AEI-DFG: Very-Stability, Hecke transforms, and Representation Varieties in the Presence of symmetry

• Applicants: Professor Dr. Florent Schaffhauser; Professorin Dr. Anna Wienhard
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541679129

Moduli spaces of Higgs bundles on compact Riemann surfaces play an important role in several areas in mathematics. They are related to local systems on the surfaces, and to representations of their fundamental groups. Two key features of the moduli space of Higgs bundles, which do not have an equivalent on the representation variety, are the Hitchin map and the C*-action. For GL(n,C)-Higgs bundles recent developments, in particular crucial work of Hausel and Hitchin, have shed new light the interplay between the C*-action and the Hitchin map through the investigation of very-stable and wobbly Higgs bundles. In this project we plan to extend and generalise this investigation to other Lie groups, in particular real Lie groups, as well as to orbifolds and surfaces with real structures. For this we consider Higgs bundles in the presence of symmetry, focusing on very stability and wobbliness, the construction of Hecke transforms, the Byalinicki-Birula stratifications and applications to higher Teichmüller spaces.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partners: Professor Dr. Vicente Muñoz; Professorin Dr. Ana Peon-Nieto