# 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         