Discrete subgroups of Lie groups and geometric structures on manifolds

This project studies locally homogeneous geometric manifolds, which are abstract mathematical objects designed to model the physical universe. The term locally homogeneous refers to the presence of a high degree of local symmetry which is captured by an object called the (local) symmetry group. It is this symmetry group which dictates the geometry, in the following sense: the meaningful quantities we can measure in a geometric manifold, such as lengths or angles, are exactly those which are invariant under the symmetry. There are many different possible symmetry groups which lead to different types of geometric manifolds useful in many contexts across mathematics and physics. There can also be many different geometric manifolds with the same local symmetry group. These all have the same local properties but can look very different at large scale. The space of all such possibilities is called a moduli space. While the precise features, for example the shape or size, of the universe is a question for empirical physics, a moduli space is the mathematical answer to the question of what possible features the universe could have. The research goals of this project are, roughly, to better understand several special types of locally homogeneous geometric manifolds which have mysterious but tractable behavior. This project contributes to the growing base of foundational mathematical knowledge on which many innovations in science and engineering are eventually built. Broader impacts of the project focus on guided student research, contributing to training the next generation of research mathematicians. The principal investigator (PI) will train graduate students to perform research related to the main topics of the project. The PI will also advise undergraduate computation and visualization projects focused on these research themes. The project develops a new program to study the surface group representations comprising higher rank Teichmuller spaces by examining extreme behavior in the Benoist limit cone. Initial results suggest that such extreme behavior is achieved by fundamental topological structures on the surface, analogous to the maximally stretched laminations appearing in the study of Thurston's asymmetric metric. The project also continues a broad program to study convex real projective structures, building on the PI’s prior work on convex real projective Dehn filling in dimension three, and the principal investigator's prior work on a general notion of convex cocompactness in projective space (of any dimension) generalizing the well-studied notion from Kleinian groups. Finally, the project will extend themes from the PI's body of work on Margulis spacetimes to new affine geometry contexts in higher dimensions, motivated by the Auslander Conjecture. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.