The Brown University pattern theory group is working with the belief that the world is complex, and to understand it, or a part of it, requires realistic representations of knowledge about it. We create such representations using a mathematical formalism, pattern theory, that is compositional in that the representations are built from simple primitives, combined into (often) complicated structures according to rules that can be deterministic or random. This is similar to the formation of molecules from atoms connected by various forms of bonds. Pattern theory is transformational in that groups or semigroups of transformations operate on the primitives. These transformations express the invariances of the worlds we are looking at. Pattern theory is variational in that it describes the variability of the phenomena observed in different applications in terms of probability measures that are used with a Bayesian interpretation. This leads to inferences that will be realized by computer algorithms. Our aim is to realize them through codes that can be executed on currently available hardware.