Professor of Mathematics, UC Santa Cruz. Number theory, representation theory, visualization. Author: Illustrated Theory of Numbers. Guggenheim Fellow 2020.
), I thought I'd write a thread about the Langlands program. There are lots of places to *really* learn about it, but I'll try to do my best in a briefer thread... (1/n)
Prehistory: Since Euler (eulerarchive.maa.org) and Dirichlet, and really taking off in the 19th century, we have understood prime numbers through the analytic (a.k.a. fancy calculus) study of L-functions, among which the simplest example is the Riemann zeta function.
Mar 5, 2021 • 11 tweets • 2 min read
The latest great paper on the ArXiv is a solution of Hilbert's 12th problem for totally real fields, by @samit_dasgupta and Mahesh Kakde. I know Samit well, but I don't know the methods (p-adic L-functions etc.) well. In any case, I'll try to say why this is awesome. (1/n)
The first great thing is the great introduction, about 12 pages long. Section 1.4 gives a quick history and reflection/perspective, which does a better job than this thread could. (2/n)
Dec 1, 2020 • 7 tweets • 2 min read
A remarkable result on the ArXiv tonight, due to Kedlaya, Kolpakov, Poonen, and Rubinstein, at arxiv.org/pdf/2011.14232…. One of the highlights is that they fully classify tetrahedra (= triangular pyramids) with rational dihedral angles. Here's what they prove... (1/n)
A tetrahedron is a 3-dimensional shape built out of 4 triangular faces. These faces meet at 4 vertices, and 6 edges. At each edge, two faces meet at an angle. These are called the dihedral angles of the tetrahedron. (2/n)
