Monday, March 27, 2006

The Most Amazing and Mysterious Thing in All of Mathematics

I think that a good candidate is the table of the Homotopy Groups of Spheres.

For those not familiar with algebraic topology, πm(Sn) is the set of equivalence classes of continuous functions from the m-dimensional sphere to the n-dimensional sphere where two functions are considered equivalent if they are homotopic. An easy way to visualise this is that two functions are homotopic if you can interpolate a continuous animation between them. (Can you guess what industry I work in?) This set also has a group structure which is straightforward to define but which I won't go into here (unless someone requests it). That's all there is to it. How can such simplicity generate such complexity?

Monstrous Moonshine is pretty mysterious too - but it takes a lot of work to state it for a non-expert. So this wins by default.

Like John Baez I also wonder about the curious appearance of 24 on row 3.

And an aside I discovered on Anarchaia: related to an earlier post.


David R. MacIver said...

I think the number four is a pretty good candidate for the most mysterious thing in all of mathematics.

(Specifically, why 4 dimensional things are so bloody weird)

sigfpe said...

They're closely related. There's the obvious physical interest of 4. There's an algebraic side to 4 coming from the quaternions and this is part of a sequence 1,2,4,8 of division algebras. This all ties up nicely with the physics through representation theory with things like the 'spinorial chessboard'. But there's a topological dimension to this too with the division algebras closely tied to Hopf bundles. The Hopf bundles give rise to many of the interesting phenomena in this table. Notice how the size of the homotopy group is much larger for k one less than a multiple of 4.

Deep down, however, I feel that ζ(-1)=-1/12 is a more special number...

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