Owen is a third-year PhD student in the Mathematics Program. He received his BS in mathematics from Georgetown University in 2012. His main research interest is algebraic number theory, particularly the arithmetic of cyclotomic fields.

Project

The classical version of Fermat’s Last Theorem, conjectured in 1637 and proved in 1995, says that the equation x^n + y^n = z^n  has no non-trivial rational solutions when n > 2 . Attempts to solve this problem in the 19^th century lead to the investigation of not only rational solutions but solutions lying in cyclotomic fields. Kummer showed that the equation has no solutions for exponent l over the cyclotomic field of level l provided the irregularity index is 0. In the nineties, Kolyvagin proved that if the index of irregularity of the prime number is less than, then the first case of Fermat’s Last Theorem is true for the exponent over the cyclotomic field of level. In this project, we use some conditions developed by Kolyvagin to obtain similar results for the second case of Fermat’s Last Theorem over the cyclotomic field of level l.

High school students everywhere spend countless hours laboring over the coordinate plane.  This humble object is the simplest example of a geometric model: an abstraction which assists us in studying and understanding our world. Flexible yet powerful, the plane models situations ranging from classical mechanics in physics, to the dynamics of supply and demand in economics, to the population trajectory of grizzlies in Yellowstone National Park.

Despite the plane’s ubiquity, it is far from the only geometric model useful for describing our universe. For example, if you are trying to determine the most efficient route for a flight from San Francisco to London, you will be much better off modeling the Earth as a round sphere than as a flat plane. With the ever growing diversity of phenomena we encounter in our universe, mathematicians are constantly working both to create and to understand new geometric models.

My mathematics research focuses upon studying a class of geometric models called hierarchically hyperbolic spaces (HHSs). The philosophy of HHSs is that they model situations where geometric information can be obtained by looking at the projections or shadows. As a non-technical example of this philosophy, consider the problem of trying to determine the shape of the Earth without space flight. Earth-bound humans may not be able to see the entire planet directly, but they can observe the shadow of the Earth on the moon. From this projection of the Earth onto the moon, we can see that the Earth must be round.

The usefulness of the plane as a geometric model boils down to how well we can understand fundamental geometric objects such as lines, circles, and polygons (triangles, squares, hexagons, etc). Appropriately, the majority of high school geometry is spent studying the properties of these basic figures. One of the reasons these basic objects are so important is that they are all convex; a geometric object in the plane is convex if the line connecting any two points in the figure stays inside the figure. Just as is the case with the plane, understanding more exotic geometric models hinges upon understanding the convex objects in the model. In many ways, the study of geometry is really the study of convexity.

Hierarchically hyperbolic spaces are a new class of geometric models, and many of their fundamental features and properties are still poorly understood to geometers. In joint work with Davide Spriano (ETH Zürich) and Hung C. Tran (University of Georgia), we have produced some of the foundational work on HHS’s by studying their convex objects. The main result of our study is to understand how you can use projections to “see” whether or not an object in a hierarchically hyperbolic space is convex. This allows us to paint a clear picture of the properties and characteristics of convex objects in hierarchically hyperbolic spaces.