Project description DEENESFRITPL Study investigates positivity structures in higher Teichmüller spaces Lie groups, which describe the symmetries of a space or a system, are widely used in many parts of modern mathematics and physics. Total positivity is an important structure in Lie groups, and it has many applications in discrete mathematics, stochastic processes and representation theory. New positivity structures have recently been discovered that generalise the total positivity in Lie groups. The EU-funded PosLieRep project aims to study these new structures to gain new insight into the theory of higher Teichmüller spaces. Show the project objective Hide the project objective Objective Lie groups lie at the heart of mathematics. They play an important role in geometry, analysis, number theory, algebraic geometry and representation theory. As they describe symmetries of a space or a system they also appear prominently in theoretical physics. Not every system realizes the full amount of symmetry, it is therefore of key importance to investigate not only Lie groups, but also their subgroups, and in particular their discrete subgroups, which are often linked to geometric or arithmetic structures.This projects builds upon new developments in the theory of Lie groups, in particular the intro- duction of total positivity in split real Lie groups on the one hand and new exiciting phenomena in the study of discrete subgroups, in particular the emergence of higher Teichmüller spaces.Higher Teichmüller spaces generalize the classical theory of Fricke-Teichmüller space in the context of simple Lie groups of higher rank. The existence of higher Teichmu ̈ller spaces came as a surprise, and their discovery and investigation led to various other interesting developments, including an exciting interplay with the theory of Higgs bundles as well as with supersymmetric field theories in theoretical physics.In this proposal we develop a unifying framework for higher Teichmüller spaces, which comprises the two known families, Hitchin components and maximal representations, but conjecturally also two new families. The basis for this conjectural unified theory lies in a new notion of positivity in Lie groups, which generalizes Lusztig’s total positivity in the context of arbitrary real Lie groups that are not necessarily split. This generalization of total positivity is of interest in its own right and leads to many exciting questions and conjectures that will be addressed in this proposal. The three main themes of the proposed project are Positivity in Lie groups, Positive representations as higher Teichmüller spaces, and Symplectic geometry of representati Fields of science natural sciencesmathematicspure mathematicstopologysymplectic topologynatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometrynatural sciencesphysical sciencestheoretical physics Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2020-ADG - ERC ADVANCED GRANT Call for proposal ERC-2020-ADG See other projects for this call Funding Scheme ERC-ADG - Advanced Grant Coordinator RUPRECHT-KARLS-UNIVERSITAET HEIDELBERG Net EU contribution € 2 028 016,00 Address Seminarstrasse 2 69117 Heidelberg Germany See on map Region Baden-Württemberg Karlsruhe Heidelberg, Stadtkreis Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all RUPRECHT-KARLS-UNIVERSITAET HEIDELBERG Germany Net EU contribution € 2 028 016,00 Address Seminarstrasse 2 69117 Heidelberg See on map Region Baden-Württemberg Karlsruhe Heidelberg, Stadtkreis Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00
