Topology and 3-Manifolds
Normal surfaces, Heegaard splittings, topologically minimal surfaces, and related questions in knot theory and 3-manifold topology.
Research
Research Areas and Publications
My research has centered on topology, geometry, visualization, and more recent work at the intersection of computation, machine learning, and mathematical representation.
Research Areas
Geometry and Visualization
Hyperbolic geometry, minimal surfaces, surface modeling, and visual approaches to mathematical understanding.
Computation and AI
Discrete differential geometry, machine learning, reinforcement learning, and computational tools for mathematical exploration.
Books
Grasshopper: Visual Scripting for Rhinoceros 3D
Industrial Press, 2017.
A Geometric Approach to Differential Forms
Birkhäuser, first edition 2006, second edition 2012.
Advanced Calculus DeMYSTiFieD
McGraw-Hill, 2007.
Selected Publications
- “Cohomology fractals, Cannon-Thurston maps, and the geodesic flow” (with M. Goerner, S. Schleimer, and H. Segerman), Experimental Mathematics, 2022.
- “Locally Helical Surfaces have Bounded Twisting” (with R. Derby-Talbot and E. Sedgwick), Pacific Journal of Mathematics, 2018.
- “Computing Heegaard Genus is NP-Hard” (with R. Derby-Talbot and E. Sedgwick), in A Journey through Discrete Mathematics, 2017.
- “Topological Index Theory for Surfaces in 3-Manifolds,” Geometry & Topology, 2010.
- “Connected sums of unstabilized Heegaard splittings are unstabilized,” Geometry & Topology, 2008.
Selected Grants
- Pitzer College Scholar in Residence, Fall 2020.
- NSF Research Grant (PI, DMS-1207804), “Applications of Topologically Minimal Surfaces,” 2012-15.
- NSF Research Grant (PI, DMS-0906151), “Topologically Minimal Surfaces in 3-Manifolds,” 2009-12.
Recent Academic Activity
- Organizer, Semester Program on “Illustrating Mathematics,” 2026.
- Organizer, Reunion and Expansion Conference on “Illustrating Mathematics,” ICERM, 2025.
- Speaker, AMS Section Meeting, San Luis Obispo, 2025.