V Goryunov

The University of Liverpool's Department of Mathematical Sciences invites applications for a PhD project on "Multidimensional Continued Fractions in Lattice Geometry." This research sits at the intersection of number theory, dynamical systems, singularity theory, and algebraic geometry, focusing on the geometric theory of continued fractions and its extension to higher dimensions. The project aims to develop new geometric and combinatorial tools to analyze three-dimensional continued fractions, particularly through the study of conjugacy classes of matrices in SL(3,ℤ). The long-term goal is to extend Gauss Reduction Theory beyond the well-understood two-dimensional case, addressing a significant open problem in the field. A key outcome will be progress on the classification of integer angles of convex integer polyhedra in three dimensions (the IKEA problem), with broader implications for toric varieties, singularity theory, and multidimensional Diophantine approximation. The successful candidate will receive comprehensive training in lattice geometry, number theory, and geometric combinatorics, with opportunities for collaboration with leading international researchers. The project is structured to support academic development, with the first year dedicated to foundational training, the second year focusing on three-dimensional Gauss Reduction Theory, the third year on in-depth analysis and connections to related fields, and the final year on thesis completion and independent research. This studentship is funded by XTX Markets and covers full Home tuition fees (£5,238 p.a. for 2026-27) and a maintenance grant (£21,805 p.a. for 2026-27) for 3.5 years. Additional funding is available for research expenses such as consumables and conference attendance. The studentship is open to all applicants, but international students must secure funding for the difference between Home and International fees. The University of Liverpool is committed to diversity and inclusion, offering support and reasonable adaptations for students with disabilities or personal circumstances. Applicants from all backgrounds are encouraged to apply. For further reading, see: Karpenkov O., Geometry of continued fractions. Second edition. Springer, 2022. Application deadline: 3 April 2026. For more information and to apply, visit the project page .