Category Archives: Mathematics

Thurston, Selberg, and Random Polynomials, Part I.

Posted on May 21, 2014 by Persiflage

Apart from everything else, you could always count on Bill Thurston to ask interesting questions. This is the first of a small number of posts which were motivated in part by figure two from this paper, and this accompanying MO … Continue reading →

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A Preview of Barbados/Bellairs

Posted on April 21, 2014 by Persiflage

This post is probably not so interesting unless you plan to travel to the Caribbean in a few weeks. The website for the conference is offline, so I thought I might update attendees on what might be happening, at least … Continue reading →

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Are Galois deformation rings Cohen-Macaulay?

Posted on April 2, 2014 by Persiflage

Hyman Bass once wrote a paper on the ubiquity of Gorenstein rings. The first time they arose in the context of Hecke algebras, however, was Barry’s Eisenstein ideal paper, where he proves (at prime level) that the completions \(\mathbf{T}_{\mathfrak{m}}\) are … Continue reading →

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The Thick Diagonal

Posted on March 14, 2014 by Persiflage

Suppose that \(F\) is an imaginary quadratic field. Suppose that \(\pi\) is a cuspidal automorphic form for \(\mathrm{GL}(2)/F\) of cohomological type, and let us suppose that it contributes to the cohomology group \(H^1(\Gamma,\mathbf{C})\) for some congruence subgroup \(\Gamma\) of \(\mathrm{GL}_2(\mathcal{O}_F)\). … Continue reading →

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The congruence subgroup property for thin groups.

Posted on March 9, 2014 by Persiflage

I finally had a chance to visit Yale, which (by various orderings) is the fanciest US university at which I had never given a talk (nor even visited). The town itself struck me, at first, as a cross between Oxford … Continue reading →

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Local crystalline deformation rings

Posted on February 8, 2014 by Persiflage

I just returned from a very pleasant conference in Puerto Rico courtesy of the Simons Foundation (general advice: if you live in Chicago, always accept invitations to conferences in January). One thing I learnt from Toby Gee was the following … Continue reading →

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Gross Fugue

Posted on January 11, 2014 by Persiflage

Here are some variations on the theme of the last post, which is also related to a problem of Dick Gross. In this post, I want to discuss weight one modular forms where the level varies in the “vertical” aspect … Continue reading →

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The mystery of the primes

Posted on January 4, 2014 by Persiflage

No, this is not the sequel to Marcus du Sautoy’s book, but rather a curious observation regarding George Schaeffer’s tables of “ethereal” weight one Katz modular eigenforms (which you can find starting on p.64 here). Let \(N\) be a positive … Continue reading →

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Daleks

Posted on December 5, 2013 by Persiflage

I’ve wanted to write a post about the new Doctor Who series for a while, but this is not that post. Instead, this post is about a Macintosh game called Daleks, which I first played on a Mac 512 (running … Continue reading →

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Local representations occurring in cohomology

Posted on November 5, 2013 by Persiflage

Michael Harris was in town for a few days, and we chatted about the relationship between my conjectures on completed cohomology groups with Emerton and the recent work of Scholze. The brief summary is that Scholze’s results are not naively … Continue reading →

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