Gerd Faltings was recently awarded the Abel Prize, and there is now a number of nice video lectures explaining his work to more or less specialized audiences. I have chosen here three talks in increasing level of specialization, hoping that at least one of them will provide an interesting watch
One of the most important theorems of elementary number theory is Fermat's little theorem. It says that whenever an integer $a$ is not divisible by a prime $p$, the number $a^{p-1}-1$ is divisible by $p$. In terms of congruences we can write this statement as
RH is a precise statement, and in one sense what it means is clear, but what it is connected with, what it implies, where it comes from, can be very unobvious. Martin Huxley Previously in the RH Saga, we introduced the "Big Five" problems motivating the idea of
I'm working on writing up Chapter 3 on the Generalized Riemann Hypothesis, but in the meantime, for this weekend, I want to recommend a talk that complements the RH Saga E2 and the associated Chapter 2, and is delivered in much of the same spirit. It is a
A number of deep problems about prime numbers emerge from the simple idea of taking a polynomial and trying to understand which primes appear among its values, and in particular whether there is a finite or an infinite number. We speak then of primes of the form $n^2+1$
Every four years, researchers from around the world gather at the International Congress of Mathematicians. Fields medals are awarded, and many lectures are given by the greatest experts in the world, aimed at a general audience of mathematicians. A lecture at the ICM (typically given both as a live talk
The "Himalayan peaks" that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of
Yet another really nice talk hosted by the Clay Mathematics Institute is this talk from 2025 by Kannan Soundararajan on progress in the theory of zeta and L-functions motivated by the Riemann Hypothesis. It is in some ways even more accessible than the two earlier talks by Sarnak, but
I'm in the process of preparing next week's Chapter 2, on the Langlands program and the landscape of different kinds of L-functions. Most research talks on the Langlands program are incomprehensible unless you already have huge amounts of technical background. But I have found one
The logarithmic integral function $\mathrm{li}(x)$ is defined as \[ \mathrm{li}(x) = \int_0^{x} \frac{\mathrm{d}t}{\log(t)} \] where $\log$ is is the natural logarithm, and the integral is understood as the Cauchy principal value at $t=1$. The function $\mathrm{li}(x)$ is for $x&
The Riemann Hypothesis may be the greatest unsolved math problem of all time. In his book Zeta and L-functions of varieties and motives, Bruno Kahn calls it an unapproachable mystery. In a short YouTube clip, Terence Tao likened it to climbing a sheer cliff face, a mile high, with
For obvious reasons, it is hard to find very accessible introductions to L-functions. Here are three pieces of exposition where such an introduction is attempted. Perhaps you will find them interesting! And perhaps you can send them to your uncle in order to explain why you love L-functions.