Polynomial relations among kappa classes on the moduli space of stable curves – 1

In these lectures we will give an introduction to properties of naturally defined cohomology classes over the moduli space of genus $g$ stable curves with n labeled points. In particular, we will focus on the kappa classes defined first by Mumford then in the form we need by Arbarello and Cornalba. We will describe an elementary construction, from joint work with Maxim Kazarian, of an infinite collection of universal, i.e. independent of $(g,n)$, polynomials in the kappa classes which we conjecture to vanish in a range depending on $g$ and $n$. Recently half of these vanishing results were proven by Chidambaram, Garcia-Failde, and Giacchetto.