Clusters Algebras and Integrable Systems

• Grant number: 12/04707-9
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: June 01, 2012
• End date: 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

Abstract

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.

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.