Dynamical systems is an area of mathematics that seeks to understand the long term behavior of closed systems. Typical questions include how much of the space a point sees as time evolves, whether it sees parts of the space as often as one would expect and how correlated are different points. This project focussed on 3 families of dynamical systems: interval exchange transformations, flows on translation surfaces and Teichmueller geodesic flow. I proved that typical interval exchange transformations were different. As a consequence of this, I showed that the product of typical interval exchanges has that every orbit visits all parts of the product space as often as one would expect. Vaibhav Gadre and I proved the analogous result for flows on translation surfaces. Howard Masur, Yitwah Cheung and I showed that the trajectories for the Teichmueller flow which are bounded, and in particular spend all of their time in restricted subset, are large in certain sense. David Constantine and I proved a strong law of large number type result for interval exchange transformations. In particular, we showed conditions under which the typical orbit visits a family of shrinking targets about a point as often as one would naively expect. Jayadev Athreya and I proved a variety of results about special trajectories, called saddle connections, on flat surfaces. In particular, we obtained asymptotics for large and small angle gaps between special trajectories. Alex Eskin and I proved that on every translation surface, the typical flow equidistributes under so called renormalization dynamics. This strengthened numerous results that previously only held for typical translation surfaces to hold for all translation surfaces. Jon Fickenscher and I showed that for a large set of interval exchange transformations were topologically mixing. That is, the images of small balls are quite dense for all large enough iterates. During the time I was funded by the project I co-organized a conference, co-organized the University of Chicago's Geometry and Topology seminar and Dynamics seminar, started an informal dynamics seminar, participated in a student seminar, participated in reading courses and met with graduate students. Additionally, I disseminated results by writing papers and giving talks in seminars and conferences.
