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
Status | Finished |
---|---|
Effective start/end date | 7/1/20 → 6/30/24 |
Funding
- National Science Foundation: $150,000.00
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