Varieties are the solution sets of polynomial equations in complex projective space. In the 1960s, Grothendieck defined the Hilbert scheme to parameterize varieties in a fixed projective space. When the varieties being parameterized are points on a surface, the Hilbert scheme is smooth and has found significant applications beyond algebraic geometry. However, much less is known when surfaces are replaced by threefolds. I will outline a broad program to study the structure of the Hilbert scheme of points on a threefold, aiming to extend some of the connections observed in the surface case to threefolds.

On the other hand, to better parameterize higher-dimensional objects, Alexeev and Knutson proposed an alternative to the Hilbert scheme called the Branch stack and conjectured its projectivity. I will describe my work on proving this conjecture, thereby enabling the Branch stack to be used in a variety of applications.