Geometric and dynamical problems on surfaces

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The principal investigator will work on problems in the geometry, topology, and dynamics of surfaces. The objective of the first part of the proposal is to increase our understanding of the mapping class group, its action on the Teichmuller space of a surface and the boundary at infinity of Teichmuller space. Among the specific problems is one that attempts to show that from the point of view of counting orbits points for the action of the mapping class group on Teichmuller space, a random mappping class group element is pseudo-Anosov. Another problem is to study the distribution of orbits of the mapping class group on Thurston's space of measured foliations. The objective of the second part of the proposal is to study the dynamics of flows on translation surfaces. The principal investigator is interested specifically in the phenomenon of minimal but not uniquely ergodic directions for the linear flow on the translation surface.The principal investigator works in several diverse areas of theoretical mathematics. These fields are topology, geometry, and dynamical systems. They come together in the broad area of studying properties of surfaces. This is a subject of mathematics that goes back hundreds of years, and yet remains a vibrant area of modern research. The work of the principal investigator involves expanding our knowledge of these areas and also involves the training of graduate students to become research mathematicians.