Most intricate and most beautiful structures in mathematics - MathOverflow

Most intricate and most beautiful structures in mathematics

2010-12-12T17:52:17Z

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...

Answer by Steven Landsburg for Most intricate and most beautiful structures in mathematics

2010-12-12T18:03:48Z

I believe the natural numbers are the most intricate and beautiful structure in all of mathematics. Particularly insofar as all of the other intricate and beautiful structure we actually work with can be encoded via the natural numbers.

Answer by Michael Hardy for Most intricate and most beautiful structures in mathematics

2010-12-12T18:04:39Z

The class of all ordinals. The class of cardinals is embedded within it (if AC holds) since one identifies a cardinal with the smallest ordinal such that the set of all smaller ordinals has that cardinality. ( $\aleph_0$ is the cardinality of the set of all finite ordinals, $\aleph_1$ is the cardinality of the set of all countable ordinals, etc. $\aleph_\omega$ is the cardinality of the set of all ordinals whose cardinality is $\aleph_n$ for some finite $n$. ($\omega$ is the ordinal that gets identified with $\aleph_0$ in the aforementioned identification) $\aleph_{\omega+1}$ is the set of all ordinals of cardinality $\aleph_\omega$ , and so on. $\aleph_\omega$ is the smallest cardinal greater than $\aleph_0$ that is known not to be equal to $2^{\aleph_0}$ .)

But if grading is only based on "intricacy", maybe the class of all sets, conventionally denoted "V" because it looks like the letter V (?) might be in first place. Some people have tried to embed all of mathematics within this thing.

Later edit: The "\aleph"s and the "\omega"s are failing to get rendered when I view this thing. Look at the code and you'll see them.

Answer by Andrew L for Most intricate and most beautiful structures in mathematics

2010-12-12T18:24:21Z

Ok, I'll throw my hat in the ring: I like the classical Cantor set.

Not only does it demonstrate the complexity that relatively simple subsets of the real line have, it illustrates an important property of measures on the real line - namely, that measurability has nothing to do with cardinality of the set (i.e. this is an uncountable set with measure zero!)

It also gives an example of a completely disconnected subset of $\mathbb{R}$ that literally has no components - it contains no open intervals of $\mathbb{R}$ in its power set.

There are many, many more observations one can make about the Cantor set, but I think the obvious ones make my point very nicely. When I teach real analysis, this is an example I think I'll be using a great deal to illustrate properties of the real line.

Answer by Kristal Cantwell for Most intricate and most beautiful structures in mathematics

2010-12-12T19:04:42Z

The monster vertex algebra.

It is (to date) the central object in monstrous moonshine, since its character is the $q$-expansion of the modular $J$-function, its automorphism group is the monster simple group, and the graded trace of any element of the monster is the $q$-expansion of a genus zero modular function. The construction of this structure (by Frenkel, Lepowsky, and Meurman) involves ascending a hierarchy of objects that are by themselves quite intricate and beautiful.

One begins with the extended binary Golay code of length 24. Up to symmetries, it is the unique copy of $\mathbb{F}_2^{12}$ in $\mathbb{F}_2^{24}$ , for which any five basis vectors are contained in a unique codeword (i.e., it forms a Steiner $(5,8,24)$ system). The codewords are separated by Hamming distance at least 8, so even if 3 bits in a code word are changed the error can be corrected. The automorphism group of the Golay code is the sporadic simple group $M_{24}$ of order 244823040.
Using the Golay code to produce coordinates of generators, one constructs the Leech lattice $\Lambda$, which is a rather densely packed copy of $\mathbb{Z}^{24}$ in $\mathbb{R}^{24}$. One can also make the Leech lattice as a subquotient of the even unimodular lattice $I\!I_{25,1}$ , which has its own exceptional properties. Peter Shor mentioned the Leech lattice in another answer, so I'll just note that its automorphism group is a double cover of Conway's sporadic simple group $Co_1$, which has order 4157776806543360000.
For any positive definite even lattice $L$, there is a canonical construction of a vertex operator algebra graded by that lattice, called the lattice vertex algebra $V_L$. I think physicists say that it is the algebra of chiral symmetries of a conformal field theory describing a bosonic string propagating in the torus $L \otimes \mathbb{R}/L$ (but I may have mixed up the words). It has an action of the holomorph of the algebraic torus $L \otimes \mathbb{C}^\times$.
The "-1" automorphism of the Leech lattice induces an automorphism $\theta$ of $V_\Lambda$, and there is a unique irreducible $\theta$-twisted module $V_\Lambda(\theta)$ that inherits an action of the centralizer $2^{1+24}.Co_1$ of $\theta$ in the automorphism group of $V_\Lambda$. The monster vertex algebra is formed by taking the direct sum of fixed points: $V^\natural = (V_\Lambda)^\theta \oplus (V_\Lambda(\theta))^\theta$.

Apparently, the hard part was proving that the monster acts on $V^\natural$ by automorphisms.

There are some additional conjectural reasons for considering it beautiful:

In the same paper where it was constructed, it was conjectured to be the unique vertex operator algebra with central charge 24, character equal to the modular $J$ function, and representation category equivalent to $Vect$. (Naturally, this does not account for higher structure like twisted modules.)
Witten suggested that it is dual to pure 3-dimensional quantum gravity with minimal cosmological constant by AdS/CFT correspondence.

Answer by Deane Yang for Most intricate and most beautiful structures in mathematics

2010-12-13T18:41:44Z

In my view it is difficult to come up with an alternative to any of the exotic Lie groups, which are unquestionably quite intricate but are also beautiful because they express the properties of certain geometric spaces using both fundamental algebra (i.e., groups) and geometric structures of their own (i.e., Riemannian geometry). I don't know $E_8$ particularly well, but I still have vivid memories of Robert Bryant's lectures describing the structure of $G_2$.

Answer by Thierry Zell for Most intricate and most beautiful structures in mathematics

2010-12-13T19:26:21Z

Since one of the questions is Are these sensible statements?, allow me to answer that one with a resounding NO, and be on record against the size-ism inherent in the statements of Scientific American. As the largest simple exceptional Lie group, E8 deserves credit for both intricacy and beauty. but the author seems to imply that the size record makes E8 not only more intricate but also more beautiful than the other simple exceptional Lie groups.

Granted I don't know much about Lie groups, but it really bothers me that an aesthetic judgment can be based on size alone. Last I checked, paintings are not judged on their size.

Answer by Buschi Sergio for Most intricate and most beautiful structures in mathematics

2010-12-13T20:05:16Z

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

Answer by Matthew Kahle for Most intricate and most beautiful structures in mathematics

2010-12-13T20:08:51Z

The (stable or unstable) homotopy groups of spheres are certainly considered intricate and beautiful by topologists.

Here is an interesting (obvious) fact about the stable homotopy groups of spheres that I learned from Vigelik:

In the category of commutative rings (with unit) there is an initial object, $\mathbb{Z}$. This seems to be one reason the integers are important or rather fundamental. But there is something more fundamental! There is a functor from commutative rings to ring spectra, the Eilenberg-MacLane functor. $H\mathbb{Z}$ is no longer initial in this category, the sphere spectrum is! So somehow the stable homotopy groups of spheres are a pretty cool/fundamental ring.

I do not know a lot about the unstable setting, but there is a lot of extra data that it has.

I think that Lennart's point about intricacy and complexity showing up when you start to try to compute the thing sounds like a confusion of what one means by intricate and complex. But it is not, it is not the messiness of the computation that makes it intricate, it is the way of teasing apart the knowledge we do have in meaningful ways that lead me to believe that it is a very intricate object. Especially all the number theory hidden in the chromatic picture, which is part of what Lennart is referring to when he mentions the moduli stack of formal group laws.

Edit: My advisor pointed out another reason that the stable homotopy groups of spheres are cool: $\pi^S_{*}(S^0)=\pi_{*}(B\Sigma_{\infty})$, so the stable homotopy groups of spheres are the homtopy groups of the classifying space of the the category of finite sets and bijections. This is essentially the Barratt-Priddy-Quillen theorem (I am told, I do not know the precise statement). That is pretty cool too! All that information about finite sets sitting there has to be something.

(seems to look fine now, please ignore the bump)

Answer by Harry Gindi for Most intricate and most beautiful structures in mathematics

2010-12-13T22:44:18Z

Consider the canonical pointed symmetric sequence of simplicial sets $S$ defined such that $S_0=S^0$ and $S_n=S^n$. This is an ordered sequence of simplicial sets with a natural symmetric group action defined by permutation of the suspensions. By abstract nonsense, we can show that this category has a symmetric monoidal closed product. The object $S$ admits the natural structure of a monoid for this tensor product. The category of symmetric spectra becomes the category of modules over this monoid.

What's deep and intricate about this object? Well, I just read a paper by A. Salch that shows that the category of commutative $S$-algebras models the proposed theory of the field with one element!

The category of modules over this monoid is the category of symmetric spectra!

Answer by Gerhard Paseman for Most intricate and most beautiful structures in mathematics

2010-12-13T22:48:35Z

The relationship between the discrete order and the multiplication on the natural numbers leads to, among other things, the study of gaps between primes. I would nominate a class of structures S(n), which are the sets of integers relatively prime to the nth primorial (p_1p_2...p_n) as a collection worthy of the labels beautiful and intricate. The symmetry and self-similar nature appeal to many, and while the construction is simple, there are many simple facts remaining to be established about the S(n). For one, the largest gap between consecutive members of S(n) seems to be unknown. (Cf http://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update for a weak upper bound; I hope to post an improvement soon.)

Gerhard "Ask Me About System Design" Paseman, 2010.12.13

Answer by Lennart Meier for Most intricate and most beautiful structures in mathematics

2010-12-13T23:03:28Z

The absolute Galois group of $\mathbb{Q}$. It contains the information of all algebraic extensions of the rationals - and is therefore the most important single object of algebraic number theory. Representations of the absolute Galois group are central to many diophantine questions; see for example the Taniyama-Shimura conjecture (aka modularity theorem) which led to a solution of Fermat's last theorem and states in some form that certain Galois representations associated to elliptic curves come from modular forms.

One of the most intricate set of conjectures is dedicated (partly) to the study of representations of the absolute Galois group of $\mathbb{Q}$: the Langlands program.

Answer by Douglas Zare for Most intricate and most beautiful structures in mathematics

2010-12-14T00:51:04Z

The Mandelbrot Set is widely viewed as beautiful and intricate, although I can't give a mathematical definition for those.

The imperfect self-similarities are no accident. Many of the pieces correspond to the behaviors of the critical point $0$ under iteration of $z \to z^2 + c$.To each point in the plane, there is a corresponding Julia set, and the relationship of the point to the Mandelbrot set indicates some of the structure of the Julia set.

Answer by Christian Blatter for Most intricate and most beautiful structures in mathematics

2010-12-14T11:28:16Z

I have voted for ${\mathbb N}$; but let me nevertheless propose an object living in the analytical realm, namely the Schwartz space ${\cal S}$ of infinitely differentiable functions $f:{\mathbb R}\to{\mathbb C}$ that for $|x|\to\infty$ together with their derivatives go to zero faster than any power $1/|x|^n$. The "intricateness" of this space stems from the many operations you can perform in it and from the fact that these operations are intertwined with each other in miraculous ways. $$ $$ Responding to a comment: You have (a) ordinary multiplication and convolution, (b) "multiplication" with arbitrary polynomials $p(x)$ and operations $p(D)$, (c) multiplication with functions of the form $x\mapsto e^{iax}$ and the translation operator $T_a: f(\cdot)\mapsto f(\cdot-a)$ and (d) scaling of the variable $x$ resp. $\xi$. The Fourier transform $\Phi$ interchanges in each of these three cases the respective operations; and at heart of it all is Gauss' normal distribution $x\mapsto {1\over \sqrt{2\pi}}\int e^{-x^2/2} dx$ which stays fixed under $\Phi$. And, last not least, there is a scalar product which is preserved by $\Phi$.

Answer by Spiro Karigiannis for Most intricate and most beautiful structures in mathematics

2010-12-14T17:02:00Z

One of the most beautiful structures, in my mind, is the classical Hopf fibration, which allows you to visualize the $3$-sphere $S^3$ as a smooth circle bundle over the $2$-sphere. When you view $S^3$ minus a point as $\mathbb R^3$, one can actually draw very nice pictures of this fibration. It's doubly interesting to me because it involves the isomorphism of $S^2$ with $\mathbb C\mathbb P^1$ from complex analysis.

There are actually 4 such Hopf fibrations (spheres which are total spaces of fibre bundles whose base and fibre are also both spheres):

1) $S^1$ is an $S^0$ bundle over $\mathbb R \mathbb P^1 \cong S^1$.

2) $S^3$ is an $S^1$ bundle over $\mathbb C \mathbb P^1 \cong S^2$.

3) $S^7$ is an $S^3$ bundle over $\mathbb H \mathbb P^1 \cong S^4$, the quaternionic projective line.

4) $S^{15}$ is an $S^7$ bundle over $\mathbb O \mathbb P^1 \cong S^8$, the octonionic projective line.

Answer by Andrew D. King for Most intricate and most beautiful structures in mathematics

2010-12-14T20:29:17Z

I don't know if it's the most beautiful or the most intricate, but I certainly think the random graph $G(n,p)$ deserves consideration, if only for philosophical reasons.

Answer by none for Most intricate and most beautiful structures in mathematics

2010-12-15T00:37:59Z

What about the complex numbers, from the way all the theorems of complex analysis fit together so well.

Also: NGB set theory (I mean the formal object Th NBG, not the general informal topic): finitely axiomatizable through an intricate argument, and proves just about everything in mathematics.

Answer by none for Most intricate and most beautiful structures in mathematics

2010-12-15T00:41:46Z

Another one: Chaitin's Omega constant.

Original answer by: none

Answer by Terry Tao for Most intricate and most beautiful structures in mathematics

2010-12-16T07:46:11Z

The Littlewood-Richardson coefficients. (Or, if one wants a single object, as per the rules of the game: the representation ring of $S_n$ or $GL_n({\bf C})$. Or the ring of symmetric functions. Or the cohomology ring of the Grassmannian with the Schubert variety basis. Etc., etc.)

On the one hand, the Littlewood-Richardson coefficients have fairly simple geometric descriptions (using such combinatorial gadgets as Young tableaux, honeycombs, or puzzles), but on the other hand obey a number of deep recursive properties. (See for instance my Notices article with Allen Knutson on one aspect of these coefficients.) Last, but not least, they are connected to an amazing number of areas of mathematics (see e.g. Fulton's survey article).

Answer by Hailong Dao for Most intricate and most beautiful structures in mathematics

2010-12-16T07:59:09Z

I heard good things about F_un!

Answer by Leo Alonso for Most intricate and most beautiful structures in mathematics

2010-12-16T15:46:14Z

Let $g$ and $n$ be positive integers such that $3g-3 + n > 0$ Let $\mathcal{M}_{g,n}$ the moduli stack of genus $g$ nodal curves with $n$ marked points. There are two obvious families of maps

forgetting a point

$$\mathcal{M}_{g,n+1} \rightarrow \mathcal{M}_g{}_n$$

and identifying two marked points

$$\mathcal{M}_{g_1,n_1} \times \mathcal{M}_{g_2,}{}_{n_2} \rightarrow \mathcal{M}_{g_1 + g_2,}{}_{n_1 + n_2 - 2}$$

This system constitutes the so-called Grothendieck-Teichmüller tower. It is indeed intricate and in my opinion, also beautiful. Moreover, it is a conjecture of Grotehndieck that its automorphism group is naturally isomorphic to the absolute Galois group over $\mathbb{Q}$.

Answer by Johannes Ebert for Most intricate and most beautiful structures in mathematics

2010-12-17T10:32:38Z

This has been forgotten so far: http://en.wikipedia.org/wiki/Riemann_zeta_function

Answer by Tony Huynh for Most intricate and most beautiful structures in mathematics

2010-12-17T17:18:33Z

The Turing degrees are an immensely intricate poset $\mathcal{D}$. Here are some of their remarkable properites:

Every countable poset is embeddable in $\mathcal{D}$.
$\mathcal{D}$ contains minimal degrees. (a non-zero degree $\mathbf{m}$ with no degree between $\mathbf{0}$ and $\mathbf{m}$)
For every non-zero degree $\mathbf{d}$, there is a degree that is incomparible with $\mathbf{d}$.
$\mathcal{D}$ contains an antichain of size $2^{\aleph_0}$.
No infinite strictly increasing chain in $\mathcal{D}$ has a least upper bound.
For every non-zero degree $\mathbf{d}$, there is a degree $\mathbf{c} < \mathbf{d}$ such that $\mathbf{c}'=\mathbf{d}$. (Here $\mathbf{c}'$ denotes the set of indices of oracle Turing machines that halt when using $\mathbf{c}$ as an oracle. Note that one must check that this is well-defined on degrees.)
For any two recursively enumerable degrees, there is a recursively enumerable degree strictly between them.
Any finite distributive lattice can be embedded in the recursively enumerable degrees.

Answer by Peter Shor for Most intricate and most beautiful structures in mathematics

2010-12-17T18:10:30Z

How about the Leech lattice. This is a 24-dimensional packing of unit spheres where each one touches 196560 others. It is the densest 24-dimensional lattice packing (and very likely the densest 24-dimensional sphere packing, although this has not been proved). It has a remarkable amount of symmetry, and most of the densest sphere packings known in dimensions < 24 are derived from it (and known sphere packings in dimensions > 24 are nowhere near as dense when normalized for the dimension).

Maybe this is already implicitly included in the list, as it is closely related to the monster vertex algebra.

Answer by Qfwfq for Most intricate and most beautiful structures in mathematics

2010-12-19T02:06:15Z

$\mathrm{Spec}(\mathbb{Z})$.

It can also be thought as the set of prime numbers. I don't know if it can really be considered "intricate"...

Answer by Michael Blackmon for Most intricate and most beautiful structures in mathematics

2011-01-26T02:39:05Z

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