EAGL is looking for graduate and undergraduate researchers.We have 3 principal research advisors: Drs. Huber, Lawton, & Peirce, associated with the lab.
For graduate students, please contact us about prospective thesis topics.
The hallmark of a good undergraduate 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 potential research projects carried on by professors for any interested to join.
Future Projects with Dr. Lawton
We will begin a weekly research seminar on some of these topics for prospective, current, and future research students. If you are interested, please let Dr. Lawton know.
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.
Students involved in this project can get started
straight away. They will, begin by writing
exploratory code to count the umber of points over
finite fields (in Mathematica) and look for patterns in
the data. Once patterns are deduced, they will make
conjectures. All the while, students will be assigned
reading and exercises to help them understand what
they are doing, and they will give presentations on
what they are learning to their mentors and their
peers working in the lab. As they write code they
will document it, and write up summaries of their
investigations. Once they have sufficiently many
conjectured formulated, they will begin to try to
discover proofs (with the help of Dr. Lawton).
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 know, 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. This project, as it will be long, is
appropriate for 1 graduate student or 2-3 undergraduate students.
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.
Future Projects with Dr. Huber
Location and distribution of zeros of Eisenstein series. The starting point for this project
is a solid understanding of the intermediate value principle from calculus. In the 1970s, Rankin and
Swinnerton-Dyer first established the location of the zeros of Eisenstein series associated with SL(2,
Z). The authors employed a clever parameterization for the Eisenstein series and expressed the series as
a sum of a trigonometric term and a remainder. They estimate the remainder term with the aid of the
error term from the Integral Test. Combined with elementary facts from complex analysis, the authors
established that the zeros of Eisenstein series all lie on a certain segment of the unit circle. This
project will generalize Rankin and Swinnerton-Dyer’s method to other types of Eisenstein series. In
particular, we will study Eisenstein series twisted by Dirichlet characters, also known as the Eisenstein
series on the Fricke subgroups. Initial computer calculations will involve approximating the
zeros and visualizing their locations. Based on this work, a change of variable should be made in the
Eisenstein series so that Rankin and Swinnerton-Dyer’s idea can be applied.
Bijective Correspondences for Analytic Relations. This research project is broadly con-
cerned with exploring, refining, and proving combinatorial relations resulting from analytic identities.
Extending the Classical. In the early 20th Century, the Indian Mathematician S. Ramanujan
significantly extended the classical theory of elliptic functions. His work has interesting applications in
combinatorics and theoretical physics. Students involved in this research program will study and extend
Ramanujan’s work.
Future Projects with Dr. Pierce
Geometry and Topology of Flocks of Birds. In this research opportunity, students will implement
statistical models for the motion of groups of animals exhibiting a flocking, schooling or herding
behavior. The models will then be analyzed to determine general laws for the distribution of individuals
in the group, and the overall geometry and topology of the group of individuals. Of particular interest is
the transition of the group between a connected and disconnected masses of animals. Parameters will
be chosen to match the models with observations from the local bird populations of south Texas.
Analysis and simulation of modified Brownian motions. Another research opportunity with Dr. Pierce.
The density of paths of a family of
Brownian motions in the plane, constrained so that the N paths cannot cross, is an interesting problem
in statistical physics. An important question is how the constrained Brownian motion problem
must be modified to produce a given density of paths. One approach to answering this question is to
understand the effect of external forces on the density of paths. Students will create simulations of
conditioned Brownian motions in the presence of various
conditioning (varying the number and position of starting and
ending points) and including barriers. Models will include
evolutionary models, and Boltzmann style samplers. Students
will then analyze the effect of the conditioning and barriers
on the density of paths in the diagram.
Generating Random Combinatoric Structures. Students will
implement growth and Boltzmann methods of sampling
random combinatoric objects such as chord diagrams, trees
and graphs. Interesting questions about such random objects
are: what are the limiting shapes of a typical object and are
there frozen boundaries or other features for large
configuations.
