PhD students and postdocs
past, present and future
Prospective PhD students
I am happy to supervise PhD projects in: tropical geometry and algebra, topological data analysis, and low-dimensional topology.
PhD 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 (tropical geometry, topological data analysis, other aspects of topology). The application deadline for funding is mid-September each year.
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.
International students: If you are interested in using topological data analysis to study problems in physics, there is an opportunity to work with Biagio Lucini (at Swansea) and me. https://www.swansea.ac.uk/postgraduate/scholarships/research/suipres-2024.php Please get in touch with Prof. Lucini or me as soon as possible. The closing date is 15 July. Note: This award is worth approximately £20k/year and reduces the international fees down to the UK fee level. It does not include a stipend.
Tropical algebra and geometry
Tropical geometry is a young subject halfway between discrete mathematics (combinatorics) and algebraic geometry. My contribution has been to introduce a scheme-theoretic foundation 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.
Topological data analysis
I am 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. The aim of this centre is to build on current persistent homology tools, extending them theoretically, computationally, and adapting them for practical applications.
Current students
Ziva is working with Yue Ren and myself in on topics at the interface of machine learning, topological data analysis, and tropical geometry. Current projects include work on directed topology, tropical geometry of feed-forward neural networks, and analysis of gene expression data.
Z. Urbancic and J. Giansiracusa, Ladder decomposition for morphisms of persistence modules, to appear in Journal of Applied and Computational Topology. arXiv:2307.03409.
Gangl, H., Ren, Y., Urbančič, Ž. Hands-on Tropical Geometry. Computeralgebra Rundbrief 72 (2023), 10-14. arXiv:2304.10130
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. Current projects include differential geometry of data, and tropical geometry of feed-forward neural networks.
I. Jones, and J. Swan and J. Giansiracusa, Algebraic dynamical systems in machine learning, to appear in Applied Categorical Structures, arXiv:2311.03118
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, arXiv:2403.07739.
David Lanners (PhD, started October 2023)
David studied physics and mathematics at ETH in Zurich. He will be joining the collaboration with Biagio Lucini and co-supervised by Tin Sulejmanpasic. He will continue our exploration of persistent homology applied to study the dynamics of extended topological objects in quantum field theoretic systems and their role in phase transitions.
Current postdocs and research fellows
Past members
Xuan Gao (PhD, submitted December 2023)
Xuan worked on aspects of tropical geometry with higher rank valuations.
Ximena came from University of Beunos Aires. She 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. She will soon be a lecturer at City University of London.
Nick is currently working in ML 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 a Heilbronn research fellow at Bristol. 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. He is currently thinking about tropical ideals, and convexity over hyperfields.
James Maxwell, Generalising Kapranov's Theorem For Tropical Geometry Over Hyperfields, arXiv:2108.01524
Stefano started 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 is now 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 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)
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.