{"id":13920,"date":"2025-03-12T11:38:07","date_gmt":"2025-03-12T18:38:07","guid":{"rendered":"https:\/\/www.hmc.edu\/about\/?p=13920"},"modified":"2025-03-12T11:55:33","modified_gmt":"2025-03-12T18:55:33","slug":"vindas-melendez-co-authors-paper-on-polynomial-properties-in-polyhedral-geometry","status":"publish","type":"post","link":"https:\/\/www.hmc.edu\/about\/2025\/03\/12\/vindas-melendez-co-authors-paper-on-polynomial-properties-in-polyhedral-geometry\/","title":{"rendered":"Vindas Mel\u00e9ndez Co-Authors Paper on Polynomial Properties in Polyhedral Geometry"},"content":{"rendered":"\n<p>Andr\u00e9s R. Vindas Mel\u00e9ndez, assistant professor of mathematics at Harvey Mudd College, 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, \u201c<a href=\"https:\/\/link.springer.com\/article\/10.1007\/s13366-025-00784-z\" target=\"_blank\">Local h-polynomials for one-row Hermite normal form simplices<\/a>,\u201d was published in <em>Beitr\u00e4ge zur Algebra<\/em> <em>und<\/em> <em>Geometrie\/Contributions to Algebra and Geometry<\/em> and is the result of a collaboration that followed a 2022 workshop hosted by the American Institute of Mathematics.<\/p>\n\n\n\n<p>Lattice polytopes are geometric shapes that sit within a structured grid, making them useful in combinatorial and discrete geometry. Vindas Mel\u00e9ndez\u2019s 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\u00e9ndez 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.<\/p>\n\n\n\n<p>\u201cThis kind of research is useful in the mathematical areas of combinatorics and discrete geometry, particularly in understanding how shapes relate to counting problems,\u201d says Vindas Mel\u00e9ndez. \u201cThis research direction will give students who work with me an opportunity to pursue computational and experimental avenues to theoretical mathematics research.\u201d<\/p>\n\n\n\n<p>The results of this study contribute to a growing area of research examining how polynomial invariants\u2014mathematical properties that remain unchanged under transformations\u2014are distributed within lattice polytopes. Vindas Mel\u00e9ndez 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.<\/p>\n\n\n\n<p>This summer, Vindas Mel\u00e9ndez anticipates working with about six Harvey Mudd students on related projects to continue exploring combinatorially defined polytopes.<\/p>\n\n\n\n<p>Through this research, Vindas Mel\u00e9ndez, his collaborators and student researchers are expanding the understanding of fundamental mathematical properties, reinforcing Harvey Mudd\u2019s commitment to innovative and evolving education.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Andr\u00e9s R. Vindas Mel\u00e9ndez, assistant professor of mathematics at Harvey Mudd College, co-authored a research paper that explores the behavior [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":13922,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14,967,22,26],"class_list":["post-13920","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-faculty","category-general-feed","category-mathematics","category-research"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/posts\/13920","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/comments?post=13920"}],"version-history":[{"count":3,"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/posts\/13920\/revisions"}],"predecessor-version":[{"id":13925,"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/posts\/13920\/revisions\/13925"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/media\/13922"}],"wp:attachment":[{"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/media?parent=13920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.hmc.edu\/about\/wp-json\/wp\/v2\/categories?post=13920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}