Leaps-MPS: limits, levelness, and lattice points

This $170,379 federal Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences program (CFDA 47.049) will support research by the Rose-Hulman Institute of Technology focused on extending the body of knowledge in Ehrhart theory, a mathematical discipline studying the connections between integer points and volumes.

The key products and services to be delivered under this award include:

• Investigating the algebraic and enumerative properties of lattice polytopes and rational polyhedral cones, including the connection between level polytopes and h-star unimodality, as well as studying families of polytopes whose h-star polynomials converge to formal power series (Ehrhart limits).
• Studying commutative algebraic properties of a family of rational polyhedral cones called S-lecture hall cones, including investigating resolutions and associated coinvariant rings.
• Integrating the research program with undergraduate student research and training opportunities, an undergraduate mathematics conference, student travel support, a graduate student seminar series, and K-12 outreach activities to increase participation from underrepresented groups in mathematics.

This 2-year award runs from September 1, 2024 to August 31, 2026 and does not include any planned sub-awards.

Abstract

An interesting and relatively elementary result in mathematics is pick?s theorem: if you take any polygon in the two dimensional plane where the vertices all have integer coordinates, computing the area of this polygon can be done by way of a formula which simply takes in the number of integer coordinate points on the boundary of the polygon and the number of integer coordinate points in the interior of the polygon. This reduces the process of computing area to a basic counting problem, which can be more efficient when computing areas explicitly or approximating them. This is of use not only in a variety of mathematical and computing applications, but also in more general applications such as forestry surveying. If one moves beyond two dimensions, the same correspondence no longer holds, but there are many interesting connections between the integer points and volumes. This mathematical discipline is called ehrhart theory, which has been a particularly fruitful field over the past several decades. The PI will focus on extending the body of knowledge in ehrhart theory, specifically by considering the implications of algebraic property called levelness and analyzing a new concept called an ehrhart limit. The PI will integrate the research program with research and training opportunities for undergraduate students. This award also supports an undergraduate mathematics conference, undergraduate student travel to mathematics conferences and diversity-focused conferences, a seminar series featuring graduate student speakers, and k-12 outreach activities. These activities will have an emphasis on increasing participation by individuals in groups historically underrepresented in mathematics. This project will investigate algebraic and enumerative properties of lattice polytopes and rational polyhedral cones through three avenues. First, the PI will investigate the connection between level polytopes and h-star unimodality. Level polytopes are a generalization of gorenstein polytopes of which much more is known regarding connections with unimodality, including a very recent result proving a longstanding conjecture. Second, the PI will investigate families of polytopes whose h-star polynomials converge to some formal power series. Such instances are known as ehrhart limits, a relatively new notion in the literature, and the PI will focus on producing additional families which exhibit interesting ehrhart limits. Lastly, the PI will study commutative algebraic properties of a family of rational polyhedral cones, called s-lecture hall cones. Specifically, the project will investigate resolutions of these cones, as well as associated coinvariant rings. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the foundation's intellectual merit and broader impacts review criteria.- subawards are not planned for this award.