Abstract:
Suppose $\alpha$ and $\beta$ are two angles satisfying $\tan(\alpha) = r \tan(\beta) > 0$, for some rational number $r > 1$. Can both $\alpha$ and $\beta$ be rational multiples of $\pi$?

Let \[p(x,y) = x^5 - (2 y + 1) x^3 - (y^2 + 2) x^2 + y (y-1) x + y^3.\] Is it true that for every integer $y \ge 4$, $p(x, y)$ is an irreducible polynomial in $\mathbb{Q}[x]$, whereby we can determine its Galois group?

We encounter questions like this in our quest to achieve complexity classifications for counting problems. In this talk I will outline a proof to the first question (using character theory), and describe at a high level how answers to such questions lead to complexity dichotomy theorems. Then I will give an overview of the status of the Classification Program for Counting Problems, including all partition functions for graph homomorphisms, counting constraint satisfation problems, and Holant problems.