Polynomial Endoscopy: What can you know if you can't find the roots?
This talk is about polynomial equations and their roots, a topic with a long history. We will focus on the period after the 16th century discovery of how to solve equations of degree 3 and 4. It quickly became clear that degree 5 was much more difficult, leading to the question of what might be said about the roots without actually finding them. We will pay particular attention to Descartes's Rule of Signs and the polynomial discriminant.