Multiplicative structure of rational functions. This project aims to develop new methods of investigating fundamental number theoretic notions of torsion and multiplicative dependence between objects of great interest such as rational functions and their values. This includes investigating such celebrated objects as torsion points on elliptic curves and torsion subgroups on algebraic varieties. The goal is to develop new methods and make pivotal advances towards solving several fundamental probl ....Multiplicative structure of rational functions. This project aims to develop new methods of investigating fundamental number theoretic notions of torsion and multiplicative dependence between objects of great interest such as rational functions and their values. This includes investigating such celebrated objects as torsion points on elliptic curves and torsion subgroups on algebraic varieties. The goal is to develop new methods and make pivotal advances towards solving several fundamental problems where multiplicative dependence plays a crucial role. The expected outcome is to provide deeper understanding of the intriguing nature of torsion and multiplicative dependence and thus open new perspectives for their applications in number theory and beyond.Read moreRead less
Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and g ....Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and groups, and the theory of rewriting systems, together with recent breakthroughs by the team ensures that significant results can be expected.
Benefits include training research students and postdoctoral researchers in cutting-edge techniques, and advancing fundamental knowledge in mathematics and computer science.Read moreRead less
Algebraic Schubert geometry and unitary reflection groups. This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that th ....Algebraic Schubert geometry and unitary reflection groups. This project aims to generalise the recent work of Elias and Williamson to the complex case. Fundamental to the study of symmetry are the ubiquitous Coxeter groups, which have an associated set of critically important ‘Kazhdan-Lusztig polynomials’. For some Coxeter groups, these may be interpreted in terms of classical geometry, leading to deep positivity properties for their coefficients. Elias and Williamson have recently shown that this geometry may be simulated algebraically for any Coxeter group, so positivity for Kazhdan-Lusztig polynomials holds for all Coxeter groups. This result has explosive consequences in many areas of geometry and algebra. This project is designed to extend these results to complex unitary reflection groups, with potentially dramatic consequences in number theory, representation theory and topology.Read moreRead less
Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from ....Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from the field of parameterised complexity, creating powerful, practical solutions for these problems. It is expected to shed much-needed light on the vast and puzzling gap between theory and practice, and give researchers fast new software tools for large-scale experimentation and cutting-edge computer proofs.Read moreRead less