Mathematics Research Opportunity for Undergraduates
The deadline for the REU application has passed. Students accepted into the REU
can find information on Info Page for REU Students.
The Mathematics Department at the University of Houston will be
running a Research Experience
for Undergraduates (REU) in Summer 2014. The REU will be conducted over 8 weeks, and
there will be four different research
projects supervised by five faculty members. Students will work on one project under the
supervision of a faculty member, and have opportunities
to interact with the other research
groups as well as participate in professional development activities. This is a
wonderful opportunity and excellent preparation for math students who expect
to go to graduate school in mathematics or have a career that involves
mathematical research, applications, or modeling.
Dates: June 2--July 26, 2014
Salary: $300 per week ($2,400 total)
To Apply: Follow these Application Instructions
Deadline to Apply: May 10, 2014
Expectations and Description of Work:
The REU will run for 8 weeks, starting June 2, 2014 and ending July 26, 2014.
During the first 6 weeks of the REU each student will meet regularly with their
faculty supervisor (number and times of meetings to be
decided by the faculty member), and perform both individual and
group work on the research project.
We expect to obtain a large room in the
Honors College for the duration of the REU, and there may also be
some space available in the Math Department on the 6th floor of PGH, where
the work and meetings may occur.
Every other week there will be a common meeting where each group
will present an informal summary of their research progress for the other
faculty and students, listing their next research steps as well as their
final research goals. The presentation will be delivered by students
only and should be organized such that each student in the team is
given the opportunity to talk.
In the last 2 weeks of the REU, each research group will work on
preparing a written research summary of the results
obtained during the summer and also give a final oral presentation of their results.
Faculty will be actively engaged in editing the write-up, helping students prepare
for the presentation, and providing feedback.
In addition to the research performed, REU students will be expected to
participate in the following professional development activities.
Professional Development Activities
- Bi-weekly colloquium presentations about applications of scientific research.
- Social picnic.
- Participate in the SURF Brown Bag Lecture Series (once per week).
- Regular participation in SURF activities such as the writing workshop.
- Visits with summer seminar speakers.
This REU project is a full-time job, and students accepted to the position
should expect to perform 40 hours of work per week dedicated to the REU. Students
who wish to participate in the REU
should not have any additional employment or be enrolled in summer courses
during the dates of June 2--July 26, 2014.
Descriptions of the Four Research Projects
Project #1: Optimal neuronal network architectures for spatial navigation
Faculty Supervisor: Dr. Zachary Kilpatrick
Abstract: Humans' brains can store memories of spatial location. One
remarkable example is that London taxi cab drivers have much more
growth in memory portions of their brains than the average person.
We will model the memory process using a mathematical model of a
neuronal network that is an integrodifferential equation. The
integral terms in this system describe the architecture of
synaptic connections in a spatial memory network. Solutions
to this equation are "activity bumps" whose position represents a
remembered spatial position.
We will try to understand how the structure of the
network impacts its ability to robustly store spatial
position, which ultimately helps an animal explore its
environment effectively. Mathematical techniques we will
develop will include perturbation theory, asymptotics, and
stochastic analysis. This will include both analytic and
numerical techniques. The results should be applicable to a
broad range of problems in spatiotemporal pattern formation.
Project #2: Active control of differential equations
Faculty Supervisor: Dr. Daniel Onofrei
Abstract: Differential equations are the mathematical instruments
modelling almost all evolution phenomena in our world. Thus, understanding how to
solve differential equations is very important for anybody curious
about how "stuff" works or how nature "acts". For example: in the heat flow
processes the temperature is the main variable, it depends on time and position
and it satisfies the heat differential equation; or in the sound propagation
phenomena, the acoustic pressure is as well a function of time and position
and it satisfies the wave differential equation.
In many situations, one desires to control certain phenomena modeled by
differential equations in order to obtain desired results, i.e., use a
set of inputs (force actions, initial temperature, sound or electric sources, etc.)
so that the outputs (temperature, acoustic pressure, electric or magnetic field, etc.)
will have desired properties (cool or heat a certain domain, sound proof a
certain region of space, cancel a certain electromagnetic field in a
desired region of space, etc.)
In this project we will try to understand the wave phenomena in
one and two dimensions and formulate and work on several
control problems related to it. The project will help
the students learn more about differential equation modeling
wave propagation and enrich their research experience by working
on real life applied wave control problems.
Project #3: Nonequilibrium statistical mechanics: a dynamical systems perspective
Faculty Supervisors: Dr. Vaughn Climenhaga and Dr. William Ott
Abstract: Many physical processes consist of an interconnected
network of local dynamical systems that interact via the exchange of
particles, energy, or information. Nonequilibrium statistical mechanics
seeks to understand the global behavior of such systems. We will approach
these systems by focusing on the local dynamics: How do energy, particles,
and information escape from the local subsystems and how does this escape
affect neighboring subsystems?
This approach leads to the study of open dynamical systems. Here we
study escape rates, hitting times, and other probabilistic concepts.
Our project is an example of what we call a micro to macro scientific
problem. How does behavior on the microscopic scale translate to
dynamics on the macroscopic scale?
Project #4: Linear Algebra aspects of Google's PageRank algorithm
Faculty Supervisor: Dr. Mark Tomforde
Abstract: Google uses a method called PageRank to measure
the importance of websites and to determine the order to
list pages when a search is performed. Roughly speaking,
a site has a high PageRank if pages with high PageRank link to it.
This seemingly circular definition can be made rigorous using
linear algebra. In fact, a matrix can be used to describe the
probability that a surfer starting on one page will end up at
another page after a certain number of clicks, and an analysis
of this matrix provides the PageRank of webpages.
In this project we will develop an understanding of the
mathematics behind the PageRank algorithm and compare it
to other ranking algorithms for webpages (e.g., the
HITS algorithm, the IBM CLEVER project, the
TrustRank algorithm, the hummingbird algorithm).
We will then seek to modify the PageRank algorithm or describe
a new algorithm that can be used to give alternate
(and hopefully better) ranking of webpages.