Geometry of Surfaces

Convex $\mathbb{R}\mathsf{P}^2$ Structures

Hyperbolic triangle groups are rigid, a fact easily believed once one remembers that a hyperbolic triangle is completely determined by its angles. However, these groups admit nontrivial deformations into $\mathsf{SL}(3;\mathbb{R})$, geometrically describing convex real projective structures.

Flat Tori and the Hopf Map

The preimage of a simple closed curve on the two sphere under the Hopf map is an intrisically Euclidean torus in $\mathbb{S}^3$. In fact Ulrich Pinkall showed that all Euclidean tori may be constructed in this way, and gives an explicit formula relating the moduli of such a torus to the length/enclosed area of the cuve on $\mathbb{S}^2$.

A Willmore Torus in S3

Using Pinkall's construction from above, we can seek a hopf torus which additionally minimizes mean curvature in S3, via a differential equation on the space of curves on S2. This shows a tours near such a minimum, visualizes via a perspective (not conformal) projection of R4.

A Closed Geodesic on the Modular Surface

A closed geodesic on the moduli space of tori, illustrated not through conformal embeddings into R3 as above, but ratherr by drawing the associated loop of Weierstrass p-functions.

A transition of conemanifold structures on S2

A geometric transition from the (2,3,7) hyperbolic trianagle tiling to the (2,3,6) euclidean one, computed as the developing maps for a unit area geometric structures on the (2,3,x) spherical conemanifold.

3-Manifold Topology

Seifert Fibrations of $\mathbb{S}^3$

The three sphere is a circle bundle over the two sphere, giving a fibration of $\mathbb{S}^3$ as well as a system of toroidal coordiantes. Furthermore, this fibration is but the simplest in an infinite sequence of Seifert fibrations of $\mathbb{S}^3$ which in general fill the sphere with $(p,q)$ torus knots.

A boundary slope of the figure 8 knot complement.

An (arbitrarily chosen) boundary slope for the figure 8 knot complement, rendered intrinsically in the incomplete Euclidean metric it inhertis under stereographic projection from a point on the knot.

Rotations of the Hypercube.

A hypercube tiling of S3 with totally geodesic faces, rotating under a 1-parameter subgroup of SO(4).

3-Manifold Geometry

Rotations of the Hypercube.

Really Lorentzian (3+1) manifold geometry: an accretion disk surrounding a Schwarzschild black hole.

Algebraic Geometry

Finding Lines on a Cubic

Every (complex) cubic surface contains exactly 27 (projective) lines. One way of starting to approach this fact (which was introduced to me at a summer school of Benson Farb and Jesse Wolfson) involves a construction taking in a single line on a cubic and producing more, by considering the pencil of planes containing that line in projetive space.

27 Lines on a Cubic Suface

The lines on a cubic surface, all of which are real, animated as the choice of affine patch for RP3 is smoothly varied.

Cubic Sufaces II

A nice animation of a cubic surface as teh affine patch is slowly changed, that arose from work on the animation above.

Elliptic Fibrations

The real points of an elliptically fibered surface, animating the fibration by elliptic curves and their degenerations.

Orientation Double Cover of a Cubic Surface

By a theorem of Klein, real cubic surfaces with all real lines are diffeomorphic to the connect sum of seven projective planes. This visualizes one such surfaces orientation double cover in the 3-sphere.

Complex Analysis

Braid Monodromy of Complex Polynomials

The space $\Sigma$ of complex polynomials of degree $n$ with distinct roots has as its fundamental group the $n$-strand Braid group. Thus subvarieties of $\Sigma$ come equipped with braid monodromy, or homomorphisms from their fundamental groups into the braid groups.

Taylor Series

The taylor series of an analytic function has a radius of convergence, but often the corresponding cirlce is hidden from view when considering only real valued functions. Here, we use domain coloring of $\mathbb{C}$ to visualize various functions and their series expansions.

Immersive Geometry in the Allosphere

The Allosphere is a 30ft diameter spherical screen enclosing a wide walkway which can accomodate 30+ visitors for simultaneous fully immersive virtual reality experiences. Designed and operated by Jo Ann Kuchera-Morin of UCSB's Media Arts and Technology (MAT) department (site), it has been used for both artistic and data-visualization purposes. In collaboration with MAT graduate students Kenny Kim and Dennis Adderton, I have worked with the Allosphere to simulate the interior of geometric 3-manifolds and give visual lectures on various topics in low dimensional topology.

Below are some photos from an exihibition featuring in person 30+ person virtual reality lectures on spherical and hyperbolic geometry