Over the past ten years, lecture hall partitions have emerged as fundamental structures in combinatorics and number theory, leading to new generalizations and new interpretations of several classical theorems. This project takes a geometric view of lecture hall partitions and uses polyhedral geometry to investigate their remarkable properties.

The solution of linear diophantine inequalities is at the heart of problems in many areas of mathematics. In recent years there has been an evolution of mathematical techniques and software tools to solve them. Nevertheless, many important problems remain beyond the scope of these methods. The focus of this proposal is the enumeration of combinatorial structures defined by linear diophantine constraints. It includes an investigation of the structure of these families and their relationship to equinumerious families with intrinsically different characterizations. A primary goal is the development of mathematical techniques for diophantine enumeration that are able to exploit the symmetry, recursion, and structure appearing in combinatorial families. One aspect will be the implementation of the techniques developed in software that can be downloaded from the web.