Many mathematicians report that they chose their career as a result of some invigorating mathematical experience, often a course taken from an especially stimulating teacher. One of the goals of our VIGRE program is to get students excited early about mathematics as a career option. Through our VIGRE seminar courses, we hope to give students opportunities to discover mathematics for themselves and to gain an understanding of and appreciation for the rewards of research.
Such intangible goals are not subject to quantitive analysis. Although we do track objective data, as indicated in our assessment page, we think that subjective comments from participants are important too in evaluating the success of our VIGRE program. The following are some reflections by VIGRE participants on how VIGRE has made a difference to them personally.
As a freshman, I enjoyed working with advanced students and hearing their ideas. Possibly most beneficial to me was the experience of formally writing on a math topic. This helped me see the value of rigorous exploration and evaluation. ... This VIGRE course has stimulated in me some interest for applied mathematics and the subsequent career opportunities. In line with that, I plan to pursue an APMS [applied mathematics] major with the hopes of working in industry or research.
I really enjoyed my VIGRE experience .... this has been a glimpse as to what I might do with my math degree. Although I am used to small classes, I've never [before] really had one-on-one talks with a professor.
This course has given me a good chance to see how the professor/student relationship works when doing research. The problem I worked on was pretty interesting, and I learned a few new methods for approaches to problem solving. It was also a good experience because of the presentation and paper we did. ... This fall I am starting on my Master's in Computational Math .... After that I am planning on going into industry to do computer-intensive math or problem solving.
This experience helped me understand what research is like. [A VIGRE postdoc] is helping me understand how to write up my results, and also taught me things about the research process along the way. ... My future plans are ... to be a professor of mathematics at the university level. Near-term, I am ... working this summer with Johns-Hopkins CTY program, teaching a course of my own.
As an undergraduate, I [worked] hand in hand with a graduate student. It was exciting as we solved and invented mathematics never before done. This relationship I made through this course has stretched outside of the course, and he has helped me mature in mathematics and as a person. ... We plan to publish our results in a refereed journal, and we are currently in the process of writing our paper. ... My decision to attend graduate school has been solidified by this experience.
This past year has been great since I got my first taste of mathematical research [in a VIGRE seminar]. This year I have researched and studied the topics of wavelets, wavelet sets, frame theory, and sampling theory. ... Just recently I completed a project for my oceanography class [in which] I used my theoretical knowledge of wavelet analysis and applied it to solve problems dealing with data compression and signal processing of sea surface temperature data. So this VIGRE seminar has really opened my eyes to a whole new world. I feel that I am one step closer to being ready for graduate study.
This has been the most engaging course I have ever been in. I have engaged in original research on the construction and classification of all wavelet sets which are finite unions of intervals, leading to three talks, one at the AMS conference in San Francisco in Fall 2000, a [Graduate Student Organization] talk here at Texas A&M, and a talk at the approximation theory seminar. ... [An undergraduate VIGRE participant] and I are an example of a graduate-undergraduate team. We have asked original research questions and have answered some of them. We have used all of the languages Maple, Matlab, and C to perform integer programming and other computations. ... I feel that this course ... and my relationship with [a senior faculty member] was pivotal in my acceptance to the University of California at Berkeley in Mathematics.
I gave a talk on [my VIGRE seminar work] in the [Graduate Student Organization] series. ... I also collaborated with [another graduate student] to write a Java applet that can automatically give the interpolation map between two arbitrary bounded finitely-many-intervals wavelet sets. ... I took a leave of one week to the IMA for a workshop in Geometric Design.
Until I took these VIGRE courses and tried a couple of the problems, I didn't have a good feel for what research in mathematics really was. Thus these courses helped me decide that I would like to continue graduate work in mathematics, and I will enter the PhD program in math at the University of Washington next fall.
My experience as a VIGRE student modeling arterial wall elasticity was enjoyable. I did try to coordinate with the other students in our group (about 5 of us) and organize the work load. I helped upgrade the Matlab tools that we developed, and showed the students how to solve non-linear-least squares optimization. So, it helped me in that regard. ... I presented a poster at a SIAM Math in Industry Workshop back in October. ... I have an opportunity to work at Lawrence Livermore National Lab for the summer.
I was able to be more of a mentor to the more junior graduate students. Out of the students in the class, I probably knew the most about the material coming in. As an undergraduate, I participated in an REU [Research Experiences for Undergraduates] program in which we studied this program. Therefore, I was able to lend some expertise to the other students in the class. I presented many times and several times helped others understand the material. ... Next semester, I will be a faculty member in a small department.
For the postdoctoral researchers at the beginning of their career, VIGRE has helped foster their excitement about teaching as well as enhance their interactions with senior faculty. It has also helped advance their careers by providing flexibility and support -- for instance for travel. Here are some of their remarks.
Each of the students [in the VIGRE seminar I ran] showed noticeable improvement over the course of the semester, and the feedback I received from the students was quite encouraging. This was my first professional experience in the role of advisor and I found it rewarding. I enjoyed the opportunity to give students one-on-one instruction and closely monitor their progress. I hope to do some further work with one of the participants to prepare a paper for a college mathematics journal describing our results. ... The summer VIGRE support is allowing me to maintain my research program and to travel to visit a colleague.
The VIGRE travel money has been very valuable. For most of the conferences and seminars where I was invited to give a talk, I was given support from the institution that invited me. The VIGRE travel money allowed me to travel to almost every conference in the US that related to [my field], even if I was not invited. It also paid for my plane ticket to [Europe] for 3 weeks where I have an ongoing collaboration with [an eminent senior researcher]. ... I am also part of a group NSF grant.
I participated in VIGRE seminars on Mathematical Models in Biomedical Engineering. In these seminars I gave many of the lectures on background material in continuum mechanics and nonlinear elasticity. Preparing these lectures helped me greatly to consolidate my knowledge in these areas. ... My participation ... led to research in collaboration with [senior faculty in mathematics and in engineering] on some of the mathematical issues that arise when modeling soft tissues as nonlinear elastic materials. [One of these collaborators] and I have had one paper accepted for publication ... I will attend the SIAM Annual Meeting in July to present a talk on this research. Also ... co-organizing a minisymposium on Biomechanics and Nonlinear Elasticity.
There was much excitement during the ... seminar. ... First and foremost we had many students involved with original research; most of the students worked in small groups. ... This work has resulted in a manuscript and several student presentations at meetings and seminars.
Some of the faculty involved with VIGRE remarked on its contribution to enhancing interdisciplinary aspects of their own research programs.
Two VIGRE classes were devoted to modeling soft tissue with the help of colleagues in Bioengineering. ... The students investigated the existence archival literature and then studied new models. They had access to experimental data from the lab of [a Bioengineering professor]. ... Though dealing with very messy data, we were able to do a better job of modeling it than previous researchers did. The group modeling the lamina cribrosa had no data with with to work since such data is much more difficult to collect. However, they did develop a model showing the correct qualitative properties. The goal of the eye model was to test a conjecture of [another Bioengineering professor] as to why the elevated pressure difference inside the eye causes damage for patients suffering from glaucoma. Her conjecture requires knowing how much the lamina cribrosa bulges out. That's where nonlinear elasticity enters the picture. Previous researchers used linear elasticity to do the estimates, whereas it's pretty clear that the lamina cribrosa is a very complicated structure exhibiting highly nonlinear behavior. [A postdoc] ... and I discovered some interesting mathematical questions about the nonlinear elastic models some people had previously proposed for soft tissue that had not been adequately addressed in the literature. We've now gotten one paper accepted and are finishing a second paper on the subject. This work came directly out of the VIGRE courses.
The students showed great initiative and interest in the problems. They put in many hours outside of class working on the models and writing and testing computer codes to do simulations of their models. I've never had students in a regular class put in such dedicated effort. They were motivated by the (clinical) importance of the applications.
[A summer VIGRE seminar] was devoted to studying a new method for ultrasound modulated laser tomography developed by a colleague in Bioengineering. The students, which included an REU student along with graduate students from Math and Bioeng, studied models for ultrasound wave propagation in soft tissue, laser light diffusion through soft tissue, and how the two interact with the former being able to "focus" the latter. The problems proved to be too difficult to make much progress in just five weeks. However, one of the students has ended up doing her Ph.D. dissertation ... on the tomography problem coming from this VIGRE class.
What was rewarding: The topic intertwined very different mathematical subjects, which both gave the students an appreciation for breadth and allowed the course to play to different students' strength. For instance, the class rediscovered a certain combinatorial equality on cell complexes through the work of computer-science-oriented students using a computational viewpoint to get lower bounds and topologically-motivated students getting upper bounds, which eventually converged. Each group came away with a new appreciation for the other set of tools.
What could have been improved: at the time we hadn't really offered appropriate courses to build up an audience for the course. Now that the department's regularly offering combinatorics courses that are popular and successful, I expect future VIGREs in the area to be able to focus more time on research rather than background.