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My research group
My research group
I work closely with Yue Ren in Durham, and Biagio Lucini at QMUL. Yue and I regularly co-supervise PhD students and postdocs. We have an active group with several group activities per week. Our meetings are often hybrid and we are joined by past group members and current collaborators.
Tropical algebra and geometry
Tropical algebra and geometry
Tropical geometry is a young subject halfway between discrete mathematics (combinatorics) and algebraic geometry. One of my contributions was to introduce a scheme-theoretic approach for the subject, where we study the geometry of systems of polynomial equations defined over idempotent semirings such as the real numbers with the max-plus operations. This is a very exciting young area, with lots of activity and hundreds of interesting open questions. My current interests here range from deeply theoretical questions on foundations to applications in data science.
Topological and geometric data analysis
Topological and geometric data analysis
I am co-durector of the EPSRC-funded Erlangen AI Hub for mathematical foundations of AI. Before this, I was part of the EPSRC-funded Oxford-Liverpool-Swansea Centre for Topological Data Analysis.
Modern science and technology generate data at an unprecedented rate. A major challenge is that this data is often complex, high dimensional and may include temporal and/or spatial information. The 'shape' of the data can be important but it is difficult to extract and quantify it using standard machine learning or statistical techniques. For example, an image of blood vessels near a tumour looks very different to an image of healthy blood vessels; statistics alone cannot quantify this difference. New shape analysis methods are required. My interests focus on: aspects of persistent homology, other diffusion-based approaches to the geometry of data, non-archimedean and tropical methods in ML and data science, working with hierarchical data sets, and applying TDA tools to study phase transitions and the dynamics of topolical features in quantum field theory systems (particularly vortices and monopoles in QCD and their relations with quark confinement).
Prospective PhD students
Prospective PhD students
I am happy to supervise PhD projects in: tropical geometry and algebra, topological data analysis, and low-dimensional topology.
Funding: We regularly have fully-funded PhD scholarships in Durham (although these are usually restricted to UK students only). Please contact me if you are interested in working with me in an area related to my research interests The application deadline for funding is mid-Sep
Externally funded and self-funded students: I am always happy to receive applications from self-funded and externally funded prospective PhD students. If you are interested in working with me then please get in touch. For UK residents, the tuition fees for a PhD are currently approximately £4k per year. For non-UK students, the fees are approximately £25k per year.
Prospective postdocs
Prospective postdocs
Please get in touch if you are interested in applying for a postdoctoral fellowship. Options include:
Current students in our group
Current students in our group
Iolo is working on several projects: applications of tropical geometry to deep learning, and category theoretic perspectives in symbolic deep learning. He has been collaborating closely with Hylomorph Solutions. His current main project is about the computational differential geometry of data. He provides a new suite of numerical tools for exploring geometric features of data sets, with applications to machine learning.
I. Jones, and J. Swan and J. Giansiracusa, Algebraic dynamical systems in machine learning, to appear in Applied Categorical Structures, arXiv:2311.03118
I. Jones, Diffusion Geometry, arXiv:2405.10858.
I. Jones, Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension, arXiv:2411.04100
Xavier was an undergraduate in physics at Durham. He is doing his PhD based at Swansea, co-supervised between Biagio Lucini and me. He will be exploring applications of topological data analysis tools to studying dynamical aspects of topological defects in quantum field theory, particularly abelian monopoles.
X. Crean, J. Giansiracusa, and B. Lucini, Topological Data Analysis of Monopoles in U(1) Lattice Gauge Theory, to appear in SciPost Physics, arXiv:2403.07739.
X. Crean, J. Giansiracusa, and B. Lucini, Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theories, arXiv:2501.19320.
David Lanners (PhD, started October 2023)
David Lanners (PhD, started October 2023)
David studied physics and mathematics at ETH in Zurich. He is co-supervised by Tin Sulejmanpasic and joins the collaboration with Biagio Lucini. He has been exploring the role of monopoles in the phase structure of Z/n lattice gauge theories and structure of emergent symmetries. He is also developing highly efficient algorithms for computing directed and zig-zag persistent homology, and he is working with Iolo to further develop diffusion geometry.
Current postdocs and research fellows in our group
Current postdocs and research fellows in our group
Ollie was previously a postdoc with Dimitra Kosta in Edinburgh and has a background in computational aspects of tropical and algebraic geometry. He will be working as part of the EPSRC-funded Erlangen Hub for the Mathematical Foundations of AI. Together, we plan to investigate tropical and non-archimedean geometry in machine learning.
Victoria will be funded by Yue Ren's UKRI Future Leaders Fellowship and will be working on computational aspects of tropical and non-archimedean geometry.
Past members
Past members
Ziva worked with Yue Ren and me on topics at the interface of machine learning, topological data analysis, and tropical geometry. Projects Her projects included work on directed topology, tropical geometry of feed-forward neural networks, and analysis of gene expression data. She is now back in Slovenia working in data science and consultancy at AFLabs.
L. Fajstrup, B. Fasy, W. Li, L. Mezrag, T. Rask, F. Tombari, Ž. Urbančič, Gromov--Hausdorff Distance for Directed Spaces, arxiv:2408.14394
Ž. Urbančič and J. Giansiracusa, Ladder decomposition for morphisms of persistence modules, to appear in Journal of Applied and Computational Topology. arXiv:2307.03409.
H. Gangl, Y. Ren, Ž.Urbančič, Hands-on Tropical Geometry. Computeralgebra Rundbrief 72 (2023), 10-14. arXiv:2304.10130
Xuan Gao (PhD, submitted December 2023)
Xuan Gao (PhD, submitted December 2023)
Xuan worked on aspects of tropical geometry with higher rank valuations.
Ximena is now a permanent lecturer at City University of London. She came to Durham from University of Beunos Aires and has interests in topological data analysis, dynamical systems, combinatorial topology, and applications to neuroscience. She has been using topological data analysis to study the function of grid cells in brains and to detect the onset of epileptic seizures.
Nick is currently working in machine learning R&D for ASML. He did his PhD at Swansea co-supervised by Biagio Lucini and me. His work examined the use of persistent homology in the analysis of phase transitions in statistical physics systems. He has studied the Kosterlitz-Thouless transition in the XY model and then the deconfinement transition in SU(2) and SU(3) gauge theory.
N. Sale, J. Giansiracusa, and B. Lucini, Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology, Phys. Rev. E. Phys. Rev. E 105, 024121, ArXiv:2109.10960.
N. Sale, J. Giansiracusa, and B. Lucini, Probing center vortices and deconfinement in SU(2) lattice gauge theory with persistent homology, ArXiv:2207.13392
James is currently working in data science. He did his MSc thesis under my supervision in 2018, and continued on to do a PhD with me. His work examines various aspects of hyperfields and tropical geometry. As a postdoctoral Heilbronn fellow at Bristol, he worked on tropical ideals, hyperfield convexity, and connections between rigidity and tropical geometry.
James Maxwell, Generalising Kapranov's Theorem For Tropical Geometry Over Hyperfields, arXiv:2108.01524
From January 2026 Stefano will be an assistant professor at CUNEF in Madrid. He started in October 2018, jointly supervised by Andrea Pulita in Grenoble. His project developed links between the theory of tropical algebraic differential equations and p-adic differential equations. He went on to a postdoc at the MPI in Leipzig and then a postdoc at KTH in Stockhom. He wrote these two papers while a student with me.
J. Giansiracusa and S. Mereta, A general framework for tropical differential equations, ArXiv:2111.03925
T. Brzeziński, S. Mereta, and B. Rybołowicz, From pre-trusses to skew-braces, to appear in Publicacions Matemàtiques, ArXiv:2007.05761
Tak-Shing was a postdoc in our Topological Data Analysis Group. He worked on applications of TDA and machine learning to statistical physics systems and phase transitions, as well as developing software tools in TDA.
Niklas Hellmer (PhD, based in Swansea January to September 2020)
Niklas Hellmer (PhD, based in Swansea January to September 2020)
Niklas did his masters degree in topology at Bonn. He was with our group for 9 months, co-supervised by Pawel Dlotko and me. Pawel moved to the Dioscuri Institute in Warsaw to lead a new TDA centre there, and Niklas transferred to continue working with Pawel. His work has been primarily on metrics for persistent homology coming from probability theory. He also devoted significant effort to a Covid19 database project.
Kayleigh worked in tropical geometry. Her project was to investigate the information content of tropical ideals and tropical schemes (of Maclagan-Rincon and Giansiracusa-Giansiracusa), and in particular, to study the extent to which these structures encode the valuation of the j-invariant for elliptic curves and the valuation of the discriminant for low degree hypersurfaces. Kayleigh passed her viva on 27 March 2020.
Ramses Fernandez Valencia (PhD, 2012 - 2015)
Ramses Fernandez Valencia (PhD, 2012 - 2015)
On 6 August 2015 my first PhD student, Ramses Fernandez-Valencia, successfully defended his thesis Topics in 2-dimensional topological conformal field theories. His work at Swansea was funded by my EPSRC grant. While working with me he wrote four papers:
R. Fernandez Valencia, On the structure of unoriented topological conformal field theories. Geom. Dedicata 189 (2017), 113–138, arxiv/1503.02465.
R Fernandez Valencia and J. Giansiracusa, On the Hochschild homology of involutive algebras. Glasg. Math. J. 60 (2018), no. 1, 187–198, arXiv:1505.02219.
R. Fernandez Valencia, On the structure of open equivariant topological conformal field theories. arXiv:1512.03471.
R. Fernandez Valencia, Hochschild homology and cohomology for involutive A∞-algebras. arXiv:1601.00269.
Ramses is currently working as a researcher in post-quantum cryptography within the IT Security Department at Eurecat in Barcelona.
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