The mathematics of generalised dualities in M-theory

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Whilst many mysteries remain about M-theory, the leading candidate for the quantum resolution of Einstein's general relativity, its most provocative features are U-dualities. Dualities are deep relationships and equivalences between seemingly distinct systems. Our central motivation is to illuminate the richness of dualities by exposing their mathematical structures. Goals1. Develop a refined understanding of quantum aspects of generalised T-dualities in string theory employing the mathematical tools of geometric and/or deformation quantisation2. Provide a mathematical underpinning of novel Exceptional Drinfel'd Algebras (EDAs) postulated to described generalised dualities of M-theory.3. Exploit the relation between Poisson-Lie dualities and Quantum Groups to give a quantum perspective on string dualities 4. Develop a linkage between the generalised Yang-Baxter equations that arise in EDAs and theory of integrable modelsThis work fits in a number of research areas within the Mathematical Science theme of EPSRC. In particular progress will be relevant to Mathematical Physics (through the potential applications to the mathematical structures sitting behind the dualities of string theory and through the relationships to integrable models), Geometry and Topology (through the development of generalised parallelisations with Hitchin's generalised geometry and its extensions, including non-commutative geometry), Algebra (through understanding the quantum algebras related to the classical exceptional Drinfel'd algebra and algebraic structures related to the Yang-Baxter equation)