*The Poincaré-extended $\mathbf{ab}$-index*, with Galen Dorpalen-Barry, Christian Stump, submitted.Abstract ###### Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $\mathbf{ab}$-index, which generalizes both the $\mathbf{ab}$-index and the Poincaré polynomial. For posets admitting $R$-labelings, we give a combinatorial description of the coefficients of the extended $\mathbf{ab}$-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback $\mathbf{ab}$-index generalizing the cd-index of face posets for oriented matroids. This recovers, generalizes and unifies results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg.

*Smooth cuboids in group theory*, with Mima Stanojkovski, submitted.Abstract ###### A smooth cuboid can be identified with a $3 \times 3$ matrix of linear forms, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over $K$. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also give a description of the automorphism group. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite $p$-groups of class $2$ and exponent $p$ arising in this way.

*Isomorphism invariant metrics*, with P. A. Brooksbank, E. A. O'Brien, J. B. Wilson, submitted.Abstract ###### Within a category $\mathtt{C}$, having objects $\mathtt{C}_0$, it may be instructive to know not only that two objects are non-isomorphic, but also how far from being isomorphic they are. We introduce pseudo-metrics $d:\mathtt{C}_0 \times \mathtt{C}_0 \to [0,\infty]$ with the property that $x\cong y$ implies $d(x,y)=0$. We also give a canonical construction that associates to each isomorphism invariant a pseudo-metric satisfying that condition. This guarantees a large source of isomorphism invariant pseudo-metrics. We examine such pseudo-metrics for invariants in various categories.

*A spectral theory for transverse tensor operators*, with Uriya A. First, James B. Wilson, submitted.Abstract ###### Tensors are multiway arrays of data, and transverse operators are the operators that change the frame of reference. We develop the spectral theory of transverse tensor operators and apply it to problems closely related to classifying quantum states of matter, isomorphism in algebra, clustering in data, and the design of high performance tensor type-systems. We prove the existence and uniqueness of the optimally-compressed tensor product spaces over algebras, called *densors*. This gives structural insights for tensors and improves how we recognize tensors in arbitrary reference frames. Using work of Eisenbud--Sturmfels on binomial ideals, we classify the maximal groups and categories of transverse operators, leading us to general tensor data types and categorical tensor decompositions, amenable to theorems like Jordan--Hölder and Krull--Schmidt. All categorical tensor substructure is detected by transverse operators whose spectra contain a Stanley--Reisner ideal, which can be analyzed with combinatorial and geometrical tools via their simplicial complexes. Underpinning this is a ternary Galois correspondence between tensor spaces, multivariable polynomial ideals, and transverse operators. This correspondence can be computed in polynomial time. We give an implementation in the computer algebra system $\textsf{Magma}$.

*Flag Hilbert–Poincaré series of hyperplane arrangements and Igusa zeta functions*, with Christopher Voll, to appear in Israel J. Math.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 $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ 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.

*On the geometry of flag Hilbert–Poincaré series for matroids*, with Lukas Kühne, Algebr. Comb.,**6**(2023), no. 3, 623–638.Abstract ###### We extend the definition of coarse flag Hilbert–Poincaré series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by applying geometric and combinatorial tools related to their topes. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all topes are simplicial. Moreover this yields a sufficient criterion for non-orientability of matroids of arbitrary rank.

*Tensor Isomorphism by conjugacy of Lie algebras*, with Peter A. Brooksbank, James B. Wilson, J. Algebra,**604**(2022), 790–807.Abstract ###### We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable we give a polynomial-time algorithm to solve a generalization of module isomorphism for a common class of Lie modules. As a consequence, we show that isomorphism testing is in polynomial time for tensors whose derivation algebras are classical Lie algebras and whose densor spaces are 1-dimensional. The method has been implemented in the Magma computer algebra system.

*Compatible filters with isomorphism testing*, J. Pure Appl. Algebra,**225**(2021), no. 3, 106528–106555.Abstract ###### Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.

*Exact sequences of inner automorphisms of tensors*, with Peter A. Brooksbank, James B. Wilson, J. Algebra,**545**(2020), 43–63.Abstract ###### We produce a long exact sequence of unit groups of associative algebras that behave as automorphisms of tensors in a manner similar to inner automorphisms for associative algebras. Analogues for Lie algebras of derivations of a tensor are also derived. These sequences, which are basis invariants of the tensor, generalize similar ones used for associative and non-associative algebras; they similarly facilitate inductive reasoning about, and calculation of the groups of symmetries of a tensor. The sequences can be used for problems as diverse as understanding algebraic structures to distinguishing entangled states in particle physics.

*Enumerating isoclinism classes of semi-extraspecial groups*, with Mark L. Lewis, Proc. Edinb. Math. Soc. (2),**63**(2020), no. 2, 426–442.Abstract ###### We enumerate the number of isoclinism classes of semi-extraspecial $p$-groups with derived subgroup of order $p^2$. To do this, we enumerate $\text{GL}(2, p)$-orbits of sets of irreducible, monic polynomials in $\mathbb{F}_p[x]$. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

*Most small $p$-groups have an automorphism of order 2*, Arch. Math. (Basel),**108**(2017), no. 3, 225–232.Abstract ###### Let $f(p, n)$ be the number of pairwise nonisomorphic $p$-groups of order $p^n$ , and let $g(p, n)$ be the number of groups of order $p^n$ whose automorphism group is a $p$-group. We prove that the limit, as $p$ grows to infinity, of the ratio $g(p, n) / f(p, n)$ equals $1/3$ for $n = 6,7$.

*Efficient characteristic refinements for finite groups*, J. Symbolic Comput.,**80**(2017), part 2, 511–520.Abstract ###### Filters were introduced by J.B. Wilson in 2013 to generalize work of Lazard with associated graded Lie rings. It holds promise in improving isomorphism tests, but the formulas introduced then were impractical for computation. Here, we provide an efficient algorithm for these formulas, and we demonstrate their usefulness on several examples of $p$-groups.

*A fast isomorphism test for groups whose Lie algebra has genus 2*, with Peter A. Brooksbank, James B. Wilson, J. Algebra,**473**(2017), 545–590.Abstract ###### Motivated by the desire for better isomorphism tests for finite groups, we present a polynomial-time algorithm for deciding isomorphism within a class of $p$-groups that is well-suited to studying local properties of general groups. We also report on the performance of an implementation of the algorithm in the computer algebra system Magma.

*Longer nilpotent series for classical unipotent groups*, J. Grp. Theory,**18**(2015), no. 4, 569–585.Abstract ###### In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing $\mathbb{N}^d$-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field $\mathbb{Z}/p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $\Theta(d^2)$ with nearly all factors having $p$-bounded order.

*Economical generating sets for the symmetric and alternating groups consisting of cycles of a fixed length*, with Scott Annin, J. Algebra Appl.,**11**(2012), no. 6, 1250110–1250118.Abstract ###### The symmetric group $S_n$ and the alternating group $A_n$ are groups of permutations on the set $\{0, 1, 2, \ldots , n - 1\}$ whose elements can be represented as products of disjoint cycles (the representation is unique up to the order of the cycles). In this paper, we show that whenever $n \geq k \geq 2$, the collection of all $k$-cycles generates $S_n$ if $k$ is even, and generates $A_n$ if $k$ is odd. Furthermore, we algorithmically construct generating sets for these groups of smallest possible size consisting exclusively of $k$-cycles, thereby strengthening results in [O. Ben-Shimol, The minimal number of cyclic generators of the symmetric and alternating groups, Commun. Algebra 35 (10) (2007) 3034–3037]. In so doing, our results find importance in the context of theoretical computer science, where efficient generating sets play an important role.

- ${\sf Densor}$,
with
James B. Wilson,
for Magma,
version 1.0 (2019).
Description ###### A Magma package, built on top of TensorSpace, to compute densor subspaces of bilinear maps.

- ${\sf EGroups}$,
for Magma,
version 1.0 (2022).
Description ###### A Magma implementation for algorithms concerning isomorphisms of elliptic $p$-groups.

- ${\sf ExceptionAlge}$,
with
James B. Wilson,
for Magma,
version 1.0 (2019).
Description ###### A Magma package for exceptional nonassociative algberas. Constructors for composition algebras and Jordan algebras are provided along with standard tools to analyze them.

- ${\sf Filters}$,
for Magma,
version 1.0 (2017).
Description ###### A Magma package for data structures and algorithms for filters for groups.

- ${\sf HypIgu}$,
for SageMath,
version 1.1 (2021).
Description ###### A SageMath package that provides functions to compute the Igusa local zeta function associated with hyperplane arrangements and matroids.

- ${\sf Sylver}$,
with
Peter A. Brooksbank,
James B. Wilson,
for Magma,
version 1.0 (2019).
Description ###### A Magma package to compute various algebras associated to tensors. At the core of the algorithms is a solver for a system of Sylvester-like equations.

- ${\sf TameGenus}$,
with
Peter A. Brooksbank,
James B. Wilson,
for Magma,
version 2.0 (2020).
Description ###### A Magma package to decide isomorphism, construct automorphisms, and assign canonical labels to groups whose Lie algebra has genus 2.

- ${\sf TensorSpace}$,
with
Peter A. Brooksbank,
James B. Wilson,
for Magma,
version 2.3 (2020).
Description ###### A Magma package for data structures, constructors, and low-level algorithms concerning tensors. Functions include constructing tensors from algebraic objects, slicing tensors to create new ones, and applying morphisms from various categories to tensors.

- Flag Hilbert--Poincaré series and Igusa zeta functions of hyperplane arrangements, New trends around profinite groups, Bella Vista Relax Hotel, Levico Terme, Italy, September 2021.
- Isomorphism via derivations, Groups in Galway 2020, National University of Ireland Galway, Galway Ireland, September 2020.
- Isomorphism, derivations, and Lie representations, Groups, representations and applications: new perspectives, Isaac Newton Institute, Cambridge, United Kingdom, February 2020.
- Multilinear tools for groups, Groups, representations and applications: computational and algorithmic methods, Isaac Newton Institute, Cambridge, United Kingdom, January 2020.
- Computing order zeta functions via resolution of singularities, Buildings, Varieties, and Applications, MPI für Mathematik in den Naturwissenschaften Leipzig, Leipzig, Germany, November 2019.
- A Tensor Playground: a demonstration of TensorSpace, Tensors: Algebra-Computation-Applications, University of Colorado Boulder, Boulder, Colorado, June 2019.
- Multilinear tools through filters on groups, Groups and Geometries, University of Auckland, Auckland, New Zealand, January 2019.
- Refinements and general filters for groups, Logic and Algorithms in Group Theory, Hausdorff Institute for Mathematics, Bonn, Germany, November 2018.