Deformation Spaces of Geometric Structures

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

In this project the PI will investigate deformation spaces of geometric structures. A surface is a space which looks locally like the two-dimensional plane. Examples are the surface of a football or a donut. Surfaces arise naturally in the mathematical fields of topology, complex analysis, and dynamics. Classical Teichmuller theory is an area of mathematics that studies the space of all geometric shapes on a fixed surface. It also interacts with other scientific fields, e.g. through its connections with string theory in theoretical physics. The PI's research lies in two types of generalizations of Teichmuller theory. Three-dimensional generalizations of surfaces are called three-manifolds, and these are locally like the three-dimensional space we live in. The PI will continue a long-term project to classify and understand possible shapes of three-dimensional manifolds. In Higher Teichmuller Theory, the PI will study deformation spaces of geometric structures on higher-dimensional spaces with an emphasis on understanding the metrics, or distance functions, on these spaces. The PI will also contribute to the mathematical community through involvement in the Inquiry Based Learning Center at the University of Michigan, curriculum development for undergraduate courses, serving as editor of mathematical journals, organizing conferences, and mentoring undergraduate students, graduate students and postdoctoral assistant professors. Higher Teichmuller theory studies geometric representations of hyperbolic groups into semi-simple Lie groups, usually of rank at least two. It is guided by inspiration from Teichmuller theory and the overall goal is to create a general theory with some of the beauty and depth of classical Teichmuller theory. The PI will use dynamical and geometric tools to study pressure metrics on Hitchin components. In particular, the PI will investigate an analogy between augmented Teichmuller space and an augmented Hitchin component introduced by Loftin. The PI will also study deformation spaces of finite area projective surfaces, Liouville currents associated to Hitchin representations, the class of groups which admit Anosov representations of specified type and the structure of spaces of Anosov representations. In the field of Kleinian groups, the PI will use tools developed in the proof of the Ending Lamination Conjecture to build combinatorial model manifolds for hyperbolic three-manifolds with freely indecomposable fundamental group. This project is expected to yield a finer understanding of Thurston's skinning map and to allow the PI and his co-authors to establish an iterated bounded image theorem, which was used by Thurston in the original proof of the Geometrization Theorem for Haken 3-manifolds, but whose proof remains elusive. The PI will also use dynamical tools to study deformation spaces of geometrically finite groups, with the goal of proving analyticity of the Hausdorff dimension of the limit set and producing pressure metrics. 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.