At ThinkTank Maths, my research spans a variety of areas, including mathematical modelling and optimisation, machine learning and deep learning, and stochastic differential equations.
Prior to joining ThinkTank Maths, my research was in geometric group theory and low-dimensional topology. Geometric group theory is an area of mathematics devoted to studying groups as geometric objects in order to solve algebraic and algorithmic problems, as well as problems in other fields.
The methods of geometric group theory have also had extraordinary success outside of mathematics, with Carlsson’s development of persistent homology in topological data analysis, Ghrist’s applications of braid groups in robotics, and recent exciting applications of graph braid groups in topological quantum computing by Maciazek et al.
My publications are listed below.
2025
Random quotients preserve acylindrical and hierarchical hyperbolicity
We show that random quotients of acylindrically hyperbolic groups, obtained by taking a quotient of the group by the nth steps of a finite collection of independent random walks, are again acylindrically hyperbolic asymptotically almost surely. Our main tools come from spinning families and projection complexes, which we relate to random walks and develop further. Furthermore, we show that a random quotient of a hierarchically hyperbolic group is again hierarchically hyperbolic asymptotically almost surely. The same techniques also yield that a random quotient of a non-elementary hyperbolic group (relative to a finite collection of peripheral subgroups) is asymptotically almost surely hyperbolic (relative to isomorphic peripheral subgroups).
@article{random_quotients,author={Abbott, Carolyn and Berlyne, Daniel and Mangioni, Giorgio and Ng, Thomas and Rasmussen, Alexander},title={Random quotients preserve acylindrical and hierarchical hyperbolicity},journal={arXiv},fjournal={},volume={},number={},year={2025},pages={},doi={10.48550/arXiv.2507.16677},note={}}
2023
Graph of groups decompositions of graph braid groups
Daniel Berlyne
International Journal of Algebra and Computation, 2023
We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a non-trivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.
@article{gog_gbg,author={Berlyne, Daniel},title={Graph of groups decompositions of graph braid groups},journal={International Journal of Algebra and Computation},fjournal={},volume={33},number={8},year={2023},pages={1531--1569},doi={10.1142/S0218196723500583},note={}}
We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.
@article{graph_prod,author={Berlyne, Daniel and Russell, Jacob},title={Hierarchical hyperbolicity of graph products},journal={Groups, Geometry, and Dynamics},fjournal={},volume={16},number={2},year={2022},pages={523--580},doi={10.4171/GGD/652},note={}}
2021
Appendix to “Largest acylindrical actions and stability in hierarchically hyperbolic groups”
We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces, so it is natural to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure; in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity outside the context of hyperbolic groups. We provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known for mapping class groups and right-angled Artin groups. We also provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an "almost hierarchically hyperbolic space" is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.
@article{ABD_appendix,author={Berlyne, Daniel and Russell, Jacob},title={Appendix to ``{L}argest acylindrical actions and stability in hierarchically hyperbolic groups''},journal={Transactions of the American Mathematical Society, Series B},fjournal={},volume={8},year={2021},pages={94--104},doi={10.1090/btran/50},note={Primary article by C. Abbott, J. Behrstock, and M. G. Durham}}
Hierarchical hyperbolicity of graph products and graph braid groups
@article{Berlyne_Thesis,author={Berlyne, Daniel},title={Hierarchical hyperbolicity of graph products and graph braid groups},journal={CUNY Academic Works},fjournal={},volume={},number={},year={2021},pages={},doi={},note={Ph.D. thesis, City University of New York}}
@article{master_proj,author={Berlyne, Daniel},title={Teichmüller's theorem and its applications},journal={University of Warwick},fjournal={},volume={},number={},year={2015},pages={},doi={},note={Master's thesis}}
2014
Ideal Theory in Rings (Translation of “Idealtheorie in Ringbereichen” by Emmy Noether)
This paper is a translation of the paper "Idealtheorie in Ringbereichen", written by Emmy Noether in 1920, from the original German into English. It in particular brings the language used into the modern world so that it is easily understandable by the mathematicians of today. The paper itself deals with ideal theory, and was revolutionary in its field, that is modern algebra. Topics covered include: the representation of an ideal as the least common multiple of irreducible ideals; the representation of an ideal as the least common multiple of maximal primary ideals; the association of prime ideals with primary ideals; the representation of an ideal as the least common multiple of relatively prime irreducible ideals; isolated ideals; the representation of an ideal as the product of coprime irreducible ideals; equivalent concepts regarding modules.
@article{Noether_translation,author={Berlyne, Daniel},title={Ideal {T}heory in {R}ings ({T}ranslation of ``{I}dealtheorie in {R}ingbereichen'' by {E}mmy {N}oether)},journal={arXiv},fjournal={},volume={},number={},year={2014},pages={},doi={}}