Representations of operators in MADNESS (Multiresolution ADaptive Numerical Environment for Scientific Simulation) Gregory Beylkin University of Colorado Boulder The high level software environment for the solution of integral and differential equations, MADNESS, was developed at ORNL (with main contributions by Robert Harrison and George Fann) and successfully used for accurate computations in quantum chemistry and nuclear physics. This software was recognized by the R&D 100 Awards in 2011. I will discuss the mathematical foundation on which this package is build (i.e., representations of operators that allow efficient computations in high dimensions within user-selected accuracy) and briefly indicate some new developments.