Questions and Experiments in Geometric Combinatorics

Grants and Contracts Details

Description

Lattice polytopes are of perennial interest in mathematics, serving as a source of connections between combinatorics, discrete geometry, number theory, geometry of numbers, optimization, commutative algebra, algebraic geometry, coding theory, and social choice theory. This proposal contains projects that are motivated by enumeration problems in combinatorics in which polytopes play a central role. The PI has an established record of mentoring, advising, and outreach that will continue during this award, contributing to human resources development in the mathematical sciences. Intellectual Merit: The first project in this proposal focuses on unimodality problems for Ehrhart h-star-polynomials, with particular emphasis on challenging conjectures due to Stanley, Hibi-Ohsugi, and De Loera-Haws- Koeppe. The study of Ehrhart h-star-unimodality has led to significant increases in our understanding of lattice polytopes, and focusing on these difficult conjectures will lead to further advances. The second project in this proposal involves a comprehensive study of geometric and algebraic aspects of a family of lattice simplices related to weighted projective spaces that has been the subject of intense recent study. By investigating Hilbert bases, Ehrhart h-star-vectors, Poincare series, and geometric hstar- polynomial factorizations for these lattice simplices, a deep understanding of their properties will emerge that can inform our study of lattice simplices in general. The third project in this proposal focuses on the class of generalized permutahedra and their relationship to Hopf algebras. The combinatorial enumeration of faces of these polytopes is known to be a subtle and interesting problem, and this project will create an extension of traditional face enumeration to refined face enumeration using symmetric functions and combinatorial Hopf algebras for this class of polytopes. Broader Impact: Through support for members of the PI''s research group, this proposal will contribute to the development of mathematicians with an understanding of the complexities and depth of mathematical practice. The PI will continue to support members of demographic groups under-represented in mathematics, e.g., women, members of racial groups minoritized in mathematics, individuals with disabilities, 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 via his involvement with the Mathematical Association of America and other organizations, 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 improvement of K-12 education
StatusActive
Effective start/end date7/1/206/30/23

Funding

  • National Science Foundation: $150,000.00

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