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…
Best Answer
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.