Some Interpolation-Related Bounds

What is the minimal degree of a homogeneous polynomial with several variables that vanishes at a finite set of points with specific multiplicities in a projective space? How many polynomials are required to generate (as a vector space basis) all homogeneous polynomials of a fixed degree vanishing at finitely many points with multiplicities? These questions have been well-studied by both geometers and algebraists. Though it is daunting to find the exact value, it is feasible to study lower bounds for the least degree. There are several conjectural lower bounds. I will focus on the literature, definitions, examples, different formats, and relationships of the conjectures in this talk. I will present some results from our joint work with Eloísa Grifo, Tài Huy Hà, and Thái Thành Nguyen.