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
Fields of science (EuroSciVoc)
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CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
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Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
ERC-ADG - Advanced Grant
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2020-ADG
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80539 MUNCHEN
Germany
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.