My primary research area is semigroup theory: I got interested in it as an undergraduate, its mixture of algebraic and combinatorial ideas was just right for my taste. Semigroups, being highly general algebraic structures, crop up virtually everywhere, and from early on in my work, I've been focusing on interactions with different areas of mathematics. I am particularly interested in inverse semigroups and their applications, with my main focus being étale groupoids and C*-algebras these days. I also study geometric properties of inverse semigroups, and I am interested in how tools of geometric group theory can be extended to this setting.

In 2022, I organized a workshop on Topics in Geometric Semigroup Theory in York, exploring the connections between semigroups and large scale geometry, and in 2024, a conference on SEmigroups, grOUpoids and C*-aLgebras in Seoul. I am the central coordinator of the North British Semigroups and Applications Network.

I enjoy collaborative work and (capacity allowing) keen to talk to, or work with people in similar or complementary research areas.

1. E-unitary and F-inverse monoids, and closure operators on group Cayley graphs
   Acta Mathematica Hungarica 173 (2024) 297–316
   [abstract] [arXiv] [journal]
   We show that the category of \(X\)-generated E-unitary inverse monoids with greatest group image \(G\) is equivalent to the category of \(G\)-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of \(G\). Analogously, we study F-inverse monoids in the extended signature \((\cdot, 1, ^{-1}, ^{\mathfrak m})\), and show that the category of X-generated F-inverse monoids with greatest group image \(G\) is equivalent to the category of \(G\)-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of \(G\). As an application, we show that presentations of F-inverse monoids in the extended signature can be studied by tools analogous to Stephen's procedure in inverse monoids, in particular, we introduce the notions of F-Schützenberger graphs and P-expansions.
2. Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups (with D. Martínez and Y.C. Chung)
   to appear in Groups, Geometry and Dynamics
   [abstract] [arXiv]
   Given any quasi-countable, in particular any countable inverse semigroup \(S\), we introduce a way to equip \(S\) with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on \(S\). This allows us to unambiguously define the uniform Roe algebra of \(S\), which we prove can be realized as a canonical crossed product of \(\ell^\infty(S)\) and \(S\). We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups, and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by \(S\) being locally \(\mathcal L\)-finite, and equivalently sparse as a metric space.
3. On the simplicity of Nekrashevych algebras of contracting self-similar groups (with B. Steinberg)
   Mathematische Annalen 386 (2023) 1391–1428
   [abstract] [arXiv] [journal]
   Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych C*-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the C*-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta-Sidki groups, GGS-groups, multi-edge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk-Erschler group and Fabrykowski-Gupta group) and self-replicating spinal automaton groups.
4. Algorithmic properties of inverse monoids with hyperbolic and tree-like Schützenberger graphs (with R. Gray and P. V. Silva)
   Journal of Algebra 611 (2022) 651-687
   [abstract] [arXiv] [journal]
   We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.
5. Simplicity of inverse semigroup and étale groupoid algebras (with B. Steinberg)
   Advances in Mathematics 386 (2021) 107611
   [abstract] [arXiv] [journal]
   In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an étale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential C*-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic 0. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.
6. Inverse monoids and immersions of Δ-complexes (with J. Meakin)
   International Journal of Algebra and Computation 31 (2021) 1243-1268
   [abstract] [arXiv] [journal]
   An immersion \(f : {\mathcal D} \rightarrow \mathcal C\) between \(\Delta\)-complexes is a \(\Delta\)-map that induces injections from star sets of \(\mathcal D\) to star sets of \(\mathcal C\). We study immersions between finite-dimensional connected \(\Delta\)-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into 2-dimensional CW-complexes.
7. Inverse monoids of partial graph automorphisms (with R. Jajcay, T. Jajcayova, and M. Szendrei)
   Journal of Algebraic Combinatorics 53 (2021) 829-849
   [abstract] [arXiv] [journal]
   A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.
8. On F-inverse covers of finite-above inverse monoids (with M. Szendrei)
   Journal of Algebra 452 (2016) 42-65
   [abstract] [arXiv] [journal]
   Finite-above inverse monoids are a common generalization of finite inverse monoids and Margolis-Meakin expansions of groups. Given a finite-above E-unitary inverse monoid \(M\) and a group variety \(\mathcal U\), we find a condition for \(M\) and \(\mathcal U\), involving a construction of descending chains of graphs, which is equivalent to \(M\) having an F-inverse cover via \(\mathcal U\). In the special case where \(\mathcal U\)=\(\mathcal Ab\), the variety of Abelian groups, we apply this condition to get a simple sufficient condition for \(M\)to have no F-inverse cover via \(\mathcal Ab\), formulated by means of the natural partial order and the least group congruence of \(M\).
9. On the graph condition regarding the F-inverse cover problem
   Semigroup Forum 92 (2016) 551-558
   [abstract] [arXiv] [journal]
   In their paper titled "On F-inverse covers of inverse monoids", Auinger and Szendrei have shown that every finite inverse monoid has an F-inverse cover if and only if each finite graph admits a locally finite group variety with a certain property. We study this property and prove that the class of graphs for which a given group variety has the required property is closed downwards in the minor ordering, and can therefore be described by forbidden minors. We find these forbidden minors for all varieties of Abelian groups, thus describing the graphs for which such a group variety satisfies the above mentioned condition.
10. On the number of slim, semimodular lattices (with G. Czédli, T. Dékány, L. Ozsvárt, and B. Udvari)
    Mathematica Slovaca 66 (2016) 5-18
    [abstract] [arXiv] [journal]
    A lattice \(L\) is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Czédli and E.T. Schmidt as the duals of the lattices consisting of the intersections of the members of two composition series in a group. Our main result determines the number of (isomorphism classes of) these lattices of a given size in a recursive way. The corresponding planar diagrams, up to similarity, are also enumerated. We prove that the number of diagrams of slim, distributive lattices of a given length \(n\) is the \(n\)th Catalan number. Besides lattice theory, the paper includes some combinatorial arguments on permutations and their inversions.
11. Inverse monoids and immersions of 2-complexes (with J. Meakin)
    International Journal of Algebra and Computation 25 (2015) 301-323
    [abstract] [arXiv] [journal]
    It is well known that under mild conditions on a connected topological space \(\mathcal X\), connected covers of \(\mathcal X\) may be classified via conjugacy classes of subgroups of the fundamental group of \(\mathcal X\). In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion \(f : {\mathcal D} \rightarrow \mathcal C\) between CW-complexes is a cellular map such that each point \(y \in {\mathcal D}\) has a neighborhood \(U\) that is mapped homeomorphically onto \(f(U)\) by \(f\). In order to classify immersions into a 2-dimensional CW-complex \(\mathcal C\), we need to replace the fundamental group of \(\mathcal C\) by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex.

Preprints

• The growth of free inverse monoids (with M. Kambites, C-F. Nyberg-Brodda, and R. Webb)
  [abstract] [arXiv]
  We compute the rate of exponential growth of the free inverse monoid of rank \(r\) (and hence an upper bound on the corresponding rate for all \(r\)-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of \(2r-1\) and \(2r\), tending to \(2r\) as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to \(\sqrt{e(2r-1)}\) as \(r \to \infty\).
• Pretzel monoids (with D. Heath and M. Kambites)
  [abstract] [arXiv]
  We introduce and study an interesting class of left adequate monoids which we call pretzel monoids. These, on the one hand, are monoids of birooted graphs with respect to a natural 'glue-and-fold' operation, and on the other hand, are shown to be defined in the category of left adequate monoids by a natural class of presentations. They are also shown to be the free idempotent-pure expansions of right cancellative monoids, making them, in some sense, the left adequate analogoues of Margolis-Meakin expansions for inverse monoids. The construction recovers the second author's geometric model of free left adequate monoids when the right cancellative monoid is free.

Selected talks at conferences and meetings

Inverse semigroup- and ample groupoid algebras: introduction and simplicity (minicourse), SEmigroups, grOUpoids and C*-aLgebras (Seoul, South Korea, July 1-5, 2024)

On the simplicity of inverse semigroup and groupoid algebras, UK Operator Algebras Conference (Newcastle, UK, June 13, 2024)

The geometry of inverse semigroups, and uniform Roe algebras, Algebra, Geometry and C*-algebras (Edinburgh, UK, June 23, 2023)

Inverse semigroups as metric spaces, and their uniform Roe algebras, XVI Barcelona Weekend in Group Theory (Barcelona, Spain, May 5, 2023)

Proper and right subinvariant metrics on inverse semigroups, AMS 2022 Spring Western Sectional Meeting, Special Session on Recent Trends in Semigroup Theory (online, May 22, 2022)

Simplicity Nekrashevych algebras of contracting self-similar groups, British Early Career Mathematicians' Colloquium (online, June 15, 2021)

Simplicity of contracted inverse semigroup algebras, International Conference on Semigroups and Applications (Cochin, India, Dec 9, 2019)

Inverse monoids of partial graph automorphisms, International Conference on Semigroups 2018 (Lisbon, Portugal, July 12, 2018)

Inverse monoids and immersions of cell complexes, Rhodesfest81 - a conference in honor of John Rhodes's 81st birthday (Ramat Gan, Israel, Jun 5, 2018)

On the graph condition regarding the F-inverse cover problem, AMS-EMS-SPM meeting, Algebraic Theory of Semigroups and Applications (Porto, Portugal, June 13, 2015)

Seminar talks

Closure operators on group Cayley graphs, and presentations of F-inverse monoids, Panglobal Algebra and Logic Seminar, University of Colorado (online, Nov 14, 2023)

Inverse semigroups as metric spaces, and their uniform Roe algebras, Pure Mathematics Colloquium, University of Southampton (UK, October 27, 2023)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Functional Analysis seminar, University of Oxford (UK, June 6, 2023)

The large scale geometry of inverse semigroups, Geometry and Topology seminar, University of Bristol (UK, May 2, 2023)

Simplicity of Nekrashevych algebras of contracting self-similar groups, C*-algebras Oberseminar, University of Münster (Germany, Jan 10, 2023)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Analysis seminar, University of Glasgow (UK, Nov 24, 2022)

The large scale geometry of inverse semigroups, York Semigroup seminar, University of York (UK, Nov 2, 2022)

Inverse semigroups as metric spaces, and their uniform Roe algebras, Algebra seminar, University of Manchester, (UK, Nov 1, 2022)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Group Theory seminar, ICMAT, Madrid (Spain, May 6, 2022)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Pure Mathematics Colloquium, University of Manchester (UK, April 29, 2022)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Colloquium of the Mathematical Institute, Göttingen (online, Jan 14, 2021)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Western Sydney University Abend Seminars (online, Oct 22, 2020)

Simplicity of Nekrashevych algebras of contracting self-similar groups, Algebra seminar, Alfréd Rényi Institute of Mathematics, (online, Sept 28, 2020)

Simplicity of contracted inverse semigroup algebras, Algebra seminar, University of Szeged, (Hungary, Sept 2, 2020)

Simplicity of contracted inverse semigroup algebras, York Semigroup seminar, University of York, (online, May 20, 2020)

Hyperbolic and tree-like inverse monoids, York Semigroup seminar, University of York, (UK, Oct 17, 2019)

Algorithmic properties of tree-like inverse monoids, Algebra seminar, Heriot-Watt University, (Edinburgh, UK, May 22, 2019)

Algorithmic properties of tree-like inverse monoids, City University of New York, (USA, May 10, 2019)

Algorithmic properties of tree-like inverse monoids, Pure Mathematics Colloquium, University of St Andrews, (UK, Apr 4, 2019)

Inverse monoids and immersions of cell complexes, York Semigroup seminar, University of York, (UK, March 13, 2019)

Hyperbolic groups and generalizations I-II., Algebra Seminar, University of Szeged, (Hungary, March 14 & 21, 2018)

Inverse monoids and immersions of cell complexes, Algebra seminar, University of Porto, (Portugal, Nov 24, 2017)

Inverse monoids and immersions of cell complexes, NOVA University Lisbon, (Portugal, Nov 20, 2017)

Inverse monoids and immersions of cell complexes, University of Coimbra, (Portugal, Oct 17, 2017)

The algebraic structure of partial automorphisms of graphs, York Semigroup seminar, University of York, (UK, July 7, 2017)

The algebraic structure of partial automorphisms of graphs, Algebra seminar, Alfréd Rényi Institute of Mathematics, (Budapest, Hungary, May 8, 2017)

Inverse monoids and immersions of 2-complexes, Geometry seminar, Alfréd Rényi Institute of Mathematics, (Budapest, Hungary, Apr 25, 2014)

Inverse monoids and immersions of 2-complexes, Algebra seminar, University of Szeged, (Hungary, March 5, 2014)

F-inverse covers of inverse monoids, Groups, Semigroups and Topology seminar, University of Nebraska-Lincoln, (USA, Nov 5, 2013)

F-inverse covers of E-unitary inverse monoids, Algebra seminar, University of Szeged, (Hungary, Apr 17, 2013)