Spring 2020

A 3-manifold is a space that locally looks like the 3-space but globally can be much more complicated -it can have "holes". One example of 3-manifold is the unit tangent bundle of a surface, i.e., the set of all pairs of a point in a surface and a unit vector based at that point. A knot is an embedding of a circle inside a 3-manifold. One way to obtain a knot inside the unit tangent bundle of a surface is by taking a closed curve on the surface, and endowing it with a tangent unit vector at each point. We call this the canonical lift of the curve. Drilling a knot from a 3-manifold produces a new 3-manifold. Under some conditions on the knot, this resulting 3-manifold has a special geometry, called hyperbolic geometry. It then makes sense to compute the volume of the 3-manifold using that geometry. In this project we will (1) design a mathematical model to study knots coming from canonical lifts of curves in the unit tangent bundle and (2) use a software called Snap.py to compute the volume of the resulting 3-manifolds.

In this project, we will try to understand random experiments in certain geometries. There are lots of interesting theorems that hold with probability 1. Eventually we will focus on flows on hyperbolic surfaces and 3-manifolds that are chosen randomly. For example, the winding number of geodesic flow, the orbits of the unipotent flow, etc. In some cases, we also have an analogue of the Central Limit Theorem in this setting. Roughly speaking, that means that we if try many times randomly, eventually we can get a normal distribution. Using computer simulations, we may test such statements.

On the other hand, when studying such probability and geometry we expect to produce some interesting animations related to our random trials.

There are lots of things we can try in this direction if time/technology permits. For example, we may try to guess the ""error"" of certain theorems. Or we can try to guess their speed of convergence.