NSF Award
2405291
A topological surface is a space which is allowed to change its shape by stretching but without tearing. Understanding the geometry of the collection of these possible shapes, called the moduli space of the surface, has applications in a number of subfields of math and physics. Expanding on previous work, the PI will investigate properties of a particular flow on the moduli space that minimizes the distortions in a controllable way. This work is at the intersection of math and physics and is expected to lead to new connections between the two fields. The project will also introduce a new tool called the ghost algebra which allows one to track and extract properties of variations of quantities such as lengths. This grant will support graduate students in their research and travel, as well as the PI’s mentoring of graduate students and postdoctoral researchers. The PI will also co-organize conferences and workshops to support the career development of junior researchers. The findings from this research will be shared widely through conferences and seminars, fostering new connections between mathematics and physics.<br/><br/>This project has two main areas of study 1) the ghost algebra for correlation functions of Anosov representations and 2) the Weil Petersson gradient flow for renormalized volume. In the first, the PI and collaborators will study the symplectic structure of Higher Teichmüller spaces via the Hamiltonian flows of correlation functions by introducing a new combinatorial object called the ghost bracket on the ghost algebra of ghost polygons that allows one to compute Poisson brackets of correlation functions. In renormalized volume, the PI and collaborators will continue their program to use the Weil-Petersson gradient flow of renormalized volume to study the structure of hyperbolic three-manifolds. This program has been very successful culminating in recently completely describing the flow for acylindrical manifolds and relatively acylindrical manifolds. This sets the stage to attack the general case which will occupy the majority of this part of the research program.<br/><br/>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.
