天美传媒

Vindas Mel茅ndez Co-Authors Paper on Polynomial Properties in Polyhedral Geometry

Andr茅s R. Vindas Mel茅ndez, assistant professor of mathematics at 天美传媒, co-authored a research paper that explores the behavior of local h-polynomials, a mathematical tool used in the study of lattice-point enumeration of high-dimensional geometric objects known as lattice polytopes. The paper, 鈥�,鈥� was published in Beitr盲ge zur Algebra und Geometrie/Contributions to Algebra and Geometry and is the result of a collaboration that followed a 2022 workshop hosted by the American Institute of Mathematics.

Lattice polytopes are geometric shapes that sit within a structured grid, making them useful in combinatorial and discrete geometry. Vindas Mel茅ndez鈥檚 study focuses on a special class of these polytopes that can be represented through matrices in Hermite normal form, a method for simplifying integer matrices. Specifically, Vindas Mel茅ndez and his collaborators examined cases where the matrix contains only one non-zero row, analyzing how the coefficients of the associated polynomials behave as the size of the polytope increases.

鈥淭his kind of research is useful in the mathematical areas of combinatorics and discrete geometry, particularly in understanding how shapes relate to counting problems,鈥� says Vindas Mel茅ndez. 鈥淭his research direction will give students who work with me an opportunity to pursue computational and experimental avenues to theoretical mathematics research.鈥�

The results of this study contribute to a growing area of research examining how polynomial invariants鈥攎athematical properties that remain unchanged under transformations鈥攁re distributed within lattice polytopes. Vindas Mel茅ndez recently co-organized a workshop at the Institute for Pure and Applied Mathematics at UCLA, where researchers discussed broader questions related to computational interactions between algebra, combinatorics and discrete geometry.

This summer, Vindas Mel茅ndez anticipates working with about six Harvey Mudd students on related projects to continue exploring combinatorially defined polytopes.

Through this research, Vindas Mel茅ndez, his collaborators and student researchers are expanding the understanding of fundamental mathematical properties, reinforcing Harvey Mudd鈥檚 commitment to innovative and evolving education.