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 $\mathrm{\Sigma }$ be a differential algebraic variety of dimension $d$ By a long gap chain we mean a chain of irreducible differential subvarieties ${\mathrm{\Sigma }}_0\subset {\mathrm{\Sigma }}_1\subset {\mathrm{\Sigma }}_2\subset \cdots$ of ordinal number length ${\omega }^m\cdot d$."

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

The positive answer to this question is called the Kolchin catenary conjecture.