What is the shape of the universe?
No one knows. What makes the question especially hard is that, unlike planets or moons, we can't step back to view things from afar. We are stuck inside the universe.
Doesn't the universe just go off infinitely far in all directions? Maybe. But it also could be finite, like a higher dimensional sphere or torus. Indeed, there are infinite upon infinite possibilities for the what is called the topology of the universe –– its spatial nature independent of size. Some of those possibilities exhibit sheering, others knottedness, and others still decompositions into simpler shapes.
My mathematical research studies potential universes, called 3-manifolds (the "3" standing for three dimensions), not empirically by building or viewing anything, but by considering the possible mathematical structures such spaces can admit. In particular, I am interested in the relationships between different kinds of substructures that embed in 3-manifolds, particularly essential and Heegaard surfaces.
Mathematical Research Publications
Locally Helical Surfaces Have Bounded Twisting
An extension of the analogy between topologically minimal and geometrically minimal surfaces.
With D. Bachman and E. Sedgwick, Pacific Journal of Mathematics 292-2 (2018), 257-272Computing Heegaard Genus Is NP-hard
The problem of determining the Heegaard genus of a 3-manifold is NP-hard.
With D. Bachman and E. Sedgwick, included in A Journey Through Discrete Mathematics, edited by M./ Loebl, J. Nešetril, R. Thomas, Springer (2017), 59-88Heegaard Structure Respects Complicated JSJ Decompositions
The Heegaard splittings of generic toroidal 3-manifolds are amalgamations of splittings of the components in the JSJ decomposition.
With D. Bachman and E. Sedgwick, Mathematische Annalen 365(3) (2016), 1137-1154Surfaces That Become Isotopic After Dehn Filling
The set of essential surfaces in a 3-manifold with torus boundary remains unchanged under generic Dehn fillings.
With D. Bachman and E. Sedgwick, Communications in Analysis and Geometry 23, no. 2 (2015), 363-376Almost Normal Surfaces with Boundary
Strongly irreducible and boundary-strongly irreducible surfaces can be isotoped to be almost normal.
With D. Bachman and E. Sedgwick, Geometry & Topology Down Under, American Mathematics Society Contemporary Mathematics Series 597 (2013), 177-194Stabilization, Amalgamation and Curves of Intersection of Heegaard Splittings
A stabilization bound for Heegaard splittings in Haken 3-manifolds, and an example of surfaces that must intersect in at least n curves.
Algebraic and Geometric Topology 9 (2009) 811-832Degeneration of Heegaard Genus, a Survey
A survey of when Heegaard genus is lower than expected upon gluing 3-manifolds together.
With D. Bachman, Geometry & Topology Monographs 12 (2007) 1-15Stabilizing Heegaard Splittings of Toroidal 3-Manifolds
An upper bound on the number of stabilizations required for Heegaard splittings differing by a Dehn twist along an essential torus to become isotopic.
Topology and Its Applications 154, no. 9 (2007) 1841-1853Non-Isotopic Heegaard Splittings of Seifert Fibered Spaces
A characterization of precisely which Seifert fibered spaces yield infinitely many Heegaard splittings of the same genus.
With D. Bachman, Algebraic and Geometric Topology 6 (2006) 351-372Stabilizations of Heegaard Splittings of Graph Manifolds
Strongly irreducible Heegaard splittings of sufficient genus of graph manifolds become stabilized after one stabilization.
This is a main result from my Ph.D. Thesis, Heegaard Splittings of Toroidal 3-Manifolds, The University of Texas at Austin (2006)