The realizability problem is a long open problem in the theory of liquids and quantum chemistry.

Conditional on its solution several results have been derived in the theory of heterogenous materials.

The realizability problem is to realise given functions as the correlation functions of a point process.

The zero dimensional version is the truncated moment problem on N, that is given numbers m_1, .., m_n find a probability measure \mu on N such that \sum_{x \in N} x^k \mu(\{x\}) = m_k.

This seemingly easy problem was completely unsolved for n \geq 4. The results are based on the truncated moment problem on the non negative axis and the classification of the convex cone of non-negative polynomials. This is a joint work with M. Infusino, J. Lebowitz and E. Speer.