Overview
This project investigates three-dimensional continued fractions through lattice geometry, aiming to extend Gauss Reduction Theory to SL(3,ℤ). By connecting number theory, combinatorics, and algebraic geometry, it advances the classification of integer polyhedra and links lattice reduction to cuspidal singularities.
About this opportunity
The geometry of lattices is a rapidly developing area at the intersection of number theory, dynamical systems, singularity theory, and algebraic geometry. Central to this field is the geometric theory of continued fractions, which combines approximation theory with algorithmic and combinatorial methods. While two-dimensional continued fractions are well understood through Gauss Reduction Theory, their higher-dimensional analogues remain largely unexplored.
The core aim of this project is to study three-dimensional continued fractions via the structure of conjugacy classes of matrices in SL(3,ℤ), with the long-term goal of extending Gauss Reduction Theory to three dimensions. This represents a new and challenging direction of research, as no complete reduction theory currently exists beyond the classical two-dimensional case. The candidate will develop new geometric and combinatorial tools to analyse lattice bases, reduction procedures, and equivalence classes in dimension three.
A major outcome of this work will be progress on the classification of integer angles of convex integer polyhedra in three dimensions, known as the IKEA problem. Results in this direction are expected to yield global relations with cuspidal singularities of toric varieties, thereby creating a bridge between lattice geometry and singularity theory in algebraic geometry. The project also has implications for multidimensional Diophantine approximation and the theory of algorithms for lattice reduction.
The candidate will receive comprehensive training in lattice geometry, number theory, and geometric combinatorics. This will be complemented by exposure to related areas such as toric geometry and singularity theory. The project is well positioned for collaboration with leading researchers in the field, including Simon Kristensen (Aarhus University), Iskander Aliev (Cardiff University), and Matty van Son (University of the West Indies, Barbados), among others. These collaborations will provide both mathematical breadth and international research experience.
The project is structured to support steady academic development. The first year will focus on foundational training and a systematic study of the geometry of continued fractions and lattice theory. In the second year, the candidate will concentrate on developing and analysing Gauss Reduction Theory in the three-dimensional setting, which forms the central research contribution of the thesis. The third year will be devoted to in-depth analysis of the obtained results and their connections to singularity theory, algebraic geometry, and approximation theory. The final year will focus on consolidating the research, completing the thesis, and pursuing independent research directions arising from the project.
Further reading
Karpenkov O., Geometry of continued fractions. Second edition.
Algorithms Comput. Math., 26 Springer, Berlin, [2022], ©2022. xx+451 pp.
https://link.springer.com/book/10.1007/978-3-662-65277-0