Nicolas Smoot's project was approved by the FWF

Abstract: The study of integer partitions occupies a central position in additive number theory. In the last hundred years, our understanding of the arithmetic of partition functions has increased substantially. In particular, very deep properties in the form of infinite families of congruences have been found to exist. These congruence families vary enormously with respect to the difficulty in proving them. One major contributing factor to the complexity of the problem is the genus of the underlying modular curve.  Until recently, it was thought  hat all congruence families associated with a genus 0 curve could be proved using the same classical methods.  However, counterexamples have been discovered which necessitate the application of techniques originally developed for curves of nonzero genus. This drove us to develop a more general method of proof which utilizes the algebraic structure of localized integer polynomial rings, rather than the traditional free Z[X]-module structure.  We will take a survey of all conjectured partition congruence families hitherto resistant to proof, and attempt proofs using our method. We will establish the theoretical completeness of our method with respect to modular curves of genus 0, and apply our method to various congruences over genus 1 curves. We will also develop some algorithmic machinery which is needed to assist in some of the key computations.