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.

* 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**