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Clusters Algebras and Integrable Systems

Grant number: 12/04707-9
Support Opportunities:Scholarships in Brazil - Doctorate
Effective date (Start): June 01, 2012
Effective date (End): February 29, 2016
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Igor Mencattini
Grantee:Eber Daniel Chuno Vizarreta
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated scholarship(s):14/08512-3 - Cluster algebras and integrable systems, BE.EP.DR


The theory introduced by Fock-Goncharov, called Higher Teichmüller Theory (HTT), is a huge generalization of the classical Teichmüller theory. It has as main ingredients a compact oriented surface S and a semisimple algebraic group G. The HTT provides a description of the space of the so called positive representations of the fundamental group of the surface S into G, showing that these are faithful, discrete and hyperbolic. The goal of this project is to apply the HTT to give a cluster algebras interpretation of a class of integrable systems which are naturally defined on the moduli spaces of $SL_{n}$-local systems on the genus 1 surfaces with a puncture. We also seek understanding the relevance of the HTT in the study of representations of the fundamental group of the surface S into PU(2,1) (the group of holomorphic isometries of the complex hyperbolic space). Such representation are related to the construction of complex hyperbolic geometry.

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Academic Publications
(References retrieved automatically from State of São Paulo Research Institutions)
VIZARRETA, Eber Daniel Chuno. On Coxeter-Toda lattices. 2016. Doctoral Thesis - Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) São Carlos.

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