Project Details
Projekt
Lorentzian polynomials and equality in log-concave inequalities
Subject Area
Mathematics
Term
since 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 539864509
This project builds on recent decisive progress in two different areas of mathematics. In combinatorics, Fields medalist June Huh and his collaborators have put forward the new perspective that log-concave inequalities arise from a background K\"ahler package consisting of analogs of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. This has led to the proof of long-standing conjectures in combinatorics. The theory of Lorentzian polynomials developed by Brändén and Huh can be considered as a result and further abstraction of this approach. In convex geometry, the problem of characterizing the equality cases in the Alexandrov-Fenchel inequality has been open since the foundational work the Alexandrov and Minkowski on the subject. In a recent breakthrough, Shenfeld and van Handel have completely solved this problem in the important special case of polytopes. Not only for geometric inequalities, but also for combinatorial inequalities it is natural ask for what objects equality occurs. The recently introduced powerful tools, relying on the Kähler package and Lorentzian polynomials to estabished log-concavity, turn out in many cases to be inadequate to directly provide information about the equality cases. However, the work of Shenfeld and van Handel has been recently used to characterize the equality case in Stanley's inequality for posets and a number of related problems seem now within reach. This proposal aims at a deeper understanding of several recently discovered mechanisms from the theory of Lorentzian polynomials that preserve log-concavity. Beyond the intrinsic interest, our main motivation for this investigation is to obtain precise information about the equality cases in log-concave inequalities. Primary objectives of this project are the characterization of the equality cases in the generalized Alexandrov-Fenchel inequality and Kahn-Saks inequality for convex polytopes and applications to combinatorial posets inequalities. A key feature of our proposed research is the utilization of deep connections and synergies between combinatorics, convexity, and algebraic geometry.
DFG Programme
Priority Programmes
Subproject of
SPP 2458:
Combinatorial Synergies
