This research proposal, continuing various strands of the investigator's work, is related to discrete and continuous volume computation for rational polytopes. Here continuous volume refers to the usual (relative) volume of the polytope, while discrete volume refers to the number of integral points in the polytope, often as a function of additional parameters. Examples of the latter are the Ehrhart (quasi-)polynomials and vector partition functions. Continuous and discrete volume computation for polytopes has been of great interest in recent years, partly because of applications to many mathematical fields, some of which seem distant from discrete and computational geometry: number theory, commutative algebra, algebraic geometry, optimization, representation theory, statistics, and computer science.
The goal of this project is to develop useful theoretical and computational methods for volume computation for polytopes. The PI proposes four concrete lines of problems to work on, including variations of the Birkhoff polytope, vector partition functions, growth series of lattices, and inside-out polytopes and their application to classical enumerative combinatorial problems. All of the proposed work has a computational focus. The investigator has a track record of sustained and serious effort both in outreach to students at all levels (secondary, undergraduate, and graduate), and in building institutions in which discrete and computational geometry can grow. The investigator will continue to attract students into these fields and nurture the careers of students and young researchers.
