Richard Canary
University of Michigan, MI·Mathematics & Statistics
A manifold is a shape that looks like ordinary flat space up close but can have any dimension. The surface of a donut or a pretzel is a two-dimensional example, and our universe is a three-dimensional one.
- Fields
Mathematics & Statistics
- Topics
geometric topology
hyperbolic geometry
hyperbolic 3-manifolds
- Grants
- 1 known grant, through Aug 2027
What they work on
- Study the metric completion of the Hitchin component, a higher Teichmüller space of surface group representations, with respect to a pressure metric that generalizes the Weil-Petersson metric.
- Develop a geometric theory of the augmented Hitchin component that parallels classical augmented Teichmüller space.
- Construct uniformly bilipschitz combinatorial models for every hyperbolic 3-manifold with finitely generated, freely indecomposable fundamental group, extending tools from the Ending Lamination Conjecture.
More about the research
A single manifold can carry many different geometric structures, meaning different ways of measuring distance and angle. Mathematicians collect all of these structures into a "deformation space." Such spaces come up throughout mathematics and in physics. This project studies deformation spaces in two settings: higher rank structures on surfaces, and hyperbolic structures on three-dimensional manifolds.
Why it matters
This is basic mathematics. It adds to the understanding of how geometric structures on surfaces and three-dimensional spaces can vary and degenerate. It extends classical Teichmüller theory to higher rank settings and gives concrete models of hyperbolic 3-manifolds, which could inform other areas of mathematics and physics where these deformation spaces arise.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Matthew Stover
Temple University, PA·Mathematics & Statistics
Use ideas from geometry, topology, algebraic geometry and dynamics to answer open questions, including basic existence problems, about real and complex hyperbolic lattices, which are discrete symmetry groups of curved spaces.
Alexander Wright
University of Michigan, MI·Mathematics & Statistics·Early-career
Build quasi-convex co-bounded planes in Teichmüller spaces (spaces of hyperbolic shapes on a surface) by gluing together pieces obtained from dynamics, toward convex cocompact surface subgroups of mapping class groups.
Michelle Chu
University of Minnesota, MN·Mathematics & Statistics·Early-career
Study virtual properties of the fundamental groups of hyperbolic 3-manifolds — properties held by finite-index subgroups, which correspond to finite-sheeted covering spaces of the manifold.
The school
University of Michigan: 124 labs on PI Pages, $2.1B a year on research, 963 research doctorates a year.
Reach out
A start for your email. Fill in the brackets.