Counting social interactions for discrete subsets of the plane
Author
Fairchild, Samantha Kay
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We will use dynamical, geometric, and analytic techniques to study translation surfaces. A translation surface is, informally, a collection of polygons in the plane with parallel sides identified by translation to form a surface with a singular Euclidean structure. Understanding the geometry and behavior of flows on translation surfaces and their moduli spaces has lead to the development of many new and revolutionary techniques, including the work of award winning mathematicians like Mirzakhani, McMullen, Eskin, Yoccoz, and Zorich. My work studies geodesic flows on translation surfaces by analyzing subsets of the plane corresponding to saddle connections, which are special trajectories of this flow. Saddle connections are straight line trajectories on a translation surface connecting two singular points, with none in their interior. Saddle connections are the driver for the parabolic behavior of geodesic flows on translation surfaces, as two nearby parallel lines behave differently under dynamics of the geodesic flow once a saddle connection comes between them.
This thesis will consider discrete subsets of the plane usually arising from saddle connections on translation surfaces. The following results are all partial answers to the following question: Given a surface, how are saddle connections distributed in the plane?
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