[HN Gopher] The Quintic, the Icosahedron, and Elliptic Curves [pdf]
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The Quintic, the Icosahedron, and Elliptic Curves [pdf]
Author : mathgenius
Score : 53 points
Date : 2024-04-03 16:29 UTC (6 hours ago)
(HTM) web link (www.ams.org)
(TXT) w3m dump (www.ams.org)
| empath-nirvana wrote:
| > In 1963, the Russian mathematician Vladimir Arnold gave an
| alternative topological proof of the unsolvability of the quintic
| in a series of lectures to high school kids in Moscow.
|
| ...
|
| > Arnold's insight was to show that if there is a radical formula
| for the roots of a general polynomial, then the "dance of the
| roots" cannot be overly complex, in the sense that the image of
| the monodromy map must be a solvable subgroup of . But, for
| example, the 1-parameter Brioschifamily of quintics () = 5 + 103
| + 452 + 2, [?] C has monodromy group 5 (see Figure 1), which is
| certainly not solvable since it is simple, as we will see by
| relating it to the icosahedron in the next section. Hence the
| unsolvability of the quintic.
|
| ...of course, well within the reach of most high school students.
| dhosek wrote:
| Not "most" high school students, but the more talented ones?
| Sure. Likely they were exposed to a lot of algebra before this
| lecture to bring them to a point where they could profitably
| follow along. The kind of kids who take AP Calc can learn
| abstract algebra (and, for that matter, I'd say that any kid
| who's had Algebra II and properly taught geometry (with proofs)
| can learn abstract algebra).
| ilnumyre wrote:
| I'll paraphrase a famous piece of public speaking advice "avoid
| talking down to your audience, they will catch up".
|
| Also, who knows how the lecturer presented the information.
| Perhaps it was well composed, and in fact could be understood
| by anybody.
|
| When fields highly specialize they seem to attract a certain
| personality of people who revel in their 'insider knowledge'
| and make the learning of it for others more difficult in order
| to inflate their standing in the eyes of their students, "How
| did you learn this stuff!?"
|
| The reality of these fields, like mathematics, music theory,
| computer science, and the like is that the ideas themselves are
| so ridiculously simple that they could be understood by most
| anyone with a rudimentary understanding of the 5 basic
| arithmetic operators if presented coherently.
|
| After all, "Coding is basically just ifs and for loops."
|
| https://news.ycombinator.com/item?id=29442307
| euroderf wrote:
| > When fields highly specialize they seem to attract a
| certain personality of people who revel in their 'insider
| knowledge' and make the learning of it for others more
| difficult in order to inflate their standing in the eyes of
| their students
|
| This is one reason why Wikipedia is so very valuable. A well-
| done article gives you a foot in the door, even if by doing
| little more than decoding jargon.
| wging wrote:
| The actual argument is a bit simpler than implied by the
| quote's use of the terms 'monodromy', 'solvable subgroup', etc.
| Here's an explanation of Arnold's proof that doesn't use those
| words at all, and while not precisely _easy_ is much easier
| than you 'd think from the quoted bit of the OP article:
| https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...
| nicf wrote:
| I agree that this is not within the reach of most high school
| students, and if I had to guess, I think that either these
| Moscow kids were very carefully selected to be in this group or
| most of them didn't follow it.
|
| But I do think some very talented and motivated high school
| students could at least follow the gist of this argument! I've
| worked at a summer program for mathematically talented high
| schoolers, and while I didn't teach this particular class, one
| of the years I was there someone taught what I think is
| essentially the same proof this article is talking about, and
| they did get through it in five weeks.
| wbl wrote:
| The pictures of monodromy are called dessins de enfant because
| they are little pictures that are simple to understand. The
| hard part is why we care so much.
| zyklu5 wrote:
| This was published as: Abel's Theorem in Problems and Solutions
| by VB Alekseev if you are interested in taking a look.
|
| https://link.springer.com/book/10.1007/1-4020-2187-9
| jcranmer wrote:
| But it actually is! It's just written in an excessively jargon-
| laden manner. Here's an explanation that's less jargon-laden:
|
| We can describe a polynomial by its coefficients, standard x^5
| + ax^4 + bx^3 + cx^2 + dx + e = 0 form. Actually, we're going
| to restrict a, b, c, d, and e so that they are all determined
| by a single parameter B (this is the Brioschi family of
| quintics). Remember that all values here are complex numbers,
| not real numbers.
|
| Now what happens if we move around B a little bit
| (https://duetosymmetry.com/tool/polynomial-roots-toy/ is _very_
| helpful for seeing what I mean here)? Well, the roots will
| change a little bit. There 's a continuous mapping going on.
| Now move around B in a large, closed path that includes a
| specific point in the interior. It turns out that the
| continuous motion of the roots will result in them swapping
| places--a permutation. This is the "monodromy map".
|
| Now it turns out that the set of possible permutations has
| implications on whether or not you can solve it with
| polynomials. The paper here doesn't go into details, since it's
| written for a high-math audience (this is the "A_5 is not
| solvable since it is simple" bit).
| zyklu5 wrote:
| If the terms: fundamental group, covering space etc are
| jargon then more so is the case for words like 'explanation',
| 'laden' and well, 'jargon' itself. They have precise
| definitions meant to capture mathematical concepts and ideas
| for which there can be no adequate words in natural language.
| joecarmody wrote:
| Nice to see some Riemann Academy on the front page of HN...
| ykonstant wrote:
| What a great exposition; very happy this dear topic of mine
| garnered some interest from HN folks.
| JadeNB wrote:
| See also the lovely discussion by Baez at
| https://classes.golem.ph.utexas.edu/category/2017/12/the_ico... .
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