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Cyclotomic integers, fusion categories, and subfactors (March)
Frank Calegari, Scott Morrison, and I recently uploaded to the arxiv our paper Cyclotomic integers, fusion categories, and subfactors. In this paper we give two applications of cyclotomic number theory...
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The Weil Conjectures: The approach via the Standard Conjectures
The aim of this post is to outline a strategy for proving the Weil conjectures, proposed by Grothendieck and others. This strategy is incomplete; at various stages, we will need to assume conjectures...
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Lattices and their invariants
This post started out as an exposition on the monster Lie algebra, but it grew out of control, so I’m hacking off a chunk. Here, I’ll describe the basics of integer lattices. Lattices show up in many...
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Divergent sums and the class number formula
Over on MathOverflow, we’ve had a bunch of discussions about the class number formula. In particular, Keith Conrad pointed out a paper of Orde which gives a beautiful nonsense proof of the class number...
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The Brauer Groupoid
Recall that the Brauer group of a field k consists of central simple algebras over k up to Morita equivalence with the group operation given by tensor product. For example, the Brauer group of the real...
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Farey fractions, Ford circles, and SL_2.
The topic of this post came up during a conversation with some physicists about the fractional quantum Hall effect (which is quite fascinating, but I don’t feel particularly qualified to discuss). I...
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Rationality of the zeta function mod p
Here’s a neat argument about counting points that you could present at the end of a second course in number theory. I’m sure it’s not original, but, hey, that’s what blogs are for! Let be a smooth...
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Fun with y^2=x^p-x
Here’s a basic example that comes up if you work with elliptic curves: Let be a prime which is . Let be the elliptic curve over a field of characteristic . Then has an endomorphism . It turns out that,...
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ABC conjecture rumor
There is a rumor circulating here in Japan, to the effect that S. Mochizuki has proved the ABC conjecture. My understanding is that blogs are for spreading such things. Apparently, he had predicted...
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I just can’t resist: there are infinitely many pairs of primes at most...
Everyone by now has heard about Zhang’s landmark result showing that there are infinitely many pairs of primes at most 70000000 apart. His core result is that if a set of 3.5 * 10^6 (corrected, thanks...
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