A surface can be decomposed into a union of pairs of pants, a construction known as a pants decomposition. This fundamental observation reveals many important properties of surfaces. By forming a simplicial graph whose vertices represent pants decompositions, connecting two vertices with an edge whenever the corresponding decompositions differ by a simple move, we obtain a graph that is quasi-isometric to the Weil–Petersson metric on Teichmüller space. Meanwhile, topologists often study a structure called a rose, formed by attaching multiple circles at a single point. A rose is homotopy equivalent to a compact surface with boundary. Consequently, we can define a pants decomposition of a rose as the pants decomposition of a surface homotopy equivalent to it. In this talk, we will explore the concept of pants decompositions specifically in the context of roses.

In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role to understand group actions. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures. In this talk, we will study non-linear actions on surfaces using random dynamics. We will discuss about absolutely continuity and exact dimensionality of stationary measures as well as classification of orbit closures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.