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Tommy Hofmann

Publications and source records attributed to Tommy Hofmann.

At least 19 recordsLinked to original sources
arXiv2026

Constructing free subgroups of matrix groups

The Tits alternative asserts that a finitely generated linear group is either solvable-by-finite or contains a non-abelian free subgroup. Given a non-solvable-by-finite subgroup G of GL(d, F), where F is the field of complex numbers or a non-archimedean local field, we present methods to construct explicitly a non-abelian 2-generator free subgroup of G in specific cases. Our implementations in the computeralgebra system Oscar are publicly available.

math.GR↗
arXiv2026

A counterexample to a conjecture of Plotnick on knot exteriors

We resolve a conjecture of S. Plotnick from 1983 concerning the homotopy type of four-dimensional knot complements. We show that the exteriors of two branched twist spins of the trefoil, which were shown by Plotnick not to be homotopy equivalent as boundary pairs and hence not homeomorphic, are in fact simple homotopy equivalent. The main ingredient of the proof is the construction of an explicit element of augmentation $49$ in the integral group ring of $\operatorname{SL}_2(\mathbb{F}_5)$ which is a unit modulo the group norm and commutes with the element inducing the monodromy on the fundamental group.

math.GT↗
arXiv2026

Automorphisms and canonical forms of integer lattices

We improve the Plesken--Souvignier algorithm for computing automorphism groups and isometries of positive definite integer quadratic lattices by exploiting characteristic sublattices and refining vector enumeration. For lattices with roots, we also use the Weyl group to accelerate the computation of canonical forms. Experiments with concrete implementations demonstrate substantial speedups and show that canonical forms can be computed at a cost comparable to that of computing automorphism groups.

math.NT↗
arXiv2026

Algorithms for $p$-rationality and for $p$-saturation of units

Let $K$ be a number field. We give new practical algorithms that determine whether $K$ is $p$-rational. For real cyclotomic fields $K=\mathbb{Q}(ζ_{n})^{+}$, our method avoids computing either the class number or the full unit group. We also use the notion of $p$-rationality to explain the practical efficiency of an algorithm for determining whether a subgroup of the unit group $\mathcal{O}_{K}^{\times}$ is $p$-saturated. This in turn yields a new algorithm for the unconditional verification of unit groups of number fields that substantially outperforms existing unconditional algorithms in practice. Finally, by combining our algorithms for determining $p$-rationality with work of Greenberg, we construct certain Galois representations $ρ: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_{n}(\mathbb{Z}_{p})$ with open image and further prescribed properties.

math.NT↗
arXiv2026

Swan modules and homotopy types after a single stabilisation

We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions $n \equiv 3$ mod $4$, there exist finite $n$-complexes which are homotopy equivalent after stabilising with multiple copies of $S^n$, but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with $k$-periodic cohomology which does not have free period $k$. In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction.

math.AT↗
arXiv2026

Determination of the stably free cancellation property for orders

Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $Λ$ be an $\mathcal{O}_{K}$-order in $A$. We give practical algorithms that determine whether $Λ$ has stably free cancellation (SFC). As an application, we determine all finite groups $G$ of order at most $383$ such that the integral group ring $\mathbb{Z}[G]$ has SFC.

math.NT↗
arXiv2026

Determining unit groups and $\mathrm{K}_1$ of finite rings

We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first $K$-group of finite rings.

math.NT↗
arXiv2025

Exotic presentations of quaternion groups and Wall's D2 problem

The D2 problem of C. T. C. Wall asks whether every finite cohomologically 2-dimensional CW-complex is homotopy equivalent to a finite 2-complex. Several potential counterexamples have been proposed, the longest standing of which is a CW-complex constructed by Cohen and Dyer whose fundamental group is a quaternion group of order 32. We show that this CW-complex is homotopy equivalent to the presentation 2-complex of a presentation constructed by Mannan-Popiel, thus showing it is not a counterexample to the D2 problem. We next introduce an infinite family of presentations for a quaternion group of order $4n$ and prove that they achieve homotopy types which are not achieved by the presentations of Mannan-Popiel.

math.GR↗
arXiv2025

Norm relations and computational problems in number fields

For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of a normal extension of algebraic number fields with Galois group $G$. On the algorithmic side this leads to subfield based algorithms for computing rings of integers, $S$-unit groups and class groups. For the $S$-unit group computation this yields a polynomial time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields.

math.NT↗
arXiv2024

Number Theory in OSCAR

We give a brief introduction to computational algebraic number theory in OSCAR. Our main focus is on number fields, rings of integers and their invariants. After recalling some classical results and their constructive counterparts, we showcase the functionality in two examples related to the investigation of the Cohen-Lenstra heuristic for quadratic fields and the Galois module structure of rings of integers.

math.NT↗
arXiv2024

Wafer-scale fabrication of mesoporous silicon functionalized with electrically conductive polymers

The fabrication of hybrid materials consisting of nanoporous hosts with conductive polymers is a challenging task, since the extreme spatial confinement often conflicts with the stringent physico-chemical requirements for polymerization of organic constituents. Here, several low-threshold and scalable synthesis routes for such hybrids are presented. First, the electrochemical synthesis of composites based on mesoporous silicon (pore size of 7 nm) and the polymers PANI, PPy and PEDOT is discussed and validated by scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDX). Polymer filling degrees of 74% are achieved. Second, the production of PEDOT/pSi hybrids, based on the solid-state polymerization (SSP) of DBEDOT to PEDOT is shown. The resulting amorphous structure of the nanopore-embedded PEDOT is investigated via in-situ synchrotron-based X-ray scattering. In addition, a twofold increase in the electrical conductivity of the hybrid compared to the porous silicon host is shown, making this system particularly promising for thermoelectric applications.

cond-mat.mtrl-sci↗
arXiv2023

Finite subgroups of automorphisms of K3 surfaces

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves hermitian lattices over number fields.

math.AG↗
arXiv2022

Computation of lattice isomorphisms and the integral matrix similarity problem

Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $Λ$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple $K$-algebra $A/{\mathrm{J}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on $A$, we give an algorithm that given two $Λ$-lattices $X$ and $Y$, determines whether $X$ and $Y$ are isomorphic, and if so, computes an explicit isomorphism $X \rightarrow Y$. This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: given a number field $K$, a positive integer $n$ and two matrices $A,B \in \mathrm{Mat}_{n}(\mathcal{O}_{K})$, determine whether $A$ and $B$ are similar over $\mathcal{O}_{K}$, and if so, return a matrix $C \in \mathrm{GL}_{n}(\mathcal{O}_{K})$ such that $B= CAC^{-1}$. We give explicit examples that show that the implementation of the latter algorithm for $\mathcal{O}_{K}=\mathbb{Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

math.NT↗
arXiv2022

Generation of Local Unitary Groups

Let $E$ be a two-dimensional étale algebra over a non-Archimedean local field $K$ of characteristic zero. We show that the unitary group of a non-degenerate hermitian lattice over $E$ is generated by symmetries and rescaled Eichler isometries. In the appendix we show that unless $E/K$ is a ramified dyadic field extension and the residue field has two elements, symmetries suffice.

math.NT↗
arXiv2020

Normal CM-fields with class number one

We show that assuming the generalized Riemann hypothesis there are no normal CM-fields with class number one of degree 64 and 96. This is done by constructing complete tables of normal CM-fields using discriminant bounds of Lee--Kwon. This solves the class number one problem for normal CM-fields assuming GRH. Using the same technique to solve the relative class number one problem in degrees 16, 32, 56 and 82, also the corresponding relative class number one problem is solved assuming GRH.

math.NT↗
arXiv2020

Ionic liquid dynamics in nanoporous carbon: A pore-size- and temperature-dependent neutron spectroscopy study on supercapacitor materials

The influence of spatial confinement on the thermally excited stochastic cation dynamics of the room-temperature ionic liquid 1-N-butylpyridinium bis-((trifluoromethyl)sulfonyl)imide ([BuPy][Tf_2N]) inside porous carbide-derived carbons with various pore sizes in the sub- to a few nanometer range are investigated by quasi-elastic neutron spectroscopy. Using the potential of fixed window scans, i.e. scanning a sample parameter, while observing solely one specific energy transfer value, an overview of the dynamic landscape within a wide temperature range is obtained. It is shown that already these data provide a quite comprehensive understanding of the confinement-induced alteration of the molecular mobility in comparison to the bulk. A complementary, more detailed analysis of full energy transfer spectra at selected temperatures reveals two translational diffusive processes on different time scales. Both are considerably slower than in the bulk liquid and show a decrease of the respective self-diffusion coefficients with decreasing nanopore size. Different thermal activation energies for molecular self-diffusion in nanoporous carbons with similar pore size indicate the importance of pore morphology on the molecular mobility, beyond the pure degree of confinement. In spite of the dynamic slowing down we can show that the temperature range of the liquid state upon nanoconfinement is remarkably extended to much lower temperatures, which is beneficial for potential technical applications of such systems.

physics.chem-ph↗
arXiv2020

On Gröbner bases over Dedekind domains

Gröbner bases are a fundamental tool when studying ideals in multivariate polynomial rings. More recently there has been a growing interest in transferring techniques from the field case to other coefficient rings, most notably Euclidean domains and principal ideal rings. In this paper we will consider multivariate polynomial rings over Dedekind domain. By generalizing methods from the theory of finitely generated projective modules, we show that it is possible to describe Gröbner bases over Dedekind domains in a way similar to the case of principal ideal domains, both from a theoretical and algorithmic point of view.

math.AC↗