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

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.