The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will introduce this object and review some well-known results when X is a curve or a surface. The main focus of this talk will be on the case when X is a threefold. Specifically, I will present some tantalizing open questions and delve into describing the structure of the smooth points on Hilb^d(X). If time permits, I will also discuss the mildly singular points of this Hilbert scheme. This is all joint work with Joachim Jelisiejew and Alessio Sammartano.

Speaker’s website: https://sites.google.com/view/ritvikramkumar