Timeline for Examples of undergraduate mathematics separation from what mathematicians should know
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15 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2023 at 1:35 | comment | added | Timothy Chow | The Sylow theorems were used by Drisko to obtain an important partial result on a purely combinatorial conjecture about the enumeration of Latin squares. On the Number of Even and Odd Latin Squares of Order $p+1$. | |
| Apr 4, 2011 at 3:22 | comment | added | Toby Bartels | If the Lebesgue integral is too complicated but the Riemann integral is too limited, one could try the Henstock–Kurzweil integral. This integral is no more complicated to define than Riemann's, but it's as general than Lebesgue's. (It's actually slightly more general, including also all improper Riemann integrals; a function is Lebesgue integral iff its absolute value is Henstock integrable.) | |
| Jul 8, 2010 at 14:32 | comment | added | Peter Shor | 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. | |
| Jul 7, 2010 at 17:58 | comment | added | David Corwin | Then again, we don't know all the connections that might be discovered in math one day. How do you know that the Sylow theorems won't be relevant to your field? | |
| May 26, 2010 at 8:23 | comment | added | Pete L. Clark | The Sylow theorems are important in Galois theory, since they have the following consequence: for any field $K$ and prime $p$, (a) there exists a separable extension $L(p)$ of $K$ such that (i) every finite subextension of $L(p)/K$ has order prime to $p$ and (ii) every finite subextension of $K^{sep}/L(p)$ has $p$-power order, and ( ) any two such fields $L(p)$ are conjugate over $K$. This is especially useful in Galois cohomology; it allows various devissage arguments. I would put Sylow theory in the top half of things I learned as an undergrad ordered by present-day usefulness. | |
| May 19, 2010 at 7:38 | comment | added | Alfonso Gracia-Saz | 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). | |
| May 18, 2010 at 16:54 | comment | added | Michael Hoffman | Problem with Lie Theory is that, IMO, you really need a good background in topology and differential geometry (both of which would be at least one course in and of themselves), I just started working on Lie Theory, and it is really hard for me as an undergraduate and I think I couldn't have effectively approached it til my senior year. But that's just my opinion. | |
| May 18, 2010 at 14:50 | comment | added | KConrad | 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. | |
| May 18, 2010 at 13:44 | comment | added | Nate Eldredge | Lebesgue integration is nice because it generalizes naturally to integrating functions over more general domains (i.e. measure spaces). On the other hand, the Riemann integral generalizes naturally to integrating functions taking values in more general spaces; Banach spaces, for instance. And most of the stochastic integrals that arise in probability have a Riemann-type construction underlying them. One could argue about which type of integral one should start with, but I think in the long run a mathematician needs to be familiar with both. | |
| May 18, 2010 at 3:51 | comment | added | Paul Siegel | 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). | |
| May 18, 2010 at 3:10 | comment | added | Mike Skirvin | KConrad - At the time, I undoubtedly did not appreciate J/RCF to the extent that I do now. However, given that they were the big theorems which capped off the linear algebra portion of my undergrad algebra course, and that the professor did a nice job integrating them into exercises during the subsequent field theory/Galois theory portion of the course, I think I came away with some appreciation for them. Admittedly, it was not until later that I learned of the applications I listed in my original comment. | |
| May 18, 2010 at 2:59 | comment | added | The Mathemagician | The reason Riemann integration is usually taught to undergraduates is because 1) it's conceptually a lot simpler and easier to motivate then the Lebesgue theory and 2)the Lebesgue theory is rather mysterious without a good facility with the older theory. Maybe I'm just old fashioned,but in the 1960's a lot of universities DID try teaching the Lebesgue integral straight off the bat,sometimes in CALCULUS. The result was a disaster in most cases. There's a reason for that. | |
| May 18, 2010 at 2:42 | comment | added | KConrad | Mike, at the time you learned J/RCF, were you left with an impression that it was important? | |
| May 18, 2010 at 2:18 | comment | added | Mike Skirvin | 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. | |
| May 18, 2010 at 2:08 | history | answered | Paul Siegel | CC BY-SA 2.5 |