Ordered groups and 3-manifolds

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My proposed research deals with the interplay between two fields of mathematics known as group theory and the topology of three-dimensional manifolds. Three-dimensional manifolds are spaces that model many phenomena that occur naturally in mathematics, but also phenomena in nature, such as the possible shapes of the universe. Group theory is a type of algebra that, in its early days, dealt primarily with the properties of symmetry and rigid motions of shapes, and so in its modern form this algebraic theory is well-adapted to studying the shapes of three-dimensional spaces. The goal at hand is to use group theory to better understand the possible shapes and topologies of three-dimensional manifolds. **One of the common modern techniques is to take a three-dimensional manifold, and from it, calculate what is called its `fundamental group', an algebraic object that carries with it an incredible amount of data about the symmetries, shape and structure of the manifold you started with. Having computed the fundamental group of a manifold, the challenge is to extract all the available information from the output of your calculation, a problem which has many different approaches and has been an ongoing topic of research for decades. My research program will focus on trying a new technique of investigating the information carried by the fundamental group of a three dimensional manifold, by ordering the elements of the fundamental group (an algebraic exercise) and seeing what properties of the manifolds (a topological question) are reflected by the orderings.**This line of research is particularly interesting at present, because while the idea of ordering groups has been around for years, the idea of ordering fundamental groups is new and is producing surprising results. The outputs of the calculations are seeming to indicate that this new approach may be connected to a powerful theory called Heegaard-Floer homology, and a related area of study known as foliation theory, although for the time being there is no known reason why such connections should exist. Developments linking these three fields will allow for a fruitful transfer of ideas, problem-solving techniques and theorems from one area to another, as has already been the case in some of the simpler situations where the connection is understood.