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What should I do next semester?

Once again, apologies for the lack of updates. In my defense, I am taking almost entirely graduation requirements so that I can graduate from MIT this semester, and then I plan on taking a gap semester in the spring. I have some incomplete plans for next semester, but I thought I’d throw out the following question anyway: what should I do with all of that time?

My current plans involve going through my backlog of books and papers I haven’t had time to read and writing posts about them, but I’m sure there are plenty of other ways I could mathematically enrich my life before graduate school and I’d be very interested to hear your suggestions.

I don’t trust uncountable sets

I have a mathematical confession: I don’t trust uncountable sets.

Some time ago on MathOverflow somebody asked what a reasonable definition of “infinite permutation” would be. The first answer that comes to mind is a bijection $\mathbb{N} \to \mathbb{N}$. The set of all such bijections does form a group, but not only is it uncountably generated, it contains, as Darsh observes, a copy of every countably generated group (acting on itself by left multiplication). In particular it contains a copy of the free group on countably many generators. It also doesn’t seem to carry any natural kind of topology.

On the other hand, a much nicer candidate is the set of “compactly supported” permutations, i.e. those which fix all but finitely many elements. This countable group $S_{\infty}$ is generated by transpositions and therefore has a neat presentation given by the usual relations. I believe it’s also the largest locally finite subgroup of the full group of bijections.

I find this group much more philosophically appealing than the full group of bijections, and the reason is simple: each element of the group is computable. On the other hand, only countably many elements of the full group of bijections $\mathbb{N} \to \mathbb{N}$ are computable: the rest can’t be written down by a Turing machine. And I don’t trust anything that can’t be written down by a Turing machine.

Corollary: I don’t trust the real numbers.

Instead of explaining what I mean by this, which I don’t think I have time for today, I’ll just throw a question out to the audience: how do you feel about all this?

Exceptional structures

Recently Isabel Lugo asked about problems that are hard for intermediate values of some parameter, and in discussing the question I got to thinking about exceptional structures in mathematics such as the sporadic groups. In 2006 David Corfield asked about how “natural” the sporadic simple groups are at the n-Category cafe. In that discussion and more generally there seem to be approximately two extremes in perspective:

• Exceptional structures represent a lack of room for asymptotic behavior to occur; thus they are distractions from the “generic” case. This seems to be the case for certain exceptional isomorphisms; there are only so many groups of a particular small order. It also seems to be a good way to think about objects that behave fine in characteristic zero or high characteristic but behave badly in low characteristic, characteristic $2$ usually being the worst offender.
• Exceptional structures represent the deepest part of a theory, and the exceptional structures in different fields are often related; thus understanding exceptional structures is crucial. This seems to be the case for the octonions, which can be thought of as an underlying cause of Bott periodicity. It also seems to be a good way to think about objects related to the number $24$; John Baez tells a great story about connections between the Leech lattice, the Dedekind eta function, string theory, and elliptic curves all centered around this mysterious number.

So what do you think? Are exceptional structures accidents or miracles? (Or, as a third option: am I failing to distinguish carefully enough between interesting and uninteresting exceptional structures?)

Non-canonical isomorphisms

I find non-canonical isomorphisms very interesting, but I wish I knew more examples. To be vague, an isomorphism (perhaps in a category) is said to be non-canonical if it requires making an “arbitrary choice.” One of the reasons I find them interesting is that we often think of objects only up to isomorphism, but in order for certain things to make more sense we must distinguish between objects that are non-canonically isomorphic. Here are the examples I know of.

Dynkin diagrams and the Mahler measure problem

Funnily enough, a few days after I wrote the previous post, I was linked to a graph theory paper where one of the first results cited, which was clearly well-known to the authors, is the following remarkable generalization of what I tried to do:

Theorem: The only connected simple graphs with spectral radius less than or equal to $\ 2$ are the induced subgraphs of the Dynkin diagrams $\tilde{A}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$.

I have to admit, I really didn’t suspect that the classification result I was going after was both so simple and so interesting! Certainly there are heuristic reasons why the above classification makes sense: as I forgot to note in the previous post, there really can’t be too many vertices of degree $3$ in a graph with $\rho(G) \le 2$. But I really can’t fathom why spectral radius can be used to define the Dynkin diagrams, considering their relationship to

- binary polyhedral groups and the Platonic solids,

- the octonions (okay, this one is stretching it a little).

Anyone know any good references?

In any case, I’d like to discuss the McKee-Smyth paper because it has some interesting ideas I’d thought about independently.