Proceedings of the Young Researchers Workshop on Positivity in Lie Groups

TL;DR

This workshop notes explore the concept of $ heta$-positivity in semisimple Lie groups, its connection to higher Teichmüller theory, and its potential to classify higher Teichmüller spaces, marking progress in the field.

Contribution

It introduces and discusses $ heta$-positivity as a generalization of total positivity, highlighting its significance in classifying higher Teichmüller spaces.

Findings

$ heta$-positivity generalizes Lusztig's total positivity.

It relates to higher Teichmüller theory.

Progress expected in classifying higher Teichmüller spaces.

Abstract

These notes transcribe a workshop about the notion of total positivity and $\Theta$-positivity and its relation to Higher Teichm\"uller Theory. $\Theta$-positivity is a notion of positivity in semisimple Lie groups and was recently introduced by Guichard and Wienhard as a generalization of Lusztig's total positivity. It is believed to be the cathartic notion to classify higher Teichm\"uller spaces. Without doubt, substantial progress will be achieved in the near future on the study of $\Theta$-positive structures. These notes provide an account of the state of the art as of 2021.