**First talk at 4:00 by Jennifer Balakrishnan (Boston University)**

**A tale of three curves**

We will describe variants of the Chabauty--Coleman methodand quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

**Second talk at 5:15 by Dimitris Koukoulopoulos (University of Montreal)**

**On the Duffin-Schaeffer conjecture**

Let S be a sequence of integers. We wish to understand how well we can approximate a ``typical'' real number using reduced fractions whose denominator lies in S. To this end, we associate to each q in S an acceptable error delta_q>0. When is it true that almost all real numbers (in the Lebesgue sense) admit an infinite number of reduced rational approximations a/q, q in S, within distance delta_q? In 1941, Duffin and Schaeffer proposed a simple criterion to decided whether this is case: they conjectured that the answer to the above question is affirmative precisely when the series sum_{q\in S} \phi(q)\delta_q diverges, where phi(q) denotes Euler's totient function. Otherwise, the set of ``approximable'' real numbers has null measure. In this talk, I will present recent joint work with James Maynard that settles the conjecture of Duffin and Schaeffer.