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Singularities of Lie Group Actions in Geometry and Dynamics

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Reduction theory and relative equilibria

Mathematics describes phenomena under varying conditions and provides the basis for powerful computational models. Novel frameworks should provide insight into the dynamical behaviours of numerous physical systems.

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Mathematical methods facilitate formation of predictions about behaviours that can be tested in experiments. The continuous cycle of modelling and experimentation or observation provides a more and more realistic description of just about any behaviour in the Universe. From formation of stars to the melting of plastics, mathematics explains the how and why, as long as you know the language. Lie group theory plays an increasingly important role in fundamental descriptions of modern physics, unifying many related fields. It is the basis of the modern theory of elementary particles and thus of critical importance to descriptions of the nature of the Universe. The EU-funded project 'Singularities of Lie group actions in geometry and dynamics' (SILGA) focused on two particular applications of Lie groups. One of the key strands of research in this area is essentially a simplification of these mathematical representations in a way that still encodes the underlying mechanical and physical properties of the systems of study (reduction theory). Such foundations are important to descriptions of modern mathematical concepts such as string theory and formed the global perspective of the project. The study's local perspective was on the relative equilibria of Hamiltonian systems. Mathematics utilises symmetries to provide important qualitative information, such as that concerning stabilities or bifurcations, in a small neighbourhood of a solution. SILGA used semi-local methods to obtain a mathematical form relevant to reduction theory. This was successfully applied to advance mathematical descriptions of various phenomena in several dynamical and geometrical contexts. In groundbreaking work, they provided a common framework for relative equilibria of Hamiltonian systems. It was used to prove virtually every previous result about relative equilibria, advancing the theory with new results. The project has advanced the mathematical foundations necessary to predict and describe a number of behaviours critical to modern physics. Along the way, researchers further developed their techniques and knowledge for lasting impact on their chosen careers.

Keywords

Reduction theory, relative equilibria, mathematics, computational models, physical systems, Lie group, physics, singularities, Lie group actions, geometry, Hamiltonian, symmetries

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