Wachspress Coordinates: a bridge between Algebra, Geometry and Combinatorics

• Applicant: Dr. Martin Winter
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539851419

In this project we conduct a detailed study of Wachspress coordinates and their related objects, most notably, the Wachspress variety, Wachspress map and Izmestiev matrix. Barycentric coordinates are a classical tool in geometric modelling, enabling the interpolation of data given at the vertices of a simplex. Generalized barycentric coordinated (GBCs) extend this notion to polytopes of general combinatorics, which in turn lead to novel solutions in applications to mesh parametrization/deformation, image warping and finite element analysis. Wachspress coordinates were initially introduced as a first family of rational GBCs. They were later found to be the unique rational GBCs of lowest possible degree, already hinting at their distinguished algebraic characteristic. Other characterizations, for example in terms of cone volumes or as Taylor coefficients of certain volume functionals, emphasize a parallel convex geometric nature. Over the last years Wachspress coordinates continued to re-emerge in contexts unrelated to their initial use case and found surprising applications across mathematics (for computing Segre classes of monomial scheme), statistics (in the study of moment varieties and Bayesian statistics) and physics (in the form of positive geometries and related to the amplituhedron). Recently, the principal investigator found Wachspress coordinates to also emerge from a higher rank polytope parameter that we call the Izmestiev matrix - a matrix of Lorentzian signature whose spectral properties encode the geometry and combinatorics of the polytope. This already led to new connections to and between rigidity theory, representation theory and spectral graph theory. The many non-trivially equivalent definitions of Wachspress coordinates elevated them from a mere tool to being recognized as a bridge between disciplines. It is expected that a unifying explanation for their ubiquity will reveal deep connections between algebraic and convex geometry as well as spectral graph theory. The goal of this project is therefore to develop a multifaceted understanding of these constructions, both as objects of intrinsic interest, and as tools for a wide range of applications. The project sets out with a detailed study of the algebra of Wachspress coordinates through the Wachspress variety. It follows an investigation of the Izmestiev matrix, its derived spectral quantities, and the mechanism by which those give rise to the properties observed for the other Wachspress objects. We then use our results to execute approaches to several open problems in polyhedral geometry, combinatorics and rigidity. We prove or disprove the conjectured injectivity of the Wachspress map, and we develop a new algorithm for deciding the polytopality of simplicial spheres based on the existence of the Wachspress variety. Eventually, we explore generalizations of the Wachspress objects to a wider setting, including general projections between polytopes and non-geometric contexts.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies