Rational points on rational surfaces
Anthony Várilly-Alvarado (Rice)

Abstract: I will discuss some of the driving questions in the area of Rational points on algebraic varieties (rational points can be thought of as solutions over fields of number-theoretic interest to systems of homogeneous polynomial equations). I will focus on the case of smooth rational surfaces, and discuss some results concerning the arithmetic of del Pezzo surfaces of degree 1. I will also explain how these results help complete a qualitative picture of basic arithmetic phenomena among smooth rational surfaces. Along the way I will go over concepts like weak approximation and (time permiting) Brauer-Manin obstructions; I will not assume previous knowledge of them.