NSF, REU-0243673 (Site): Tuncay Aktosun (PI) and Ratnasingham Shivaji (Co-PI)}.

Undergraduate Research in Applied Mathematics 05/01/03 - 04/30/05, $193,296.

The Department of Mathematics and Statistics hosted very successful summer REU programs during the periods of June 9 - August 8, 2003, June 9 - August 7, 2004 and June 9 - August 6, 2005. The project received 201 applications out of which 25 students were offered participation. Following is the list of the students who accepted our offer and participated in the program:

Mr. Colin Adams (Oklahoma Baptist University, Oklahoma),
Mr. Michael Borkowski (Simon's Rock College, Massachusetts,
participated in 2003 and 2005),
Ms. Alyssa Cramer (University of Louisville, Kentucky),
Ms. Libby Day (Brown University, Rhode Island),
Ms. Autumn von Flotow (University of Oregon, Oregon),
Mr.Austin Ford (Massachusetts Institute of Technology, Massachusetts),
Mr. Mathew Gilliland (Murray State University, Kentucky),
Ms. CarmenHenderson (Jarvis Christian College, Texas),
Ms. Sylvia Kober(Binghamton University, New York),
Ms. Shelly Koone (Northwesternstate University, Louisiana),
Ms. Tammy Landers (Millsaps College,Mississippi),
Ms. Anna Little (Samford University, Alabama),
Mr. KenMarks (Millersville University, Pennsylvania),
Ms. Lisa McFerrin(Virginia Tech, Virginia),
Mr. Dave Perry (New York University, NewYork),
Mr. Lance Pittman (Mississippi State University,Mississippi),
Ms. Amber Russell (Mississippi State University,Mississippi),
Ms. Jessica Schaefer (Northern Arizona University,Arizona),
Mr. Brian L. Schilling (Mississippi State University,Mississippi),
Mr. Kellen Wampler (Rose-Hulman Institute ofTechnology, Indiana),
Mr. Matt Williams (Clarkson University, NewYork),
Mr. Brian Wilson (University of Oklahoma, Oklahoma),
Mr. DanWolf (Augsburg College, Minnesota),
Ms. Anastasia Wong (MillsCollege, California).

It is note worthy that 12 out of the 25 participants were women. The participants working on wave propagation in inhomogeneous media were supervised by Aktosun and those working on population dynamics were supervised by Shivaji.

Some of the work in the wave propagation research (with Aktosun) resulted in the following publication:

T. Aktosun, M. H. Borkowski, A. J. Cramer, and L. C. Pittman. Inverse scattering with rational scattering coefficients and wave propagation in nonhomogeneous media. In: R. Kaashoek, S. Seatzu, and C. van der Mee (eds.), Recent advances in operator theory and its applications, Operator Theory: Advances and Applications, 160, Birkḧauser,Basel:1–20, 2005.

The population dynamics participants (with Shivaji) worked on Logistic as well as Allee effect models with diffusion and constant yield harvesting. They also considered the effects of nonlinear diffusion (arising due to aggregative population movements) as well as nonlinear boundary conditions.

Some of this work resulted in the following publications:

Adam Collins, Matthew Gilliland, Carmen Henderson, Shelly Koone, Lisa McFerrin and Kellan Wampler. Population Models with Diffusion and Constant Yield Harvesting, Rose-Hulman Institute Of Technology Undergraduate Math Journal, Vol.5, Issue 2, 2004.

Tammy Ladner, Anna Little, Ken Marks and Amber Russell. Positive Solutions to a Diffusive Logistic Equation with Constant Yield Harvesting, Rose-Hulman Institute Of Technology Undergraduate Math Journal, Vol.6, Issue 1, 2005.

David Perry, Jessica Schaefer, Brian Schilling and Matthew Williams. Population Dynamics with Nonlinear Diffusion, Rose-Hulman Institute Of Technology Undergraduate Math Journal, Vol.7, Issue 2, 2006.

Jaffar Ali, R. Shivaji and Kellan Wampler. Population Models with Diffusion, Strong Allee Effect and Constant Yield Harvesting, Journal of Mathematical Analysis and Application,352(2009),pp.907-913.

Based on the information we have to date, the following participants are now pursuing (or have pursued) graduate studies (institutions are listed next to their names):

Ms. Libby Day (Stanford University),
Ms. Autumn von Flotow (Northern Arizona University),
Mr. Austin Ford (Northwestern University),
Ms. Tammy Landers (Mississippi State University),
Ms. Anna Little (Duke University),
Mr. Lance Pittman (University of Illinois at Urbana-Champaign),
Ms. Amber Russell (Louisiana State University at Baton Rouge),
Ms. Jessica Schaefer (Northern Arizona University),
Mr. Kellen Wampler (Indiana University),
Mr. Matt Williams (Virginia Tech).

All students filled out an evaluation of the program via the REU website. In response to question on over all evaluations, 13 participants gave a rating of 5/5, 11 participants gave a rating of 4/5, and a participant rated 3/5. We also received very favorable written responses from the participants on the REU program.

Following are some sample remarks, particularly by population dynamics students:

During the program, I learned a great deal about the field of Differential Equations. Hitherto my summer in Mississippi, I had only introductory classes in the area which I believe was not a very good representation of what is involved in the field. Also, working with Shivaji on the paper was a new experience, and I feel that I have much improved my ability to present mathematics.”

I hope to continue to do research in the field of Differential Equations. Also, I am happy to have met the professors at MSU and will probably seek their guidance when applying to graduate school, a program for next year, etc.”

The REU had a major impact on my future plans it has enhanced my desire to do advanced studies in mathematics.”

The supervision was excellent the professors were very helpful always went above and beyond the call of duty.”

I found that the supervision we received was spot on. While Dr. Shivaji was always there to answer any questions we had, he also gave us plenty of space so that we could discover things for ourselves. We worked with a few graduate students who taught us the basic skills we needed to write up in the proper form any results from our research. I found the research topics to be interesting, but mainly I was just interested in the experience of doing math research. I feel that if I wished to continue researching what I began in the REU I would have a good knowledge of how to do so.”

I attend a small private university, the professors of which are not researchers. They are employed for instruction only and this they do very well. However, until this summer I had no exposure to math research of any kind. Had I not attended this REU program this summer, I most likely would have continued working in one of the business offices at my university, as I do during each semester. I believe that this REU provides an honest and realistic view of what mathematics research is like. Because of this, I have found it to be valuable in helping me decide whether or not math research is what I want to pursue in the future. So yes, I feel that this REU is a good way to win undergrad students into mathematics etc. because without such an experience, they would be less able to make a wise decision about pursuing a career in any of these fields.”

I felt that the guidance that Shivaji gave us was excellent, and our interaction with the graduate students, especially Mark, was very helpful in planning for the future. We were able to work very closely with Shivaji--he gave us so much help but let us figure everything out for ourselves. I feel that I have been able to go to an independent status as far as research is concerned; and I do hope to continue some of the research topics that we discussed: I just hope that I have time during next year.”

This REU has given me a few problems to work on right now. More importantly, it helped me decide that I do enjoy this type of work and want to continue with research more. Before the program I had been apprehensive to pursue a higher degree. But after the program I'm sure I want to continue my education.”

My mentor was very helpful and did an excellent job of guiding our group without doing the work for us. I think all the people in my group were very hardworking, intelligent individuals and feel fortunate to have been in the group with them. The faculty and graduate students were very welcoming and helpful. As to myself, this program has definitely been helpful in giving me the direction thinker when it comes to math.”

The mentorship we received from Dr. Shivaji was excellent. He was enthusiastic about undergraduate research and had high expectations of what we could accomplish. He provided much needed guidance, but allowed us to work quite independently. Our team worked together very well; we all learned a lot.”

This REU has made me completely reconsider what I want to do in the future. I know for certain that I want a Master's in math (instead of math education) and who knows, maybe a PhD? I am now happily confused about what I want to do, where before I had one set plan in my mind that might have prevented me from realizing how many options I have.”

The level of advising was great. I liked the fact that we were given problems to work with on our own, and then later we were given some guidance, after attempting them ourselves. I enjoyed working closely with my mentor. I would have appreciated a little more communication with my group. I think we could have accomplished more if this had happened.”

The mentoring was excellent. We were able to work closely with our mentor and with each other. There was emphasis on going from more dependent research to more independent research as well. The program was a great experience and I learned a lot. It also got me excited about research and graduate school. It has already influenced my future plan of studies. I now am strongly looking at graduate school.”

I went into the program (population) with no preconception of what it would be like and left with a solid idea of what I was doing. I also noticed that my ability to focus on the problem and research process in general was quite improved. In that sense I do believe that I gained some independence when it comes to the research process.”

I genuinely had a great summer and I'm glad I got a chance to do the program. I did the program to decide whether or not to even apply to grad school and I now am confident that I will. It was a success.”

I worked in the population dynamics area. I thought that the guidance and instruction provided by the graduate students and faculty mentor allowed us to develop our own research skills. I am hoping the exposure I received during the program will help my transition into more advanced classes. I am more seriously considering graduate school in mathematics or engineering because of the REU.”

Recently (in 2007-2008) we received the following comments:

Ms. Anna Little who is now at Duke, in her letter of support for Shivaji's nomination for the Phil Oldham Mentor Award, wrote:

Shivaji served as my research mentor during the summer 2004 REU in applied differential equations at MSU. This was a very instrumental time for me, as I debated whether or not to pursue mathematics at the graduate level, and Shivaji served as a role model for the kind of mathematician I would like to be. Providing guidance when needed while allowing students plenty of independence to grapple with the unknown, he was the ideal mentor. Dynamic and personable, insightful and enthusiastic, he imparted his knowledge and drive to us.”

Mr. Matt Williams who is now at Virginia Tech finishing his Masters and moving onto the Ph.D. program, wrote that he had such a great experience at MSU three years ago and that it was a great encouragement for research in the mathematical sciences.

Ms. Amber Russell, a 2004 REU participant and currently in Graduate School at Lousiana State University - Baton Rouge, wrote the following in her nomination of Shivaji for the Oldham Mentor award:

``I could list several appropriate and flattering adjectives to describe him, but I think the impact he's had on my life and where I am now speaks more loudly. He had faith in my abilities and encouraged me throughout my undergraduate studies. I know I am only one student, but I am one student for whom he made a positive and tangible difference. That should say enough.''