Author Archives: Persiflage

Simons Annual Meeting

Posted on December 23, 2021 by Persiflage

The last time I traveled for math was when I gave the Coble lectures at UIUC pre-pandemic (at least pre-pandemic as far as the US goes). A few months ago it seemed like one could begin to start traveling again, … Continue reading →

Posted in Mathematics, Travel | 3 Comments

Schur-Siegel-Smyth-Serre-Smith

Posted on November 25, 2021 by Persiflage

If \(\alpha\) is an algebraic number, the normlized trace of \(\alpha\) is defined to be \( \displaystyle{T(\alpha):=\frac{\mathrm{Tr}(\alpha)}{[\mathbf{Q}(\alpha):\mathbf{Q}].}}\) If \(\alpha\) is an algebraic integer that is totally positive, then the normalized trace is at least one. This follows from the AM-GM … Continue reading →

Posted in Mathematics | Tagged , , , , , | 5 Comments

Polymath Proposal: 4-folds of Mumford’s type

Posted on August 24, 2021 by Persiflage

Let \(A/K\) be an abelian variety of dimension \(g\) over a number field. If \(g \not\equiv 0 \bmod 4\) and \(\mathrm{End}(A/\mathbf{C}) = \mathbf{Z}\), then Serre proved that the Galois representations associated to \(A\) have open image in \(\mathrm{GSp}_{2g}(\mathbf{Z}_p)\). The result … Continue reading →

Posted in Mathematics | Tagged , , , , , , , , , , , , | 2 Comments

59,281

Posted on August 19, 2021 by Persiflage

The target audience of this blog (especially the mathematics) is usually professional mathematicians in the Langlands program. I do sometimes, however, have posts suitable for a broader mathematical audience. Very rarely though do I have anything (possibly) interesting to say … Continue reading →

Posted in Mathematics | Tagged , , , , , , , , | 14 Comments

Divisors near sqrt(n)

Posted on June 8, 2021 by Persiflage

Analytic Number Theory Alert! An even more idle question than normal (that’s because it comes from twitter). Alex Kontorovich noted with pleasure the following pictorial representation of the integers from a Veritasium youtube video, where prime numbers are represented by … Continue reading →

Posted in Mathematics | Tagged , , , | 5 Comments

Potential Automorphy for GL(n)

Posted on April 29, 2021 by Persiflage

Fresh on the arXiv, a nice new paper by Lie Qian proving potential automorphy results for ordinary Galois representations \(\rho: G_F \rightarrow \mathrm{GL}_n(\mathbf{Q}_p)\) of regular weight \([0,1,\ldots,n-1]\) for arbitrary CM fields \(F\). The key step in light of the 10-author … Continue reading →

Posted in Mathematics | Tagged , , , , , , , , , , , , , , , , , , | 1 Comment

Don’t cite my paper!

Posted on April 26, 2021 by Persiflage

The process of publishing a paper is an extremely long one, and it is not atypical to take several years from the first submission to the paper finally being accepted. The one part of the process that happens extremely quickly, … Continue reading →

Posted in Rant | Tagged , , | 18 Comments

The Arbeitsgemeinschaft has returned!

Posted on March 26, 2021 by Persiflage

An update on this post; the Arbeitsgemeinschaft on derived Galois deformation rings and the cohomology of arithmetic groups will now be taking place the week of April 5th. Here is some practical information if you are curious. Is there somewhere … Continue reading →

Posted in Mathematics | Tagged , , | 2 Comments

Test Your Intuition: p-adic local Langlands edition

Posted on March 9, 2021 by Persiflage

Taking a page from Gil Kalai, here is a question to test your intuition about 2-dimensional crystalline deformation rings. Fix a representation: \(\rho: G_{\mathbf{Q}_p} \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)\) after twisting, let me assume that this representation has a crystalline lift of weight … Continue reading →

Posted in Mathematics | Tagged , , | 12 Comments

Fermat Challenge

Posted on February 21, 2021 by Persiflage

A challenge inspired from a question of Doron Zeilberger. Do there exist arbitrarily large integers \(n\) with the following property: There exists an ordered field \(F\) such that \(x^n+ y^n = z^n\) has solutions in \(F\) with \(xyz \ne 0\). … Continue reading →

Posted in Mathematics | Tagged , , | 6 Comments