2010 NSF REU in Applied Mathematics and Biostatistics

May 30 - August 7, 2010

 

Report

 

Application and Recruitment Process:

 

The 2010 REU program targeted undergraduate US citizens and permanent residents who completed their sophomore or junior year by the end of Spring 2010. Students who have background in multivariable calculus, linear algebra, and ordinary differential equations or probability prior to the start of the REU program were eligible to apply. Women, students from other underrepresented groups, community college students, and students from regions and institutions with limited research opportunities were actively recruited and encouraged to participate. The application deadline was February 15, 2010.

 

The selection of the participants was based on their academic performance, their aptitude in mathematical sciences, and recommendations.  Of the seventy-one applications received, the following seven students were selected:

 

John Corring, University of Southern Mississippi

Helene Duke, Providence College

Justin Hansen, University of Vermont

Emily Poole, University of Arkansas

Bonnie Roberson, Mississippi State University

Brittany Stephenson, Mississippi State University

Yicong Yong, University of Florida;   

 

 

Research and Education Activities:

 

The objectives of this 10-week REU site program were to involve the participating students in meaningful research activities, to show the students the enjoyment of doing research, to encourage them to pursue advanced degrees in mathematical sciences, and to increase the participation in research activities by women and other underrepresented groups.

 

There were four groups working on four different research areas: Image Processing (mentor: Hyeona Lim, graduate student mentor:  Arundhati Bagchi Misra, participants: John Corring, Helene Duke), Population Dynamics (mentor: Ratnasingham Shivaji, graduate student mentors: Jerome Goddard II, Dagny Grillis, participants: Emily Poole, Bonnie Roberson, Brittany Stephenson), Computational Mathematical Biology (mentor: Xingzhou Yang, participant: Yicong Yong), Highly Stratified Model in Biostatistics (mentor: Haimeng Zhang, participant: Justin Hansen). There were activities to improve the participants' written and oral presentation skills as well as computational skills. By closely working with a team of four faculty members and three doctoral-level graduate students, the participating undergraduate students had the opportunity to perform research in a Ph.D. Granting department/interdisciplinary research center, gain insight into the roles and responsibilities of active researchers, discuss graduate school choices, and be in an active research environment. All four groups have developed new methods and findings for their projects. 

 

We also invited five colloquium speakers (3 in academia and 2 in industry) from different disciplines. The following is a list of speakers and their title of talks:

1.        Seong-Gon Kim, Department of Physics and Astronomy, MSU, June 25. Title: Ranking Billions of  Web Pages Using Diodes: How Google works and how to beat it

2.        Michael Neumann, Department of Mathematics and Statistics, MSU, July 8. Title: The fundamental theorem for finite Markov chains revisited.

3.        Chris Brooks, Department of Biological Sciences, MSU, July 13. Title: Using graphs to model biological invasions.

4.        Lalitha Venkataramanan, Schlumberger Doll Research, July 22. Title: Research in applied mathematics at Schlumberger.

5.        Stan Dickens, Southern Farm Bureau Life Insurance Company, July 28. Title: Pursuing a  career in the actuarial field

 

We had two educational trips. On Friday, July 9, we visited the Department of Mechanical Engineering at the University of Alabama at Birmingham, where the participants listened to presentations on Computational Fluid Dynamics, Computational Geometry/Visualization, and Computational Methods in Impact Engineering: Finite Element Method and Multibody Dynamics and their Applications. On July 26-27, we visited the Department of Mathematics at Tulane University. The Tulane faculty presented the following three lectures to our REU students:

An interesting sequence coming from an integral (by Victor Moll)

Mathematics and the modeling of swimming microorganisms (by Lisa Fauci)

A gambler's ruin is a random walk (by Ricardo Cortez).

 

We also had several social activities such as movie nights, frequent coffee hours, and lunch and dinner meetings. We had a joint picnic with the REU group in the Department of Computer Science and Engineering on August 1.

 

 

Findings:

 

For Image Processing project, new partial differential equation (PDE)-based denoising models which are modified Euler-Lagrange equations derived from noise constrained functional minimization problems were developed. These PDE models are designed to rectify speckle noise present in ultrasound images. Non-standard numerical schemes of the corresponding speckle denoising models were also derived. Numerical results from a variety of computational strategies and models for ultrasound images were compared.

For Population Dynamics project, a one-dimensional reaction-diffusion model arising in population dynamics where the per capita growth rate is a weak Allee type was studied. In particular, the effects of grazing on the steady states were considered and the complete evolution of the bifurcation curve of positive solutions as the grazing parameter varies was discussed. The results were obtained via the quadrature method and Mathematica computations. In particular, it was established that the bifurcation curve is S-shaped for certain ranges of the grazing parameter. This occurrence of an S-shaped bifurcation curve was also proven analytically.

For Computational Mathematical Biology project, a computational model to understand the particle dynamics around the choanoflagellates was considered. The flow patterns were shown by visualizing how the flagellum, microvilli, suspended particles interact with the surrounding fluid. The regularized Stokeslets method, a grid free method, was used to solve the governing equations. The Runge-Kutta method was employed to solve the related ODE system. The numerical results were compared with the data in the published biological experiments.

For Highly Stratified Model in Biostatistics project, the efficiency of the MPLE for nested case-control sampling under the highly stratified situation was considered, where the covariate values are increasingly less dependent upon the past and there is no censoring. In particular, through numerical study under the parametric distribution for the failure time, the efficiency of the MPLE was investigated and its performance was compared with the MLE when the sample size is finite.

 

 

Research Presentations and Publications: 

 

On Thursday, July 29, all 4 groups gave a poster presentation at the Summer 2010 Undergraduate Research Symposium which was hosted by the Shackouls Honors College at Mississippi State University. Two students from the population dynamics group gave presentations at the 30th Southeastern-Atlantic Regional Conference on Differential Equations in October 1-2, 2010. Also the participants plan to present their results at the 2011 Annual American Mathematical Society meeting in New Orleans in January 6- 9.  Three female participants are planning to participate in the Nebraska Conference for Undergraduate Women in Mathematics which will be held in January 28-30, 2011. Currently, manuscripts are being prepared for submission to journals.

 

 

Training and Development:

 

The first two weeks were spent as an orientation and training period. 3 Ph.D. students gave lectures on basics of MATLAB, Mathematica, Metric Spaces, and LaTeX. These graduate students helped the mentors to guide the REU participants so that they can have the opportunity to perform research on their projects.

 

 

Outreach Activities:

 

The website for our REU program has been created (http://www.msstate.edu/dept/math/events/reu) and it is open to the public. In the fall semester after the 2011 REU program, we plan to conduct a one-day Research Experiences for Teachers (RET) workshop for teachers from neighboring high schools. We will invite teachers and have a workshop to disseminate new findings from the REU program and help them to have their students prepare for undergraduate research.

 

 

Contributions to Human Resource Development:

 

Among the 7 participants we selected this year, 5 were women. Also, 3 out of 4 graduate student mentors were women. We will continue to make every effort to reach out to women and other underrepresented groups and encourage their participation.

 

 

Overall rating given by the students: 4.7/5

 

Extracts from REU participants' comments:

 

The mentoring and guidance in this specific program were amazing and a lot was learned during research in highly stratified models in biostatistics.  I learned how to carry out research on my own as well as worked closely with my faculty advisor.  The research in general was very informative and taught me a lot.

 

The graduate mentors, faculty mentor, and fellow students in my research group were amazing.  I have no complaints about any of these people.  I worked very closely with my mentors, but we were also able to do our own research.

 

My mentor is a fantastic advisor, and I really learned a lot from him this summer. He challenged me and the research was difficult, but he also made sure i did not feel overwhelmed or incapable. The graduate mentors were also a really big help- it was fun to work with them and they did a great job. I enjoyed the laid back structure our group had for meetings and meeting times. I'm looking forward to working with my mentor more in the fall.

 

My mentor gave me very effective instructions every time we met and I was able to do some research on my own following his directions and hints. This program definitely has trained me how to do independent research.

 

My mentor was amazing. His vast knowledge related to this area of study made our research run a lot more smoothly.  He guided us down all the right paths anytime we were stuck or frustrated and always knew how to lead us in other directions to obtain different results.  He ensured our research was thorough and accurate and compelled us to check all of our work.  The graduate mentors were also an immense help.  Any problems with proofreading, from things such as our paper to codes in Mathematica, they could solve.  They were always willing to meet with us at any time and help us in any way possible, including over the phone when they weren't available in person.  My mentor did a great job of giving us a very limited amount of information to start with and forcing us to work independently to obtain results. He has done so well that I now find myself asking new questions and coming up with new ideas to research.

 

The application process seemed very efficient.  It was short enough that it did not seem tedious and overdone, but it was also long enough to receive what I feel was all the necessary information about each applicant.

 

I'd suggest playing up the facilities- I had no idea the caliber of research centers Mississippi State had until I arrived in Starkville. I think information about HPCC and all that goes on there would really increase the pool.

 

I am way beyond satisfied with the stipend and other benefits.  The advisors outdid themselves.  We always felt welcomed and very well taken care of.  Honestly, we often felt very spoiled. It was very well planned and very well carried out.

 

Really good benefits- we were well taken care of throughout the program. I really enjoyed the variety of colloquium speakers, and the fieldtrips were fun as well. Herbert was clean and a decent size room, but it might be better to put next years participants in the newer dorms if possible. Also, maybe more social activities at the beginning of the summer

 

 The entire research program was awesome.  There was nothing that should be changed.  My only suggestion would be to move the social activity with the other REU students to an earlier date.  This would give the students time to make friends with the other students earlier.

 

Before this program, I did not even know what research in mathematics entailed.  I now know how to take one simple differential equation and study it incredibly thoroughly.  I now also feel very, very comfortable using Mathematica; whereas, before my knowledge in that area was also limited.  Another invaluable skill I gained is the use of LaTex to write mathematics papers.  I am very excited to now know how to do this. Furthermore, I gained knowledge on myself and how to manage my time and stop procrastinating.  I find myself more motivated to complete tasks since I do not know what the results may be.

 

The program has changed my entire view of mathematics.  It has also helped me realize what I may want to do with my mathematics major in the future.  It helped a lot to be around graduate students because I may want to follow in their footsteps, and it was nice to see firsthand what that may be like.  I used to be somewhat nervous about being a math major because I had no idea what I wanted to do, but at this point I feel very confident in my decision to stick with this major.  This program has helped me to love it more.

 

This program has taught me lots of skills that I can use right now for my studies, such as latex and matlab.  Since I have some taste of what real research is like, in the future, I will be ready for any difficulty that I will face when doing research.

 

My REU experience has confirmed that I want to continue my mathematical studies on the graduate level. I think I learned a lot about challenging myself academically, and how to push through frustration to achieve a rewarding result.

 

This program has taught me about future studies and what is required in grad school.  It has thus given me a step forward in this area as well.