Log-concave Inequalities in Combinatorics and Order Theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

This project focuses on the fundamental mathematical phenomena that emerge as a result of the underlying structure of objects that are often difficult to discern. Unimodality is a remarkable example of such a phenomenon, which is characterized by the presence of a single maximum or mode in a statistical distribution. This phenomenon has been observed in various objects, including student grade distributions and the frequency of earthquakes at a specific location. The project aims to analyze the precise mechanism of the emergence of such phenomena using approaches based on recent advances and techniques in combinatorics, probability, and order theory. The PI will mentor students as part of this project. More technically, the project deals with log-concave inequalities and correlation inequalities in combinatorial objects and their connections to the underlying combinatorial structure. Significant advancements have been made in this field in recent years, particularly with the solutions to the Heron-Rota-Welsh Conjecture and Mason's Conjecture in matroid theory being the most prominent examples of progress. The project aims to explore the combinatorial nature of these inequalities by developing purely combinatorial tools to generalize and strengthen them to match their equality conditions. On the order theory side, the project seeks to use these new insights to establish log-concave and correlation inequalities, which historically were crucial elements in deriving the best-known bound to the 1/3-2/3 Conjecture in order theory. The employed tools include a combination of classical tools from these fields, such as FKG-type inequalities in probability and mixed volume inequalities in geometry, along with some new tools such as Lorentzian polynomials and the combinatorial atlas method. 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.