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Positivity in Lie groups and representation varieties

Project description

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.

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

Host institution

MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Net EU contribution
€ 1 820 078,59
Address
HOFGARTENSTRASSE 8
80539 Munchen
Germany

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Region
Bayern Oberbayern München, Kreisfreie Stadt
Activity type
Research Organisations
Links
Total cost
€ 1 820 078,59

Beneficiaries (2)