Project description
Higgs bundle geometry seen in a new light
Higgs bundles are the subject of research in several areas in mathematics and physics. A central feature of Higgs bundles is that they come in collections parametrised by the points of a quasi-projective variety: the moduli spaces of Higgs bundles, which have been found to play a central role in the geometric Langlands programme. Research is currently focussing on inducing the full Langlands correspondence from the Higgs bundle abelianised version. The EU-funded GoH project will expand on the research by studying central elements of the geometry of Higgs bundles from a new perspective. In particular, it will explore the Bialynicki-Birula stratification through advanced algebraic techniques and carefully study the irreducible components of the nilpotent cone by applying the theory of SU(p,q)-Higgs bundles.
Objective
Higgs bundles play a fundamental role in the current panorama of mathematics and theoretical physics through their many connections. Amongst the latter is the link with the geometric Langlands programme, a suitable generalization of the relation between a curve and its Picard variety, which moreover admits a natural quantum field theoretical interpretation. According to this, any G- local system on a curve yields a perverse sheaf on the moduli stack of G*-bundles (where G* is the Langlands dual to G). A simpler (abelianised) version of the geometric Langlands programme has been proven for Higgs bundles by Donagi and Pantev. A programme initiated by these two scientists aims at inducing the full Langlands correspondence from its abelianised version. Building on the work of the researcher and the hosts, we will fill in the gaps of this program and provide alternative tools broadening the current state of the art also beyond this action. In doing so, we will study central elements of the geometry of Higgs bundles from a new perspective. More precisely, we will give a way to understand the Bialynicki-Birula stratification via algebraic techniques, and, related to that, carefully study the irreducible components of the nilpotent cone, applying also the theory of SU(p,q)-Higgs bundles. Finally, we will explore the case of positive characteristic, with the aim to shed light on the Hecke eigenproperty in this setting.
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Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
06100 Nice
France