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May 27, 2024

Lanthanides and the Exceptional Lie Group G2

Posted by John Baez

The lanthanides are the 14 elements defined by the fact that their electrons fill up, one by one, the 14 orbitals in the so-called f subshell. Here they are:

lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium.

They are also called ‘rare earths’, but that term is often also applied to 3 more elements. Why? That’s a fascinating puzzle in its own right. But what matters to me now is something else: an apparent connection between the lanthanides and the exceptional Lie group G2!

Alas, this connection remains profoundly mysterious to me, so I’m pleading for your help.

Posted at 10:15 PM UTC | Permalink | Followups (9)

May 26, 2024

Wild Knots are Wildly Difficult to Classify

Posted by John Baez

In the real world, the rope in a knot has some nonzero thickness. In math, knots are made of infinitely thin stuff. This allows mathematical knots to be tied in infinitely complicated ways — ways that are impossible for knots with nonzero thickness! These are called ‘wild’ knots.

Check out the wild knot in this video by Henry Segerman. There’s just one point where it needs to have zero thickness. So we say it’s wild at just one point. But some knots are wild at many points.

There are even knots that are wild at every point! To build these you need to recursively put in wildness at more and more places, forever. I would like to see a good picture of such an everywhere wild knot. I haven’t seen one.

Wild knots are extremely hard to classify. This is not just a feeling — it’s a theorem. Vadim Kulikov showed that wild knots are harder to classify than any sort of countable structure that you can describe using first-order classical logic with just countably many symbols!

Very roughly speaking, this means wild knots are so complicated that we can’t classify them using anything we can write down. This makes them very different from ‘tame’ knots: knots that aren’t wild. Yeah, tame knots are hard to classify, but nowhere near that hard.

Posted at 12:23 PM UTC | Permalink | Followups (5)

May 15, 2024

3d Rotations and the 7d Cross Product (Part 1)

Posted by John Baez

There’s a dot product and cross product of vectors in 3 dimensions. But there’s also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There’s nothing really like this in other dimensions.

The following stuff is well-known: the group of linear transformations of n\mathbb{R}^n preserving the dot and cross product is called SO(3)SO(3). It consists of rotations. We say SO(3)SO(3) has an ‘irreducible representation’ on 3\mathbb{R}^3 because there’s no linear subspace of 3\mathbb{R}^3 that’s mapped to itself by every transformation in SO(3)SO(3), except for {0}\{0\} and the whole space.

Ho hum. But here’s something more surprising: it seems that SO(3)SO(3) also has an irreducible representation on 7\mathbb{R}^7 where every transformation preserves the dot product and cross product in 7 dimensions!

That’s right—no typo there. There is not an irreducible representation of SO(7)SO(7) on 7\mathbb{R}^7 that preserves the dot product and cross product. Preserving the dot product is easy. But the cross product in 7 dimensions is a strange thing that breaks rotation symmetry.

There is, apparently, an irreducible representation of the much smaller group SO(3)SO(3) on 7\mathbb{R}^7 that preserves the dot and cross product. But I only know this because people say Dynkin proved it! More technically, it seems Dynkin said there’s an SO(3)SO(3) subgroup of G 2G_2 for which the irreducible representation of G 2\mathrm{G}_2 on 7\mathbb{R}^7 remains irreducible when restricted to this subgroup. I want to see one explicitly.

Posted at 4:36 PM UTC | Permalink | Followups (39)