CAREER: Locally Homogeneous Geometric Manifolds and Their Moduli Spaces

University Of Texas At Austin, Austin TX

Investigators

Abstract

This project concerns locally homogeneous geometric manifolds. These are abstract mathematical objects designed to model the universe we live in. The term locally homogeneous refers to the presence of a high degree of local symmetry which is captured by a Lie group called the (local) symmetry group. It is this symmetry group which governs the geometry, in the following sense: the meaningful quantities we can measure in a geometric manifold (e.g. lengths or angles) are exactly those which are invariant under the symmetry group. 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 scales. The space of all such possibilities is called a moduli space; it is a topological space whose points are the possible geometric manifolds of a certain type and whose topology organizes those geometric manifolds into families whose features vary continuously. While the precise features (e.g. shape, size, etc.) of our universe is a question for empirical physics, a moduli space is the mathematical answer to the question of what possible features could the universe have. The research component of this project studies geometric manifolds of many types, with focus on flat affine geometry, real projective geometry, and constant curvature pseudo-Riemannian geometry. These geometries have just the right amount of symmetry to allow for large interesting moduli spaces with mysterious but tractable behavior. Core elements of the educational component include training Ph.D. students, a new Texas Experimental Geometry Lab for undergraduate research, and the Texas Winter Workshop on Geometric Structures for early career mathematicians. A guiding philosophy in the PI's research program is that geometric manifolds of one type may be fruitfully studied by deforming, or transitioning, to a different type of geometry. For example, the PI’s work studying Margulis spacetimes as geometric limits of anti de Sitter (AdS) spacetimes, has yielded many results about the geometry, topology, and deformation theory of these affine Lorentzian three-manifolds. The PI will develop new tools, following the geometric limit point of view, for new affine geometry contexts in higher dimensions, in order to address questions surrounding important open problems such as the Auslander Conjecture. The PI will also pursue a broad program to study convex real projective structures on manifolds. One focus is to identify which three-manifolds admit such structures and to describe the moduli spaces, with applications to questions in low-dimensional topology. In a more general context, the PI will conduct a thorough investigation of a new notion of convex cocompactness in real projective geometry, generalizing the well-studied notion from the Kleinian groups setting. Educational goals of this project center on expanding and vertically integrating the research training program in Geometric Structures at UT Austin being developed by the PI. The Texas Experimental Geometry Lab (TXGL) will introduce undergraduates at UT Austin to research in geometry, topology, and/or dynamics through computational and experimental projects. TXGL will also provide opportunities for Ph.D. students and post-docs to gain experience mentoring and teaching undergraduates outside the traditional classroom setting. In addition, TXGL will produce visualizations that illustrate concepts from current mathematics research. The Texas Winter Workshops in Geometric Structures will bring together small groups of early career mathematicians in order to learn an emerging new topic at the intersection of different (but related) mathematical fields. 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.