Our purpose is two-fold: (1) to introduce promising mathematics students to real research in problems of current interest, through intensive, 8-week, mentored, summer programs; and (2) to make substantial research contributions to a variety of somewhat related, unsolved problems. The research that is necessary for graduate study in mathematics, and most eventual careers in the field, is quite different from traditional undergraduate course work. Experience has shown that an honest exposure to real research, within a group of like-ability students, is very efficient for future decision making by the students and as preparation and inspiration for likely graduate study.
The platform for this REU program is matrix analysis and applications, broadly defined, including combinatorics. This is an excellent source of accessible, worthwhile, and open-ended problems and a most valuable area as part of the knowledge base for future work in mathematics. The PI, through his own research program and wide variety of international contacts, generates many problems, progress on which will be of interest to the community. In addition, work is likely to continue on several large problems, with many parts, on which progress has been made in prior summers. This has in the past, and likely will in the future, generate many student-authored publications, in respected, traditional journals, that will be a lasting contribution to the field.
William and Mary REU 2008-2014 Over 6 summers (2009-2014), 56 bright undergraduates were introduced to high-level mathematical research by the principal investigator and several other expert senior and junior mentors. The recurring program took place in June and July and focussed upon problems in matrix analysis, combinatorics and their applications. Some of the more specific areas that were considered more frequently included: 1. Totally positive matrices 2. Completion of partial matrices to ones with desired properties 3. Structure of the field of values of a matrix and recent generalizations 4. Semipositive matrices 5. The relationships between eigenvalue multiplicities in a symmetric matrix and the graph of the matrix 6. The frequency of equality in the interlacing inequalities for eigenvalues, given matrix structure 7. Properties of linear trees 8. Interpolation problems 9. Matrix reconstruction from minors as well as several individual topics. Students presented two talks about their work and wrote a detailed report about it (averaging 20+ pages). Often the report became a published paper, with the assistance of mentors. Thus far, 44 published papers have appeared in well known journals, and perhaps 15-20 more are in some stage of the preparation/editorial process. As the students often (but not always) worked in small groups, or other coalitions, a large fraction of them have or will become published authors (frequently joint with mentors). Furthermore, a queriable data base of possible multiplicity lists for the eigenvalues of symmetric matrices whose graph is a given tree, for all trees on fewer than 13 vertices, was constructed, and this has proven to be a valuable research tool. The data base exploited both computational and theoretical work over several years. In addition, the students often attended meetings and spoke about their work in the community at large, some receiving prizes for their talk/work. In one case, a participant received the "Young Scientist" award against all graduate student competitors at a biennial international meeting on linear algebra and statistics.
