## Independent study and Research on

the Geometry and Dynamics of Surfaces

The study of surfaces is one of the most classical topics in mathematics. Today, surfaces are studied from various perspectives, including metric geometry, dynamics, topology, complex analysis, combinatorics and group theory. These different points of view are connected in deep and surprising ways.

The most obvious example of a surface is the Euclidean plane, but there are many other examples: the sphere and the torus, as well as more exotic examples like the Klein bottle, the MÃ¶bius strip and projective plane. One especially important example is the square torus, which is constructed by gluing the opposite sides of a square. It admits a natural flat metric coming from the square as a subset of the plane, and the linear flow on it can be understood using linear algebra. Other surfaces are also formed by gluing together sides of planar polygons and can be studied in a similar way to the square torus.

The goal of the independent study part of this project is to explore and understand different geometric structures on surfaces and natural induced dynamical systems. A few students will have the opportunity to continue their study by starting a research project during the summer, which should lead to an honors thesis.