Research

Projects in Progress

Thurston's famous Geometrization Conjecture (proven by Perelmen in 2006) states that a closed orientable 3-manifold decomposes into components which each have one of eight model geometries; the geometry with the richest theory being hyperbolic geometry. A family of closed n-manifolds is locally combinatorially defined (or LCD) if there are finitely many finite simplicial complexes, called models, such that a closed n-manifold is in the family if and only if it has a triangulation that is locally one of these models. 

Preliminary Theorem 1 (Cooper, M. Patel): The set of closed 3-manifolds that admit a particular Thurston geometry is LCD.

Manually constructing the models can be a tedious task, making an alternative approach using branched n-manifolds particularly valuable. The concept of a branched n-manifold was introduced by Bob Williams in his research in dynamical systems. We use a slightly stronger definition than Williams. A branched n-manifold is finite simplicial complex. It is a generalization of a smooth n-manifold that maintains a well defined tangent space at each point and is covered by coordinate charts. Unlike a smooth manifold, every point in a branched n-manifold has an open neighborhood that is the union of finitely many subspaces, called sheets, which are all homeomorphic to R^n. In dimensions 1 and 2, branched manifolds are used extensively to study the geometry of 3-manifolds. Branched n-manifolds and LCD sets are closely related. In particular: 

Preliminary Theorem 2 (Cooper, M. Patel):  Let W be a branched n-manifold. The family of closed n-manifolds that immerse into W is LCD. 

Therefore, to prove Preliminary Theorem 1, it is enough to show: 

Preliminary Theorem 3 (Cooper, M. Patel):  Let G be a Thurston geometry. There is a compact branched manifold, W(G), such that a closed 3-manifold M admits a G-structure if and only if there is a immersion of M into W(G).

For each geometry except hyperbolic geometry, we construct the branched 3-manifold that satisfies Preliminary Theorem 3. The hyperbolic case uses LCD sets and branched 3-manifolds (and is still in progress). 

Papers and Preprints

Integral Metaplectic Modular Categories  (with A. Deaton, P. Gustafson, E.C.Rowell, S. Poltorastski, S. Timmerman, B. Warren, Q. Zhang) Journal of Knot Theory and Its Ramifications Vol. 29, No. 5 (2020).

Unoriented Links and the Jones Polynomial (with S. Ganzell, J. Huffman, K. Tademy, G. Walker) Involve, a Journal of Mathematics Vol. 12 (2019) pg. 1357-1367.

Conferences & Travel