02
Aug

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- Department of Mathematics
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Dissertation defense: Canonical Decompositions of Hyperbolic 3-Orbifolds

Thurston's geometrization program revolutionized 3-manifold topology, starting in the 1970s. A key part of the program was the idea that in some sense most 3-manifolds admit a

"hyperbolic structure". If this structure has finite volume, then its geometric properties are very tightly connected to the topological properties of the manifold, and researchers are very interested in exploring this connection. One example is the "canonical", or "Epstein-Penner", decomposition of a non-compact finite-volume hyperbolic 3-manifold. It is a decompostion of the manifold into ideal polyhedra which is completely determined by the hyperbolic structure and has many useful applications. It is a famous accomplishment in our field that the computer program SnapPy can compute canonical decompositions of such manifolds. The goal of my work is to extend SnapPy's ability to compute canonical decompositions of non-compact finite-volume hyperbolic 3-manifolds to non-compact finite-volume hyperbolic 3-orbifolds, where an orbifold is a natural generalization of a manifold which is prevalent in many areas of mathematics. In this dissertation, I describe my algorithm and summarize how it has been used in a joint project with my advisor and his other students to create a census of orbifolds commensurable to the figure eight knot complement.

Committee Chair and Advisor: Dr. Jason DeBlois

Tuesday, August 2 at 10:00 a.m. to 1:00 p.m.

Benedum Hall, 226

3700 O'Hara Street, Pittsburgh, PA 15261

Dissertation defense: Canonical Decompositions of Hyperbolic 3-Orbifolds

Thurston's geometrization program revolutionized 3-manifold topology, starting in the 1970s. A key part of the program was the idea that in some sense most 3-manifolds admit a

"hyperbolic structure". If this structure has finite volume, then its geometric properties are very tightly connected to the topological properties of the manifold, and researchers are very interested in exploring this connection. One example is the "canonical", or "Epstein-Penner", decomposition of a non-compact finite-volume hyperbolic 3-manifold. It is a decompostion of the manifold into ideal polyhedra which is completely determined by the hyperbolic structure and has many useful applications. It is a famous accomplishment in our field that the computer program SnapPy can compute canonical decompositions of such manifolds. The goal of my work is to extend SnapPy's ability to compute canonical decompositions of non-compact finite-volume hyperbolic 3-manifolds to non-compact finite-volume hyperbolic 3-orbifolds, where an orbifold is a natural generalization of a manifold which is prevalent in many areas of mathematics. In this dissertation, I describe my algorithm and summarize how it has been used in a joint project with my advisor and his other students to create a census of orbifolds commensurable to the figure eight knot complement.

Committee Chair and Advisor: Dr. Jason DeBlois

Tuesday, August 2 at 10:00 a.m. to 1:00 p.m.

Benedum Hall, 226

3700 O'Hara Street, Pittsburgh, PA 15261

- Event Type
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- University Unit
- Department of Mathematics
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