Igusa zeta functions and hyperplane arrangements

We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert–Poincaré series. We show how these rational functions are connected to local Igusa zeta functions and class counting zeta functions for certain graphical group schemes studied by Rossmann and Voll. We report on a general self-reciprocity result and a non-negativity result of the numerator polynomial under a coarsening, and we explore other connections within algebraic combinatorics. We report on joint works with Christopher Voll and with Galen Dorpalen-Barry and Christian Stump.

Simultaneous Sylvester system & symmetries

Flag Hilbert-Poincaré series & related zeta functions

We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert-Poincaré series. We show how these rational functions are connected to Igusa zeta functions and class counting zeta functions for certain graphical group schemes studied by Rossmann and Voll. We report on a general self-reciprocity result and explore other connections within algebraic combinatorics. This is joint work with Christopher Voll and with Lukas Kühne.

Flag Hilbert--Poincaré series and Igusa zeta functions of hyperplane arrangements

We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert--Poincaré series, and we show how these rational functions are connected to Igusa zeta functions. We report on a general self-reciprocity result and explore other connections within algebraic combinatorics. This is joint work with Christopher Voll.

Isomorphism via derivations

By bringing in tools from multilinear algebra, we introduce a general method to aid in the computation of group isomorphism. Of particular interest are nilpotent groups where the only classically known proper nontrivial characteristic subgroup is the derived subgroup. Through structural analysis of the biadditive commutator map, we leverage the representation theory of Lie algebras to prove efficiency for families of nilpotent groups. We report on joint work with Peter A. Brooksbank, and James B. Wilson.

Computing order zeta functions via resolution of singularities

For a number field K with ring of integers O, the order zeta function of K is a Dirichlet generating series enumerating orders, i.e. unital subrings of O of finite index. In comparison with the Dedekind zeta function of K, the order zeta function of K is poorly understood: for number fields of degree larger than 5, next to nothing general is known. Encoding this Dirichlet series as a p-adic integral, we develop computational tools to repeatedly resolve singularities until it is distilled to enumerating points on polyhedra and p-rational points of algebraic varieties. This is joint work with Anne Fruehbis-Krueger, Bernd Schober, and Christopher Voll.

A Tensor Playground: a demonstration of TensorSpace

New algorithms for tensors change the way we compute with and manipulate tensors. Making room for modern theory while keeping the tried and true is the purpose of TensorSpace. We demonstrate how to build a tensor space system and apply it to known problems.