Reflexive Polytopes, Lecture Hall Partitions, and Euler-Mahonian Distributions

Grants and Contracts Details


Overview. The PI will conduct three research projects regarding Ehrhart h*-vectors for integrally closed reflexive polytopes, Ehrhart y*-vectors for reflexive polytopes, and Euler-Mahonian distributions, respectively. All of these projects fundamentally involve integer point enumeration for Gorenstein polyhedral cones, yet draw on different sets of tools and their relationships. The objectives of this project are: to clearly explain the relationship between reflexive simplices, integral closure, and Ehrhart h*-unimodality; to identify the geometric and combinatorial significance of Ehrhart- non-negativity for reflexive polytopes; and to develop a unified approach to the study of Euler- Mahonian distributions. The methods of these projects will include a combination of traditional proof techniques and computational experimentation. The PI will continue with, and build upon, his mentoring and advising of postdoctoral researchers, graduate students, and undergraduate students. Intellectual merit. While each of the three research projects in this proposal have at their heart a problem of enumeration, the core merit of these projects is the interaction of the integer point enumeration with related tools and techniques. For example, questions about unimodality of integer sequences lead to connections between discrete geometry, commutative algebra, and the representation theory of Lie algebras. The study of triangulations of cones leads to connections between commutative algebra involving toric ideals and Grobner bases. One of the projects attempts to unify, at a theoretical level in the context of Euler-Mahonian distribution identities, integer point enumeration for 0/1-cubes, colored descent representations of wreath products on coinvariant algebras, enumerative properties of lecture hall partitions, and other algebraic and combinatorial structures. The interactions of established techniques in these contexts are interesting, fundamental, and as-yet undeveloped. Broader Impact. Through support for research activity by the PI, this award will impact the mathematiciansin- training mentored by the PI, including a postdoctoral researcher, graduate students, and undergraduate students. Thus, this proposal will contribute to the development of mathematicians with an understanding of the complexities and depth of mathematical practice. To the greatest extent possible, the PI will continue to support members of demographic groups under-represented in mathematics, e.g., women, members of racial groups minoritized in mathematics, the disabled, and students from Appalachian Kentucky counties. Members of the PI's research group will also receive training from the PI on topics related to mathematics education and outreach. Through service to the mathematical community, such as his service as a member-at-large of the American Mathematical Society Committee on Education, and his service as the Editor-in-Chief of the American Mathematical Society blog On Teaching and Learning Mathematics, the PI will continue contributing to national efforts to improve postsecondary mathematics education. As a co-director of the Central Kentucky Mathematical Circles, an outreach program for K-12 students, the PI will contribute to local efforts to improve K-12 STEM education.
Effective start/end date3/11/163/10/18


  • National Security Agency: $40,000.00


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