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BRIDGES: Brazil-France interplays in Gauge Theory, extremal structures and stability


This project stands at the crossroads of complex algebraic geometry and Riemannian geometry to put together a team of Brazilian and French experts from both disciplines with a common goal: enhance our understanding of the interplays between algebraic invariant theory and special connections on bundles and build a transatlantic research group ready to weigh in on the new challenges of Geometry in the 21st century. A first evidence of such correspondence between special metric structures and algebraic conditions could already be found, more than a hundred years ago, in the Koebe-Poincaré Uniformisation Theorem. This has been fruitfully exploited in several branches of Physics, applied and pure Mathematics and lies at the core of the common intuition of modern-day geometers. In essence, this result classifies complex curves according to their unique constant scalar curvature metric, which yields a metric structure on their moduli space, thereby giving new tools for its study. That this is impossible to extend to higher dimensions has been known for many years, however hope remains that for some subclasses of connections with special holonomy, a classification may exist and, indeed, many results already point in this direction. We focus on three subfields of this very vast problem: A) Gauge Theory and slope stability; B) Canonical Kähler metrics and K-stability; C) G2-geometry and special structures. These three theories are at distinct stages of development, from the well-established to those in their nascent stages. While the second topic counts many experts in France, there are almost none in Brazil. In contrast, the third theme is essentially absent from the French mathematical community but quite an active field in Brazil. We propose to build a team of French and Brazilian mathematicians around these questions, with the aim of sharing knowledge, learning from one another and giving doctoral students and post-doctoral scholars the opportunity to take advantage of all the expertise at hand. This will require, on the one hand, transferring and adapting tools and techniques between the three theories and, on the other, transfers of technology and human capital between France and Brazil. (AU)

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