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Positivity in Lie groups and representation varieties

Projektbeschreibung

Studie erkundet Positivitätsstrukturen in höheren Teichmüller-Räumen

Lie-Gruppen, welche die Symmetrien eines Raums oder eines Systems beschreiben, kommen in vielen Bereichen der modernen Mathematik und Physik zum Einsatz. Die totale Positivität ist eine wichtige Struktur in Lie-Gruppen und weist viele Anwendungen in der diskreten Mathematik, in stochastischen Prozessen und der Darstellungstheorie auf. Erst kürzlich wurden neue Positivitätsstrukturen entdeckt, welche die totale Positivität in Lie-Gruppen verallgemeinern. Ziel des EU-finanzierten Projekts PosLieRep ist, diese neuen Strukturen zu untersuchen, um neue Einblicke in die Theorie der höheren Teichmüller-Räume zu gewinnen.

Ziel

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

Wissenschaftliches Gebiet

CORDIS klassifiziert Projekte mit EuroSciVoc, einer mehrsprachigen Taxonomie der Wissenschaftsbereiche, durch einen halbautomatischen Prozess, der auf Verfahren der Verarbeitung natürlicher Sprache beruht.

Finanzierungsplan

ERC-ADG - Advanced Grant

Gastgebende Einrichtung

MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Netto-EU-Beitrag
€ 1 820 078,59
Adresse
HOFGARTENSTRASSE 8
80539 Munchen
Deutschland

Auf der Karte ansehen

Region
Bayern Oberbayern München, Kreisfreie Stadt
Aktivitätstyp
Research Organisations
Links
Gesamtkosten
€ 1 820 078,59

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