Geometric Limits in Higher Teichmueller Theory

William Marsh Rice University, Houston TX

Investigators

Abstract

A manifold is an abstract mathematical object that in the small looks like the space we live in, but that can have different global properties in the large. A geometric structure on a manifold is a way of measuring quantities that are invariant under a certain group of symmetries, the same way distances and angles do not change under rotations or translations. In many cases, on a fixed manifold, one can consider different geometric structures with the same underlying group of symmetries. The world of possible such structures is the moduli space. One example is the classic Teichmuller space that parametrizes metrics of constant negative curvature on certain two dimensional manifolds. Higher Teichmuller theory studies more general geometric structures on more complicated manifolds and their dynamical properties. This award provides funding for the research that focuses on fundamental questions in higher Teichmuller theory: how can we describe the geometry of these objects starting from the parameters in the moduli space? In particular, can we understand how these structures degenerate when the parameters leave all compact sets in the moduli space? Some numerical experiments related to this research will be carried out as undergraduate research supervised by the PI. The PI plans to study geometric structures on low-dimensional manifolds with a rank 2 semi-simple Lie group of symmetries, such as anti-de Sitter, convex projective and Lorentzian conformally flat, using techniques from Higgs bundles, harmonic maps and representation theory. More precisely, the PI will study geometric limits of these structures when the parameters leave every compact set in the corresponding moduli spaces, with the aim of finding a (partial) compactification of these moduli spaces and give a geometric interpretation of the boundary points. A different line of research is related to a long-standing question by Gromov about existence and regularity of minimal area metrics on surfaces with a fixed systolic constraint, which he will investigate exploiting tools from convex optimization. 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.