RESEARCH GROUP

GEOMETRY

In a few days, we will participate in the conference
 Geometry meets Combinatorics in Bielefeld
an event in the crossroads of geometry, algebra, and combinatorics. Petra Schwer and Josh Maglione are speakers of the conference.
 
See you there!

The paper "Flag Hilbert--Poincaré series of hyperplane arrangements and their Igusa zeta functions" by Josh Maglione and Christopher Voll has been recently accepted for publication in the Israel Journal of Mathematics!

Link to the paper: arXiv:2103.03640

Abstract:
We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert-Poincaré series for irreducible Coxeter arrangements of types A, B, and D in terms of total partitions of the respective types. We show that a different specialization of the flag Hilbert-Poincaré series, which we call the coarse flag Hilbert-Poincaré series, exhibits intriguing nonnegativity features and - in the case of Coxeter arrangements - connections with Eulerian polynomials. For numerous classes and examples of hyperplane arrangements, we determine their (coarse) flag Hilbert-Poincaré series. Some computations were aided by a SageMath package we developed.

The paper "TriCCo -- a cubulation-based method for computing connected components on triangular grids" by Aiko Voigt, Petra Schwer, Noam von Rotberg and Nicole Knopf has been recently accepted for publication in the journal Geoscientific Model Development!

Link to the paper: arXiv:2111.13761

Abstract:
We present a new method to identify connected components on triangular grids used in atmosphere and climate models to discretize the horizontal dimension. In contrast to structured latitude-longitude grids, triangular grids are unstructured and the neighbors of a grid cell do not simply follow from the grid cell index. This complicates the identification of connected components compared to structured grids. Here, we show that this complication can be addressed by involving the mathematical tool of cubulation, which allows one to map the 2-d cells of the triangular grid onto the vertices of the 3-d cells of a cubic grid. Because the latter is structured, connected components can be readily identified by previously developed software packages for cubic grids. Computing the cubulation can be expensive, but importantly needs to be done only once for a given grid. We implement our method in a Python package that we name TriCCo and make available via pypi, gitlab and zenodo. We document the package and demonstrate its application using simulation output from the ICON atmosphere model. Finally, we characterize its computational performance and compare it to graph-based identifications of connected components using breadth-first search. The latter shows that TriCCo is ready for triangular grids with up to 500,000 cells, but that its speed and memory requirement should be improved for the application to larger grids.