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 [...]

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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 . The set of all such bijections does form a group, but not only is it uncountably generated, it [...]

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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 [...]

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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 [...]

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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 [...]

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