SILGA

Singularities of Lie Group Actions in Geometry and Dynamics

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 Obiettivo del progetto (Objective)

'This project continues and advances the research lines of my past scientific activity, in the context of a reintegration to my home scientific community and the starting of a stable academic activity with a tenure track and a subsequent permanent position.

The scientific part of the proposal is the investigation of the geometry of actions of Lie groups in several dynamical and geometrical contexts, with emphasis in their singularities, and is articulated in the following two sections:

A. Reduction Theory. Following previous research by me and other groups, I will study the reduction theory for Dirac and generalized complex geometry from both the global and local point of view. This study is a natural continuation of previous research efforts about the reduction theory of symplectic and Poisson manifolds. The main novelty is that we will focus on reduction in the case when the group action presents singularities (fixed points) in Dirac and generalized complex geometry, which is a topic yet to be explored.

B. The theory of Hamiltonian relative equilibria. In this section I intend to perform a complete reorganization of the theory of Hamiltonian relative equilibria, as well as to advance it. We will competely redesign the existing theory in a way specifically adapted to the various distinguishing features of symmetric Hamiltonian systems This is a big project started during my previous stage at the University of Manchester. We have substantially advanced this problem during that period, and we expect to have results with significant impact within the duration of this Reintegration Grant.'

Introduzione (Teaser)

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.

Descrizione progetto (Article)

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.