Counting integral points in affine thin sets

Many problems in number theory can be framed as questions about counting integral solutions to a Diophantine equation, within a box of growing size. If there are very few, or very many variables, certain methods gain an advantage, but sometimes there is extra structure that can be exploited as well. For example: let $f$ be a given polynomial with integer coefficients in $n$ variables. How many values of $f$ are a perfect square? A perfect cube? These questions arise in a variety of specific applications, and also in the context of a general conjecture of Serre on counting points in thin sets. In this talk, we will survey recent progress on counting points in thin sets, including the resolution of several central questions. In the context of affine thin sets of type II, we will describe a new sieve method that is insensitive to the singularity of the underlying hypersurface. This includes joint work with Dante Bonolis and Katharine Woo.