The goal of this award is to further the understanding of dynamical systems and geometry relating to important families of surfaces. Dynamical systems (systems that evolve over time) are ubiquitous mathematical objects that can be used to model real world phenomena such as weather systems, robotics, and the motion of particles. Surfaces are two-dimensional structures that play a fundamental role in mathematics, as well as in other science and engineering disciplines. In particular, the surfaces of interest in this project have deep connections to other areas of mathematics including algebraic geometry, number theory, and mathematical physics. As part of the broader impacts of this project, the PI will aim to expand access to mathematics research by involving undergraduate and graduate students in projects in the PI's research field. The PI will also work with local and national programs that encourage STEM engagement at the K-12 level and the preparation of students for STEM careers. This project focuses on the dynamics and geometry of translation surfaces, dilation surfaces, and their moduli spaces. For translation surfaces, there is a well-established connection between the dynamics of the natural SL(2,R) action on the moduli space of translation surfaces and the dynamics of the straight-line flow on individual translation surfaces. The PI will utilize this connection to study problems about the structure and dynamics of random translation surfaces, as well as to apply