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.

Project Report

The principal investigator worked on and completed a number of projects in mathematics. The PI works in the fields of dynamical systems, topology, geometry, and complex analysis. All are fundamental in mathematics and these subjects often overlap. In dynamical systems one studies the motion of objects as time evolves. Examples in the real world are the motion of planets and the motion of cells. The PI works on the subject of billiards in polygons and has been studying this dynamical system for years. As part of the current project the PI studied the size of the set of billiard paths that do not come too close to the vertices. Often in mathematics one studies an object and the various geometries one can put on it. A prime example is to take a surface and study all the ways one can put a hyperbolic or negatively curved metric on it. The set of all possible such metrics s called the moduli space of hyperbolic surfaces; a subject that goes back almost 200 years. It is important to study properties of this moduli space. One example is the Weil-Petersson metric on moduli space. In studying a metric on a space one wants to understand the behavior of geodesics or shortest paths. This connects the fields of geometry and dynamical systems. In one paper the PI along with collaborators proved the ergodicity of the geodesic flow on moduli space. Ergodicity is a fundamental concept. It means that the proportion of time a typical geodesic can be expected to spend in any part of the space is proportional to the volume of that part of the space. In other papers the PI studied behaviors of individual geodesics. The PI also wrote a paper in the subject of 3 dimensional topology. A fundamental object is what is called a handlebody. One can think of this as a solid donut with several holes. One can study the curves on the surface of the donut that can be filled in. In the language of topology these are called compressing discs. These compressing discs can be made into a graph and the PI showed this graph has a nice geometric property called hyperbolicity.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
0905907
Program Officer
Joanna Kania-Bartoszynska
Project Start
Project End
Budget Start
2009-07-15
Budget End
2014-06-30
Support Year
Fiscal Year
2009
Total Cost
$189,979
Indirect Cost
Name
University of Chicago
Department
Type
DUNS #
City
Chicago
State
IL
Country
United States
Zip Code
60637