Project Details
Projekt
Combinatorial and probabilistic aspects of symmetric edge and cosmological polytopes
Subject Area
Mathematics
Term
since 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 539865598
Undoubtedly, polytopes do not only belong to the central objects studied in contemporary research in geometry and combinatorics, but they are also important for applications in a multitude of areas, including optimization, theoretical physics, mathematical statistics, computational geometry, and machine learning, among others. Two prominent classes of lattice polytopes that are interesting not only for their intrinsic combinatorial and geometric properties but also due to their multiple applications to e.g., metric space theory and in particular physics, are symmetric edge polytopes and cosmological polytopes that are associated to graphs. In the proposed research program we study these two classes of polytopes (and further generalizations) from a deterministic and a probabilistic perspective. The objectives can be summarized as follows: (a) We propose several ways to extend the definition of classical symmetric edge polytopes by replacing the underlying graph by a higher-dimensional simplicial complex. We investigate fundamental geometric and combinatorial properties of the resulting lattice polytopes, such as the dimension, the f-, h-, h^*- and the gamma-vector. To achieve this we are interested in triangulations of these polytopes and ask for the existence of a regular unimodular triangulation (equivalently a squarefree Gröbner basis of the toric ideals). (b) We investigate combinatorial, geometric and probabilistic properties of randomly generated lattice polytopes such as random symmetric edge polytopes, as well as random cosmological polytopes. We study expectation and variance asymptotics of the f-vector as the size of the underlying graph or the simplicial complex tends to infinity. The considered models of random graphs encompass the classical Erdös-Rényi model, but also r-regular random graphs and random geometric graphs. (c) We study asymptotic distributional properties of randomly generated lattice polytopes in high dimensions. In particular, we establish quantitative central limit theorems for the number of facets and the logarithmic volume for random symmetric edge and cosmological polytopes using variants of Stein's method for normal approximation such as the discrete Malliavin-Stein method. (d) We contribute to a deeper understanding about the typical or expected behavior of lattice polytopes by studying such polytopes from a probabilistic angle. This complements the more traditional approach in which properties of specific classes of lattice polytopes are usually investigated. This project also opens up several new directions for future research, including the study of generalized symmetric edge polytopes for random simplicial complexes and matroids as well as the quest for more refined central limit theorems, such as moderate or large deviation principles and concentration bounds.
DFG Programme
Priority Programmes
Subproject of
SPP 2458:
Combinatorial Synergies
