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In the December 2010 issue of Scientific American, an article "A Geometric Theory of Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most intricate structure known to mathematics, the exceptional Lie group E8." Elsewhere in the article it says "... what is perhaps the most beautiful structure in all of mathematics, the largest simple exceptional Lie group. E8." Are these sensible statements? What are some other candidates for the most intricate structure and for the most beautiful structure in all of mathematics? I think the discussion should be confined to "single objects," and not such general "structures" as modern algebraic geometry.

Question asked by Richard Stanley

Here are the top candidates so far:

1) The absolute Galois group of the rationals

2) The natural numbers (and variations)

4) Homotopy groups of spheres

5) The Mandelbrot set

6) The Littlewood Richardson coefficients (representations of $S_n$ etc.)

7) The class of ordinals

8) The monster vertex algebra

9) Classical Hopf fibration

10) Exotic Lie groups

11) The Cantor set

12) The 24 dimensional packing of unit spheres with kissing number 196560 (related to 8).

13) The simplicial symmetric sphere spectrum

14) F_un (whatever it is)

15) The Grothendiek-Teichmuller tower.

16) Riemann's zeta function

17) Schwartz space of functions

And there are a few more...

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With great respect for Richard-and without going so far as to call for it's closure,because it is an interesting question-I think if anyone of lesser stature in the mathematical community had posted this question,there would have numerous calls to close it as too general and subjective. – The Mathemagician Dec 12 '10 at 18:16
The main output of this question will likely be a reinforcement of the ego of each domain/community. Not that good for the unity of mathematics. – Denis Serre Dec 12 '10 at 18:38
IMO, the statements are sensible because they include the words "arguably" and "perhaps". In other words, I think there is little objective content to them. With all due respect to the OP, I think this question is the epitome of "subjective and argumentative", and I have voted to close for that reason. – Pete L. Clark Dec 12 '10 at 19:04
I think the question is really good and deserves its place on MO. This said (and this is in no way in reference to the OP, it is just a general rant) I've noticed that often on MO when someone with no points posts a soft question/big list type of question there is always the same bunch of people who rush to close it and to say basically it's lame. When a professor posts a question virtually of the same order, he gets 80 votes up and congratulations on the "amazing question". I've seen many examples of that and it is this same "police" going to every post and deciding what's good, what's bad. – Carlo Von Schnitzel Dec 12 '10 at 19:46
Since this has already attracted a vote to reopen (at time of typing) I've opened a thread on meta MO (… ); if you feel the question should be reopened please take the discussion there. Also, please vote this comment up for visibility. – Yemon Choi Dec 12 '10 at 20:08

35 Answers 35

I like the hyperbolic plane where Escher's "circle limits" live.

In the hyperbolic plane you can turn your car (i.e. constant aceleration perpendicular to the constant speed) and not manage to close its trajectory (there are equidistant curves and horocycles).

Also, the symmetry group of the tiling by right angled hexagons contains all but a finite number of closed surface groups. So you can build almost all closed surfaces gluing these hexagons.

People in the hyperbolic plane wont agree on the angle between two stars (i.e. boundary points) but if you average the measurements of other people around you the result will agree with your own measurement (hence everyone thinks they are right).

The modular group (and its congruence subgroups) are important in number theory (which I know next to nothing about) and also in complex analysis (where, for example, the congruence subgroup $\Gamma(2)$ is the covering group of the plane minus two points and allows one to prove that if an entire function omits two values in its image it must be constant).

The list could go on (the Gauss-Bonnet theorem, Brownian motion escapes with positive speed to infinity, Anosov property the geodesic flow, quasi-geodesics are at bounded distance from geodesics, the area of a convex hull is bounded by a constant times the number of points, the strong isoperimetric inequality holds i.e. perimeter is greater then volume for all sets, etc, ...).

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Maybe this wouldn't be my first choice, but I still think it's worth being on the list: Gödel's constructible universe $L$.

I would argue that it is intricate because it can serve as a model for "all of mathematics" (i.e., ZFC), furthermore answering many combinatorial questions left open by ZFC alone. Even though most(?) set theorists will probably argue that it is not "the" true model giving the right answer to these questions, it is still undoubtedly a rich and complex structure, moreover one in which the axiom of choice and the continuum hypothesis are not only true but "explained".

But it is also beautiful because of its connections with higher computability theory (e.g., the sets of integers constructed at the level $\omega_1^{\mathrm{CK}}$ of the constructible hierarchy, where $\omega_1^{\mathrm{CK}}$ is the smallest nonrecursive ordinal, are exactly the hyperarithmetical sets, i.e., the (lightface) $\Delta^1_1$ sets of the analytic hierarchy), and, in a related manner, because of Jensen's results on the "fine structure" of $L$. In a very intuitive way, I'd say that $L$ consists of sets that are ultimately "computable" (iterating the Turing jump as far as it can be), a perfectly regular construction that prohibits any randomness.

So even if set theorists are unhappy with $L$ because it forbids really large cardinals, and even if they try to construct something better (the core model), I argue that Gödel's original $L$ is still something immensely intricate and beautiful.

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This is a less-known example. Consider the semigroup $$ A_0 = \big\{ \big[\begin{smallmatrix} 0&0\\0&0 \end{smallmatrix}\big], \big[\begin{smallmatrix} 1&0\\0&0 \end{smallmatrix}\big], \big[\begin{smallmatrix} 0&1\\0&0 \end{smallmatrix}\big], \big[\begin{smallmatrix} 0&1\\0&1 \end{smallmatrix}\big] \big\} $$ under usual matrix multiplication. The variety $\mathrm{var}A_0$ generated by $A_0$ is finitely universal in the sense that the lattice of subvarieties of $\mathrm{var}A_0$ embeds all finite lattices. The semigroup $A_0$ is a minimal example since the variety generated by any semigroup of order three or less is not finitely universal.

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I think that Stone-Cech compactification has a highly and deeply complexity.

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See my comment to Daniel Geisler's answer. – Deane Yang Dec 13 '10 at 20:08
I agree with Deane's comment, even though I think a case can be made for $\beta \mathbb N$. (The poster doesn't say what space he is taking the Stone-Cech compactification of, mind you.) – Yemon Choi Dec 13 '10 at 22:10

Shelah's Body of Work. Considering that this list of references is over 100 pages long, I think this a contender.

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No offence, Michael, but you seem to be interpreting the original question rather creatively with this answer. – Yemon Choi Jan 26 '11 at 3:04
@Choi Indeed, but aren't the best answers creative? If I had said, L, L(A), L[A], the cumulative hierarchy V, the collection of P-names generated by a partial order, or mentioned the PO constructed by essentially 'weaving' together every CCC PO in a ground model to produce a model of MA+'not CH', they would not have elicited the same feelings of respect and amazement, I get when I consider such objects. So I opted for something I could share, which everyone could appreciate, which demands the same sense of respect and amazement. – Michael Blackmon Jan 26 '11 at 4:14
@Blackmon: "aren't the best answers creative" - it depends on your question, doesn't it? As you may see from some of my previous comment s on some of the answers here, I don't find them particularly good. Moreover, although no one else seems to have mentioned things I find kewl, I haven't taken the opportunity to put "the publication record of the late N. J. Kalton" as an answer... – Yemon Choi Jan 26 '11 at 19:08
@Blackmon: Because I don't think this is a productive MO question, from the point of view of light rather than heat. I also think "the publication record of the late N. J. Kalton" would be a poor answer to the question; and I am not a fan of several other answers to this question. It doesn't help that the title of this question is not quite the same as what the question seems to actually ask for... – Yemon Choi Jan 27 '11 at 5:01
@Choi I agree, this question is nothing more than a place to vent about how "awesome you're field is." I made the choice not to do this, and just posted something everyone should be able to appreciate. With that being said, I think this particular question should be closed, for the reasons highlighted in the comments to the original question. – Michael Blackmon Jan 27 '11 at 5:13

protected by François G. Dorais May 13 '14 at 5:18

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