Timeline for Examples of undergraduate mathematics separation from what mathematicians should know
Current License: CC BY-SA 2.5
32 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 5 at 15:47 | review | Close votes | |||
| Jan 9 at 10:26 | |||||
| Dec 18, 2021 at 2:11 | comment | added | Michael Hardy | I have qualms about the word "expected". Naive students are in effect taught that a good reason for learning something is that it's "expected" of them. Much evil follows from that kind of thinking. | |
| Mar 26, 2019 at 0:00 | review | Close votes | |||
| Mar 26, 2019 at 16:34 | |||||
| Jan 27, 2017 at 14:08 | comment | added | user44143 | If this were a new question, asking for four different things, I would vote to close as "too broad". | |
| Jan 27, 2017 at 8:10 | answer | added | Aryeh Kontorovich | timeline score: 2 | |
| Jan 27, 2017 at 4:29 | answer | added | JKreft | timeline score: 2 | |
| Dec 15, 2013 at 20:29 | comment | added | Michael Hardy | (Correction: I meant $c\not\in\{0,1\}$.) | |
| Dec 15, 2013 at 6:55 | comment | added | LSpice | @MichaelHardy, for my own understanding, I didn't understand tensor products until I formulated to myself the idea that "the tensor product makes bi-" (more generally, multi-) "linear maps into linear maps." | |
| Jan 2, 2011 at 0:50 | comment | added | Michael Hardy | Here's how I might answer if an undergraduate asked me what a tensor product is. The difference between an ordered pair $(x,y)$ of vectors and a tensor product $x\otimes y$ of vectors is that if you multiply the the first component by a scalar $c\not\in{0,1}$ and the second by $1/c$, you get a different ordered pair of vectors but the same tensor product of vectors. The way it often (?) gets explained in graduate algebra courses is a lot of technicalities needed to make sure all the rules of algebra are followed, but doesn't emphasize the essential intuition. | |
| Jan 1, 2011 at 23:17 | answer | added | gowers | timeline score: 41 | |
| Jan 1, 2011 at 21:09 | comment | added | KConrad | Andrea, you don't need the Sylow theorems in full-blown form to prove the fundamental theorem of algebra via Galois theory. For that proof you need the existence of p-Sylow subgroups and that a nontrivial finite p-group has a subgroup of index p (all for p = 2). The conjugacy of Sylow subgroups or congruence conditions on the number of Sylow subgroups don't matter. | |
| Jan 1, 2011 at 19:31 | comment | added | Andrea Ferretti | @Vivek: for instance you can combine it with Galois correspondence to obtain an almost entirely algebraic proof of the fundamental theorem of algebra. | |
| Jan 1, 2011 at 19:17 | comment | added | Vivek Shende | Sylow theorems, widely applicable? To what? | |
| Jan 1, 2011 at 18:48 | answer | added | Frank Thorne | timeline score: 7 | |
| Jul 8, 2010 at 3:05 | answer | added | Michael Hutchings | timeline score: 26 | |
| Jul 7, 2010 at 23:01 | comment | added | Emerton | representation theory to the problem of non-abelian class field theory. He developed this idea in a famous letter to Weil (available on the web-page of his collected works at IAS), and further in a lecture titled "Problems in the theory of automorphic forms" and in the Yale lecture notes "Euler products". All this happened rather quickly. | |
| Jul 7, 2010 at 22:59 | comment | added | Emerton | 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 ... | |
| Jul 7, 2010 at 22:11 | answer | added | Kurt Luoto | timeline score: 6 | |
| Jul 7, 2010 at 19:08 | answer | added | Bill Dubuque | timeline score: 4 | |
| Jul 6, 2010 at 21:01 | comment | added | Peter Shor | 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. | |
| Jul 6, 2010 at 19:47 | comment | added | David Corwin | I might like to know, for example, who noticed the deep relations between representation theory and number theory known as the Langlands Program. Did this person (or people) have a working knowledge of both number theory and representation theory? Did a number theorist and a representation theorist realize this together at the conference? I don't know much about Langlands, so this question might sound naive (or simply need better phrasing), but I think this kind of question might help us (at least partially) answer this philosophical dispute. | |
| Jul 6, 2010 at 19:24 | comment | added | David Corwin | 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? | |
| Jul 6, 2010 at 18:17 | answer | added | Asaf Karagila♦ | timeline score: 11 | |
| Jul 6, 2010 at 17:07 | answer | added | muad | timeline score: 17 | |
| Jul 6, 2010 at 8:59 | answer | added | Amitesh Datta | timeline score: 11 | |
| May 25, 2010 at 19:37 | answer | added | Frank Quinn | timeline score: 32 | |
| May 18, 2010 at 12:45 | answer | added | Gerald Edgar | timeline score: 13 | |
| May 18, 2010 at 2:55 | comment | added | The Mathemagician | 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......... | |
| May 18, 2010 at 2:16 | comment | added | Sam Lichtenstein | 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,... | |
| May 18, 2010 at 2:08 | answer | added | Paul Siegel | timeline score: 6 | |
| May 18, 2010 at 0:35 | answer | added | JSE | timeline score: 27 | |
| May 17, 2010 at 21:32 | history | asked | Vipul Naik | CC BY-SA 2.5 |