#### Hall 123

#### Sums of rational cubes, Selmer groups, and cubic forms

#### Aditya Karnataki

##### CMI

*It has been known for long which integers can be written as sums of rational squares. In contrast, there does not seem to be a simple pattern followed by the integers that are sums of rational cubes. Recently, Alpöge, Bhargava, and Shnidman showed that the density of integers expressible as the sum of two rational cubes is strictly positive and strictly less than 1. We will present a short review of their beautiful techniques and make some comments on related questions.*

Done