Interpolation for points in $\mathbb{P}^N, N\geq 2$

Interpolation problems study hypersurfaces in projective space passing through prescribed sets of points. Classically, one asks how many independent conditions a collection of points imposes on hypersurfaces of a fixed degree, a question that can be studied algebraically via homogeneous ideals and their Hilbert functions. In this talk, I will begin with the classical interpolation problem for reduced points and introduce the algebraic framework used to study it. I will then move to fat point schemes, where points are assigned multiplicities and hypersurfaces are required to vanish to higher order. In this setting, interpolation problems naturally lead to symbolic powers of ideals and containment relations between symbolic and ordinary powers. I will conclude by discussing open questions, including potential connections between interpolation problems and combinatorial structures such as matroids.