Research



My main research activity is in algebraic number theory which consists of applying methods of algebra, in particular group theory, to solving problems relating to the extensions of number fields. In recent years this line of work has also found applications in mathematical physics, and in particular to the solution of the quantum Yang-Baxter equation.

I enjoy spending most of my time thinking about problems relating to Hopf-Galois structures (algebraic number theory) and skew braces (related to the solutions of the quantum Yang-Baxter equation). On the one hand, I focus on the problem of classification of skew braces, hence obtaining a classification for certain Hopf-Galois structures; on the other hand, using these results to study the rings of integers of extensions of global or local fields in Galois module theory. You can see my research statement for a detailed account of my work.

Publications and Preprints arXiv and researchgate:

  1. On Autoregressive Component of Crime Time Series, preprint
    In this paper, we study the time series behaviour of monthly crime numbers in the UK. For the aggregate monthly crime numbers, we propose several suitable dynamic linear models, with at most 6 parameter estimates, which explain over 82% of variations in the data, provide robust seasonality indicators, and perform with low mean absolute percentage forecasting error. Next, we validate the models through residual analysis, investigate the relative importance of each parameter, discuss the effect of pandemic, and make comparison to auto ARIMA models. Finally, we explore the applicability of our modelling framework to other datasets of crime and to individual crime types and show that crime patterns acquire historic behaviour proportionate to the number of crimes.

  2. Some Braces of Cardinality p4 and Related Hopf-Galois Extensions (with D. Puljic and A. Smoktunowicz), https://arxiv.org/abs/2106.13735
    We describe all certain braces of cardinality p4 which are not right nilpotent. The constructed braces are left nilpotent, solvable and prime, and they also contain a non-zero strongly nilpotent ideal. Submitted to New York Journal of Mathematics.

  3. Skew Braces and Hopf-Galois Structures of Type Certain Metacyclic p-Groups, early results presented
    In this paper I am aiming to publish a generalisation of results of sections 3.5 and 4.5 of my thesis.

  4. Skew Braces and Hopf-Galois Structures of Heisenberg Type, https://doi.org/10.1016/j.jalgebra.2019.01.012
    Materials from Sections 3.4 and 4.4 of my thesis, further enhanced, where skew braces, their automorphisms, and Hopf-Galois structures of Heisenberg type are classified; J. Algebra, Apr 2019.

  5. A Statistical Modeling Framework for Characterising Uncertainty in Large Datasets: Application to Ocean Colour, www.mdpi.com/2072-4292/10/5/695
    In collaboration with several researchers, I developed a statistical model in order to characterise uncertainty of a remote sensing procedure for a NASA satellite; J. Remote Sensing, Apr 2018.

  6. On Hopf-Galois Structures and Skew Braces of Order p3, PhD Thesis on the University of Exeter Repository
    In my PhD thesis I classified all Hopf-Galois structures and skew braces of order p3. In addition, I found suitable descriptions for automorphism groups of skew braces, which was previously unknown. The work which will lead to advancements of our understanding of Galois modules theory of p-extensions of fields, as well as solutions of the quantum Yang-Baxter, also revealed many interesting patterns which can be used as an inspiration for future investigations; Jan 2018.

  7. Continuous Cocycles Endowed with Point-Open Topology, arXiv:1804.01029
    A theorem showing continuous nonabelian 1st group cohomology commutes with inverse limits through mainly topological methods.