Student Research

The hallmark of a good research project is one where students can immediately start
work and learn about what they are doing as they go. Students will write documented computer
programs, under the advisement of their mentors, to explore ideas and make conjectures based off
observed patterns. While students are making such explorations, ideally in teams of 2-3, they will be
reading about the mathematical background that provides reason and motivation for their research.
Thereafter, research students will work on proving their conjectures with their mentors and team
members. Not all research students will obtain new results, however they will all get to have the
research experience and will make progress on problems. With that said, each student leaves behind
computer programs, visualizations, presentation slide-shows, progress reports and essays.

The following is a list of some potential research projects:

Counting Points on Moduli Spaces. One of the central goals in geometry is to classify geometric
objects and their properties. Often the objects themselves form an auxiliary space, called a moduli
space, whose points are equivalence classes of the geometric objects. Understanding such a moduli
space allows one to understand not only what geometric objects exist and the degree to which they are
unique, but also how they relate to one another (deformation theory). Many moduli spaces have the
structure of an algebraic variety, that is, a space determined as the zero set of a set of polynomials.
These spaces can be formulated over many different fields (complex numbers, real numbers, finite
fields). Understanding how to count the points over finite fields can help one understand the rich and
often mysterious geometry and topology of the moduli spaces (and therefore answering the
classification problems). Given the many moduli spaces of interest to the mathematical community,
there is no end to the “counting problems” that are
available to explore. In particular, simply changing
the structure group of the bundles defining the
moduli spaces provides many open problems.

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Local Topology of Moduli Spaces. This project will also be conducted by Dr. Lawton. The project
consists of looking at moduli spaces of various types. However, unlike the above project where the
global topology is the goal via arithmetic geometry, the goal of this project is to understand the local
topology. In other words, the topology of neighbor-hoods around singularities. The main method is
again computational. Students will begin to gather data by calculating the generators and relations of
tangent spaces, then they will make conjectures of explicit general descriptions of the objects that form
the singular loci. Thereafter, students (with the help of Dr. Lawton) will use the general description to
describe the topology.

Gluing Poisson Structures. Another available research opportunity with Dr. Lawton. The project, the
moduli space of flat bundles, deals with a rich Poisson geometry that is understood in only few cases.
Dr. Lawton has results in some cases, and in these cases, the underlying spaces are surfaces. They can
be decomposed or glued together, and this relates to the Poisson structure. Dr. Lawton relates these
structures to “doodles” and can provide a pictorial list of allowed manipulations of these “doodles” and
have students explore how the structures change under gluing and cutting the diagrams. Again, as with
the other projects, the students will start working without fully understanding what they are doing, but
will be reading and presenting on the background material while they are gathering data and looking
for patterns in their projects.

Exposition on SL(2,C) Character Varieties. Dr. Lawton is looking for a few undergraduates to help
him write an expository paper on the spaces of 2x2 representations of groups. This project will offer
students a chance to learn a subject that intersects many areas of modern mathematics (algebraic
geometry, topology, ring theory, differential geometry, to name a few). As this is an expository
project, the theorems are known, however, there is no single treaty which puts all that is known together
with proofs. Many of these results are very far from “well-known” and it is worth writing a survey on
this topic given how useful it is in modern geometry and topology.

Spaces of Chain Polygons. Chains generalize lines in Euclidean, Hyperbolic, and Spherical
geometries. The spaces of chains and their geometric subspaces have been studied in the lab. A
potentially interesting project would be to study the chain triangles, or n-gons more generally and try to
determine, up to conformal equivalence, what the topology of these spaces of chain n-gons looks like.

Here are some other project areas:

Rank 2 Lie Groups and Character Varieties

Quantizing A-Polynomials and Spin Networks

Topology of Singular Strata of Moduli Spaces

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