Research

Flat polyhedral tori in three-space with eight vertices, the fewest possible, and which shapes of flat torus they can take.

Which rotationally symmetric surfaces can be realized as surfaces of revolution in which homogeneous three-dimensional geometries, Riemannian and Lorentzian.

The last three open degree-six symplectic hypergeometric monodromy groups, settled with machine-found certificates verified in exact arithmetic, and a program to do the same for every hypergeometric group in degree six and below.

Finding isometric embeddings of discretized surfaces and manifolds into curved ambient spaces by minimizing an energy, beginning with hyperbolic surfaces drawn in three-dimensional space.

Faithful pictures of elliptic curves over the complex numbers and over finite fields, and the equivalence of categories that makes the finite field pictures work.

Rendering spacetimes with many black holes by turning general relativity into classical optics.

Drawing the space around a knot from the inside, by thickening knots in the 3-sphere and projecting from a point on the knot.

Embedding graphs and knowledge into Riemannian symmetric spaces, where Euclidean and hyperbolic pieces coexist.

Pictures of every complex algebraic number of low degree, and the hyperbolic geometry and Diophantine approximation they reveal.

Real-time, geometrically exact views from inside all eight Thurston geometries

How one geometry degenerates into another inside projective space, from the Heisenberg plane to the first limits that reach Nil.