Kolchin's problems

Kolchin's problems are a set of unsolved problems in differential algebra, outlined by Ellis Kolchin at the International Congress of Mathematicians in 1966 (Moscow)

Kolchin Catenary Conjecture

The Kolchin Catenary Conjecture is a fundamental open problem in differential algebra related to dimension theory.

Statement

"Let ${\displaystyle \Sigma }$ be a differential algebraic variety of dimension ${\displaystyle d}$. By a long gap chain we mean a chain of irreducible differential subvarieties ${\displaystyle \Sigma _{0}\subset \Sigma _{1}\subset \Sigma _{2}\subset \cdots }$ of ordinal number length ${\displaystyle \omega ^{m}\cdot d}$."

Given an irreducible differential variety ${\displaystyle \Sigma }$ of dimension ${\displaystyle d>0}$ and an arbitrary point ${\displaystyle p\in \Sigma }$ , does there exist a long gap chain beginning at ${\displaystyle p}$ and ending at ${\displaystyle \Sigma }$?

The positive answer to this question is called the Kolchin catenary conjecture.^[1][2][3][4]