Category Archives: Mathematics

Magma Instability

I had occasion to return to some magma scripts I wrote in 2012. I the script used a number of pre-computed auxiliary files with computations, and was a little complicated, but didn’t use anything particularly complicated. So I was really … Continue reading

Posted in Mathematics | Tagged , | 3 Comments

Clozel 70, Part II

Many years ago, Khare asked me (as I think he asked many others at the time) whether I believed their existed an irreducible motive \(M\) over \(\mathbf{Z}\) (so good reduction everywhere) with Hodge-Tate weights \([0,1,2,\ldots,n-1]\) for any \(n > 1\). … Continue reading

Posted in Mathematics | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 6 Comments

Clozel 70, Part I

I recently returned home from a trip to Paris for Clozel’s 70th birthday conference. Naturally I stayed in an airbnb downtown, and the RER B gods smiled on me with a hassle free commute for the entire week. Tekés was … Continue reading

Posted in Mathematics, Travel | Tagged , , , , , , , , , , , , , , | Leave a comment

Quadratic Reciprocity

I accidentally proved quadratic reciprocity in class today, or at least three quarters of a proof. Can you finish it off? Here’s the proof: start with a real quadratic field \(K\), and the sequence \(1 \rightarrow \mathcal{O}^{\times}_K \rightarrow K^{\times} \rightarrow … Continue reading

Posted in Mathematics, Teaching | Tagged , , | Leave a comment

Deciphering Quanta

Sometimes it is claimed that Quanta articles are so watered down of mathematical content that they become meaningless. That presents a challenge: do I understand the quanta article on my own work? Here goes: New Proof Distinguishes Mysterious and Powerful … Continue reading

Posted in Mathematics | Tagged , , , , , , | 4 Comments

What the slopes are

Let \(f\) be a classical modular eigenform of weight \(k\), for example, \(f = \Delta\). The Ramanujan conjecture states that the Hecke eigenvalues \(a_p\) satisfy the bound \(|a_p| \le 2 p^{(k-1)/2}.\) A slightly fancier but cleaner way of saying this … Continue reading

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

Check the arXiV regularly!

In a previous post, I discussed a new result of Smith which addressed the following question: given a measure \(\mu\) on \(\mathbf{R}\) supported on some finite union of intervals \(\Sigma\), under what conditions do there exist polynomials of arbitrarily large … Continue reading

Posted in Mathematics | Tagged , , , , , | Leave a comment

Potential Modularity of K3 surfaces

This post is to report on results of my student Chao Gu who is graduating this (academic) year. If \(A/F\) is an abelian surface, then one can associate to \(A\) a K3 surface \(X\) (the Kummer surface) by blowing up … Continue reading

Posted in Mathematics, Work of my students | Tagged , , , , , , , , | 2 Comments

The future is now; recap from Cetraro

I’ve just returned from the second Journal of Number Theory biennial conference in Italy. It’s always nice to get a chance to see slices of number theory one wouldn’t otherwise see at the conferences I usually go to (although this … Continue reading

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

30 years of modularity: number theory since the proof of Fermat

It’s probably fair to say that the target audience for this blog is close to orthogonal to the target audience for my talk, but just in case anyone wants to watch it in HD (and with the audio synced to … Continue reading

Posted in Mathematics | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 13 Comments