The interpolation problem: When can you pass a curve of a given type through N random points in space?

TL;DR

This paper investigates the conditions under which a curve of a specific type can pass through N random points in space, connecting classical interpolation problems with modern geometric concepts.

Contribution

It provides a comprehensive analysis of the interpolation problem, linking it to advanced topics like moduli spaces, deformation theory, and the study of smooth versus singular objects.

Findings

Characterization of when curves can interpolate given points

Connections between interpolation and moduli space properties

Insights into deformation and singularity in geometric objects

Abstract

The interpolation problem is a natural and fundamental question whose roots trace back to ancient Greece. The story is long and rich, with many chapters, and a complete solution has been obtained only recently. Exploring it leads us on a tour through a number of general themes in geometry. This concrete problem motivates fundamental concepts such as moduli spaces and their properties, deformation theory, normal bundles, and more. Questions about smooth objects lead us to consider singular (non-smooth) objects, and in fact these smooth objects are studied by instead focusing on somehow simpler "non-smooth" objects, and then deforming them.