The 2011 PROJECT REPORT

GueoGrantcharov, MirroslavYotov
Department of Mathematics and Statistics
Florida International University
Miami, Florida 33199

As originally planned, the 2011 MAA-NREUP summer research took place at the Department of Mathematics and Statistics of Florida International University during May 9th – June 17th, 2011.The program was extended without cost to the end of August. This was our first such program, but we have had an active student seminar for the last 4 years, and although the decision for funding our project came with some delay, the experience of working with students on research related topics helped us make the proper arrangements and organize strong and successful program. Mentors of the project were Dr. Grantcharov and Dr. Yotov. The Department of Mathematics and Statistics at FIU generously provided us with all we needed for our work: computer lab, library resources, printer support.

We chose the participants in the project carefully, following both the directions of NREUP and the current trend of non-math majors joining the Student Seminar on Algebraic Geometry and Commutative Algebra at the Department. As a result, we had four sophomores in the project: one math major, and three physics majors. None of them had any previous experience in doing research in mathematics. Because of the delay of funding decision some of our students had already enrolled in summer courses, while others had arranged part-time jobs for the summer period. In order to have them focused on the work with the project, we strongly recommended the students to drop those classes and enroll the Independent Study class accompanied the research instead or rearrange their summer jobs, and be available for our work.The students followed our advice and that worked very well for the project.

Thanks to our experience, we successfully completed our research program following closely the plan outlined in the proposal. We are happy to report that our first funded summer project was an excellent experience for the participants who were exposed to active research in mathematics for the first time in their university studies.

As explained in the proposal, the program had two phases. The first was devoted to teaching the students concepts from Linear Algebra, Category Theory, and Differential Geometry needed for the research. For this purpose, a three-week Independent Study course on Linear Algebra was organized as planned. The course was attended by the participants in the project as well as by three more students: one graduate and two graduating math majors. Full details about the course can be found on its web site at http://www2.fiu.edu/~yotovm/LieAlgGeomStr/indexSS11.html. At the beginning of the third week though, we realized that to cover all the topics we needed, we had to extend this phase by a week.This was caused by the fact that our physics students, although knowledgeable of upper level math for their physics classes, were lacking skills in understanding and making math proofs. In our proposal, we had suggested that, if needed, the fourth week of the project may have some more teaching the students. In reality, we had intensive teaching during that week as well.

To ensure enough time for the students to get acquainted with the research problems of the proposal, we combined the teaching during the fourth weekwith an introduction to those problems. The last two weeks were devoted to the research itself. During these two weeks, the students performed the main computations and obtained the main results. This was the time when we observed how effectively the students worked together and how happy and enthusiastic they were when achieving their results. For this phase, we originally planned to split the students in two groups and give each of them work on a separate problem. But it spontaneously turned out that a better organization was to let all of them work on each of the problems for a week. During the first phase, from Monday through Thursday, we had our classes starting at 9:30 a.m. having lectures till 1:00 p.m. After an hour and a half lunch break, we had our afternoon sessions devoted mainly to doing exercises on the material taught in the morning. We ended our days at 5:00 p.m.The Fridays were devoted to examinations on the material taught during the previous four days. These were combined written and oral examinations, and constituted the means to monitor students’ achievements in learning the material from the course. Our choice of examination type was partially motivated by the fact that this is the way the elite graduate schools in mathematics examine their students: we wanted to give our students an impression of what they might have in their graduate students’ life.

The schedule for the second phase of the project followed the 4-to-1 day format as well. Four days a week, from Monday till Thursday, the students were actively researching, and the Fridays were devoted to recapitulations of the week’s work, simplifications of computations or argumentations, discussions of possible generalizations of what they have obtained so far. For this phase, as a part of the Departmental support, we were provided with all the needed hard- and software, but the students managed to do the basic part of their study without using any special software.

After the sixth week, the participants began writing a paper on their results. A draft of it is published on the web site of the program(at ) and is attached bellow. It is being fine-tuned and prepared for submission for publication now.

Meanwhile, we will popularize as much as we can the achievements of our students. A presentation of their results at the Students’ Colloquium of the Department will be scheduled by the end of October. The students are planning to participate in the poster session of the JMM in Boston, January 2012. We will also try to organize their presentations in other conferences and math forums of MAA and AMS.

Overall, this was really a rewarding experience for us all. We all, students and mentors, worked very hard. We provided our participants with an excellent environment to attract them to mathematics at the graduate level and perhaps also beyond that. An evidence for that is that, by the end of the program, one student changed his major from physics to mathematics, while the other two are already taking upper division math courses and planning to become at least master’s students in Math before going into pursuing Ph.D. in Physics!

We as mentors learned from this program as well. We got help in this from the students themselves: we asked them for (anonymous) feed backs. We learned from those in particular that we should in future begin such a program with at least a couple of days devoting to basic logic and techniques of proof – to help the non-math majors understand better the material taught and prepare them for the research to follow. The students’ feed backs (we got three of them) are attached below. We set high expectations and we think that we have accomplished our goals and fulfilled the promises in our proposal. Our previous experiences in working with students were certainly helpful. We are hopeful and optimistic of continuing such work in future. Finally, we are grateful to MAA for finding a way to fund us for our 2011 MAA-NREUP at our department.

The evaluator of our project was Dr. Gulden Karakok with School of Mathematical Sciences the University of Northern Colorado.

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