Category Archives: Mathematics

Random p-adic Matrices

Posted on January 23, 2013 by Persiflage

Does anyone know if the problem of random matrices over (say) \(\mathbf{Z}_p\) have been studied? Here I mean something quite specific. One could do the following, namely, since \(\mathbf{Z}_p\) is compact with a natural measure, look at random elements in … Continue reading →

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NT Seminar: A haruspicy

Posted on January 18, 2013 by Persiflage

Following JSE’s advice, I will blog on something that I know absolutely nothing about. Apologies in advance for mathematical errors! SLM gave a number theory seminar this week about the first Betti number of \(\Gamma(n)\) — as \(n\) varies — … Continue reading →

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Number theory and 3-manifolds

Posted on January 13, 2013 by Persiflage

It used to be the case that the Langlands programme could be used to say something interesting about arithmetic 3-manifolds qua hyperbolic manifolds. Now, after the work of Agol, Wise, and others has blown the subject to smithereens, this gravy … Continue reading →

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Classic Papers in Number Theory

Posted on December 27, 2012 by Persiflage

One of my students came to me with the idea of having a reading course on “classic papers in number theory”. The idea is for everyone to spend the week reading a particular paper, and then have one student lead … Continue reading →

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The Two Cultures of Mathematics: A Rebuttal

Posted on December 12, 2012 by Persiflage

Gowers writes thoughtfully about combinatorics here, in an essay which references Snow’s famous lectures (or famous amongst mathematicians – I’ve never met anyone else who has ever heard of them). The trouble, however, starts (as it often does) with the … Continue reading →

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Hilbert Modular Forms of Partial Weight One, Part II

Posted on December 8, 2012 by Persiflage

Anyone who spends any time thinking about Hilbert modular forms of partial weight one — see part I — should, at some point, wonder whether there actually exist any examples, besides the “trivial” examples arising as inductions of Grossencharacters. Fred … Continue reading →

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There are no unramified abelian extensions of Q (almost)

Posted on November 27, 2012 by Persiflage

In my class on modularity, I decided to explain what Wiles’ argument (in the minimal case) would look like for \(\mathrm{GL}(1)/F\). There are two ways one can go with this. On the one hand, one can try to prove (say) … Continue reading →

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Jacobi by pure thought

Posted on October 26, 2012 by Persiflage

JB asks whether there is a conceptual proof of Jacobi’s formula: \(\Delta = q \prod_{n=1}^{\infty}(1 – q^n)^{24}\) Here (to me) the best proof is one that requires the least calculation, not necessarily the “easiest.” Here is my attempt. We use … Continue reading →

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En Passant

Posted on October 24, 2012 by Persiflage

Several times in NYC, I’ve had the chance to visit Eataly, an italian food court/upscale delicatessen run by Mario Batali. You can either sit down at one of the various restaurants for an antipasto plate with a glass of wine, … Continue reading →

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Remarks on Buzzard-Taylor

Posted on October 18, 2012 by Persiflage

Let \(\rho: G_{\mathbf{Q},S} \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p)\) be continuous and unramified at \(p\). The Fontaine-Mazur conjecture predicts that \(\rho\) has finite image and is automorphic. Buzzard and Taylor proved this result under the assumption the natural assumption that \(\rho\) is odd, that … Continue reading →

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