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Solving cutting-edge math problems

European researchers are expanding the boundaries of knowledge in the area of mathematics where differential geometry meets gauge theory. Collaboration has brought new insights into various aspects of the so-called parabolic construction of principal bundles over a smooth projective variety.
Solving cutting-edge math problems
The study of principal g-bundles is an area of mathematics that has important implications in the fields of topology — the study of shapes and spaces — and differential geometry. There are also important applications in physics, where they form the theoretical basis for gauge theory and provides links to conformal field and string theories.

The EU-funded research project, 'Moduli of principal g-bundles over curves' (MODPRINBUN), has enabled further investigation in this area through international collaboration between mathematicians. Research objectives included providing a direct proof of the unirationality of the moduli spaces of principal bundles over curves. Also, a closer study of stability of the principal bundle in terms of extension data stability was necessary.

Work focused on several specific problems, including ramified spectral covers, explicit examples of families of smooth curves having infinite monodromy for the Hitchin connection in rank 2, vector bundles in positive characteristic, and affine Lie algebras of the celebrated strange duality isomorphism between spaces of generalised theta functions.

Project outcomes led to the publication of a number of research papers in academic journals and international conferences. The conferences also attracted collaboration between leading European experts working in this field at a number of events.

MODPRINBUN, which finished in late 2012, has advanced knowledge and increased collaboration in this area of theoretical mathematics. Its insights will enable further developments and innovations in both theoretical and applied fields such as physics.

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