The point of this project is that, in certain situations, a complicated differential equation can be made easier to solve by using symmetry.

In particular, there are some geometric spaces that admit enough symmetries that, if one is able to find one "geodesic" in the space, then one can parlay that into the ability to find all the geodesics in the space. (Finding a geodesic amounts to solving a certain system of ordinary differential equations. I will certainly elaborate on details with any interested student.)

It is my intention to use this kind
of thinking to analyze geodesics
in spheres, in hyperbolic space, and, if time permits, in certain
relativistic space-times.
**Note:** One of Einstein's fundamental observations is the path of a
beam of light should be described by a geodesic in the space-time
in which we live. So geodesics play a fundamental role in general
relativity.

A more technical description of the project is as follows, but don't be deterred if you don't know the meaning of some of the terms appearing below. I hope to clarify everything for you by conversation, and/or by pointing you to appropriate references.

We will compute geodesics in various geometric spaces with transitive isometry group by computing a "small and easy" class of geodesics and then finding the rest through symmetry. We will begin with the 2-dimensional sphere with its constant curvature Riemannian metric. We should be able to work our way up to a constant curvature 3-dimensional Lorentz manifold, at least in terms of numerical computations. By "numerical", I mean: the code would take, as input, various numbers describing an inital point and an initial velocity vector at that point, as well as the length of a time interval, and it would produce, as output, the coordinates of the point reached by geodesic flow for the specified length of time from the specified initial point and velocity. If the student has enough computer skills (more than I have anyway), we might be able to aim for some visualization, in which one sees a trajectory traced out in some of the various spaces.

I think four weeks will be enough for such a project. You will need to learn the basics of Riemannian (and possibly Lorentzian) geometry and will need to know or learn some programming skills. If you have already done some programming, and perhaps have some curiosity about relativity, you should be a good candidate; Lorentzian geometry is the geometry of relativity.

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