Decorated Super-Teichmüller Spaces and Cluster 'Superalgebras'

Penner's decorated Teichmüller space of a punctured surface is known to have a cluster algebra structure. And more recently, a decorated version of super-Teichmüller theory is developed by Penner and Zeitlin. In this talk, we will explore the decorated super-Teichmüller theory and its connection to certain cluster 'super'algebras, in particular, Ovsienko-Shapiro's extended quiver. Special attention will be given to the case of once-punctured tori, which leads to the definition of dual Markov numbers.