Algebraic methods in Discrete Geometry: Ideals of Graphs and Polytopes

• Applicant: Dr. Antonio Macchia
• Subject Area: Mathematics
• Term: from 2020 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 454595616

The main goal of the project is to further explore and investigate the multiple connections between Combinatorics, Discrete Geometry and Commutative Algebra, showing that there are many deep connections among these areas yet to be unveiled and exploited. We will focus on three groups of problems:1) The slack realization space of polytopes: this is a new model for the realization space of polytopes that represents a polytope by its slack matrix, attained by evaluating its defining inequalities at all its vertices, and arises from a certain saturated determinantal ideal, called slack ideal. It is gaining a big importance, proving that Commutative Algebra can be a precious computational tool to study classical polytopal questions. In fact, combining the slack model with the Grassmannian model, we obtained a reduced slack model, which allows us to look at a small part of the slack matrix that contains the essential information. We have used this reduced model to prove that a certain large simplicial sphere, whose realizability was not known, is not realizable as a polytope. I am going to leverage this algebraic point of view to investigate hard (rational) realizability problems. On the other hand, a deeper algebraic understanding of slack ideals, for instance characterizing polytopes with a toric slack ideal or proving radicality of slack ideals, would give precious information on the structure and dimension of the associated slack variety. A further interesting application is the classification of combinatorial types of polytropes.2) Lattice polytopes: computing the width of lattice polytopes and, even more, finding the exact value of the flatness constant Flt(d) are hard tasks, even in very low dimension d. I will focus on the search for lattice polytopes whose width is larger than the dimension. In particular, in dimension 3, Codenotti and Santos recently proved a lower bound C for Flt(3), constructing a certain tetrahedron whose width is C. Later we proved that this tetrahedron is a local maximizer for width among all 3-dimensional hollow bodies. We aim at a global version of this result, showing that Flt(3) = C. Moreover, I would like to look for new families of hollow lattice polytopes and cyclic simplices, whose width exceeds the dimension.3) Graphs and binomial ideals: the study of binomial ideals has many diversified relations with graphs: binomial edge ideals and ideals of orthogonal representations of graphs are just two remarkable examples. I plan to push forward the understanding of these deep combinatorial connections, looking for a graph-theoretical characterization of all the Cohen-Macaulay binomial edge ideals, studying the algebraic invariants of the ideals of orthogonal representations of graphs, and analysing the relation between these classes of binomial ideals and slack ideals of polytopes.

• DFG Programme: Research Grants