else in books when real functions are integrated w.r.t a measure only by breaking them into positive and negative parts. If you do things the right way, you can just integrate the function values in one step without the breaking-up of the function values, which drove me up the wall when I first saw that. – KConrad May 18 '10 at 14:50
I agree with Mike Skirvin about the Jordan/rational canonical forms: they do appear in problems in many areas. Even if they did not appear, I still consider them important just for illustrating the more general concept of "canonical form" (i.e., the choice of a particular representative of each equivalence class in an equivalence relation). – Alfonso Gracia-Saz May 19 '10 at 7:38
Teaching Jordan canonical forms includes a very important lesson: not all matrices are diagonalizable. If we leave it out of the curriculum, we would have to include the lesson in some different way. – Peter Shor Jul 8 '10 at 14:32

One might argue that (exotic) counterexamples fall into the first category. For example, when I took Munkres' course in topology it was organized around many very carefully chosen (counter)examples showing how various properties were related. This led me me to delve into many exotic spaces listed in Steen and Seebach's Counterexamples in Topology. While I'll probably never make use of any of those exotic counterexamples, it did help me to learn better how to employ the axiomatic method, e.g. to understand how to constuct examples showing that axioms are independent, to construct pertinent examples for theoretical signposts. summits, etc. One isn't necessarily expected to remember the examples but, rather, the methodology (e.g. various ideas of completion, compactification, ...)

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