Geometry and topology are concerned with the study of shapes. In Euclidean geometry, we study objects such as circles and rectangles. A circle is highly symmetric; for example, any rotation about the center of the circle preserves the circle. A rectangle, though still symmetric in some ways, has less symmetry than the circle; its two sides may have different lengths. In low-dimensional topology we study more complicated objects or spaces of two, three, and four dimensions. A central aim of this National Science Foundation funded project is to understand these spaces through symmetries. Topological objects arise naturally in other fields, including biology, chemistry, physics, and engineering. At times, the best way to understand one is via its relationship with another through what is known as a "covering map". A second project is to analyze covering maps. A challenge in studying more intricate three or four dimensional spaces is that one cannot always visualize or draw these even using a computer. It is therefore helpful to break them into building blocks. One of the projects is to do so with objects called hyperbolic 3-manifolds. In addition to the mathematical research, the PI has demonstrated a strong dedication to outreach, mentoring, and advocating for underrepresented individuals in the STEM disciplines. With the NSF travel funds she will continue to engage in opportunities inside and outside the academia directed at promoting mathematical research and education.
The focus of this research project is to understand the finite degree covering spaces, the group of symmetries, and the combinatorics of hyperbolic manifolds in low dimensions. The PI plans to tackle the following projects during the funding period: (1) making effective the Virtually Haken Theorem, (2) quantifying separability properties to determine whether or not the fundamental groups of three-manifolds and the mapping class groups of closed surfaces are linear, (3) exploring infinite-type surfaces, their mapping class groups, and the actions of these groups on hyperbolic complexes, and (4) giving a combinatorial characterization for hyperbolic three-manifolds. Though the research project primarily focuses on questions in topology, geometry, and geometric group theory, the topics explored by the PI have deep connections to combinatorics, representation theory, dynamics, and topological quantum field theory (TQFT) as well.
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.