Strong Fermion Interactions in Fractional Quantum Hall States : Correlation Functions Introduction to Quantum Hall States The Composite Fermion Hierarchy and Justification of the CF Approach Correlation Diagrams and the Algebra of Correlation Functions Invariant Theoretic Essentials Constructions of Correlation Diagrams and Their Correlation Functions Trial Wave Functions for Systems of Fermions in IQL State Correlation Functions for Some Configurations with Quasi Electrons and Related Open Questions This monograph presents an intuitive theory of trial wave functions for strongly interacting fermions in fractional quantum Hall states. The correlation functions for the proposed fermion interactions follow a novel algebraic approach that harnesses the classical theory of invariants and semi invariants of binary forms. This approach can be viewed as a fitting and far reaching generalization of Laughlin’s approach to trial wave functions. Aesthetically viewed it illustrates an attractive symbiosis between the theory of invariants and the theory of correlations. Early research into numerical diagonalization computations for small numbers of electrons shows strong agreement with the constructed trial wave functions. The monograph offers researchers and students of condensed matter physics an accessible discussion of this interesting area of research