This VIGRE project showcases a rapidly expanding commitment by the Department of Mathematics to foster multidisciplinary education and research activities, as a natural part of our dedication to excellence in mathematics. This commitment is behind the ever increasing partnership between the Department of Mathematics of the University of Texas at Austin (UT) and the Texas Institute for Computational and Applied Mathematics (TICAM), an independent unit with participation from a large number of engineering and science departments. The VIGRE project's central goal is to improve the quality of training that future mathematical scientists, from undergraduate mathematics majors to postdoctoral researchers, experience at UT. In keeping with the department's commitment, many of the UT VIGRE activities have a pronounced multidisciplinary flavor. By placing greater emphasis on how mathematics interacts with the larger intellectual, scientific, and public communities, we hope not only to provide aspiring mathematical scientists with a firm grounding in the subject's core, as we have always done, but also to give them a glimpse of (and better prepare them for) the multitude of career opportunities that now present themselves to members of the work force commanding highly developed mathematical skills. With the aid of VIGRE funding, the department will stimulate the formation of a local educational landscape in which activities that successfully exploit the combined talents of undergraduate students, graduate students, postdoctoral fellows, and faculty members are striking features. Running the gamut from new courses and seminars to research experiences, the program will aggressively promote increased cooperation on the UT campus between mathematics and other disciplines. UT is a comprehensive research institution whose charter contains the explicit constitutional mandate to be "a university of the first class." Superimposed upon that mandate is UT's stated mission of serving all the citizens of an enormous state with an extremely diverse population. In its continuing endeavor to contribute to this mission, the Department of Mathematics has assembled a world-class team of scholars and teachers. To a large extent, this faculty represent in their research interests the traditional core areas of mathematics, where the strength of the department has historically resided. On the other hand, recent faculty appointments reflect our decision to dedicate a significant portion of the department's resources to the interdisciplinary side of mathematics. Developments in this direction received a major boost from the creation of TICAM, in whose operation the department is deeply involved. But the department's commitment to interdisciplinary mathematics goes beyond the founding strengths of TICAM, to embrace financial mathematics, computational and mathematical biology. Training of students, both graduate and undergraduate, are the focal points of this VIGRE project. The framework in which such training takes place should be a set of programs geared to move students as effectively as possible toward their degrees. A prime objective of the UT VIGRE Graduate Traineeship component is to lower our average time to the doctorate to five years. Given the high quality of our recent students, we believe that we can achieve this end by the obvious means; namely, by freeing more of our Ph.D. students from hour-consuming teaching duties. At the same time-and it requires a delicate balancing act to get the desired results here without upsetting the five-year-to-the-Ph.D. applecart-we have engineered certain VIGRE activities to increase opportunities for graduate students to explore realms of the mathematical sciences that extend well beyond their areas of specialization. Some of the specific activities that will be initiated to add scientific and professional value to the UT graduate experience of VIGRE trainees are the following: participation in a "junior" research seminar, where graduate students, undergraduates, and postdocs will give presentations under the supervision of a senior faculty member; participation in a multidisciplinary research proseminar directed at first-year graduate students that will provide them with an informal overview of the variety of mathematical research that is being done locally by faculty, postdocs, and advanced graduate students; serving as the member of a vertically integrated team supervising an individual VIGRE undergraduate research project; serving as a "facilitator" in the VIGRE summer REU program, in which our sister schools in the UT system have been invited to participate; involvement in a VIGRE computational finance research program (which will also have undergraduate and postdoctoral components). The department's baccalaureate degree programs have in great measure been shaped by the dictates of the educational agenda at a major state university (e.g., we are expected to train a large number of mathematics teachers for Texas's elementary and secondary schools). We now have in place a strong core curriculum designed to give our majors a command of mathematics at a level that allows them either to pursue graduate degrees in the mathematical sciences or to secure mathematics related employment. VIGRE funding will enable us to enrich and enliven the undergraduate experience with research opportunities of various types: wide scale involvement in individual research projects; participation in research-oriented seminars, including a new multidisciplinary research proseminar that will be a clearing house for undergraduate research projects; participation in the multidisciplinary summer REU program cited in the previous paragraph; improved access to the Dean's Scholars Program, a nationally recognized interdisciplinary honors program in the College of Natural Sciences. Each of these activities will be encased in a vertically integrated structure. The VIGRE summer REU programs, in particular, are geared to expand the mathematical horizons of students. The topics slated to be treated in these programs over the life of the award are: wavelets and signal processing, mathematical biology, mathematical finance, computational number theory and cryptography. Although UT is presently one of the largest universities in the country, the size of the department's tenured or tenure-track faculty is quite modest, numbering fifty-two in Fall 2000. The UT Administration, not wishing to see the research activity of a small but distinguished faculty totally overwhelmed by instructional duties, compensates for this disproportion in size by giving the department a generous supply of postdoctoral positions. Each year our faculty includes roughly two dozen new or very recent Ph.D. recipients who are appointed to Postdoctoral Instructorships or Research Lectureships, with teaching loads the same as those of regular faculty members (i.e., at most two classes per semester). Most of the postdocs who have held these positions over the years have departed from UT with well-established research programs and highly developed teaching skills. Through the use of VIGRE funds, we will improve the UT postdoctoral experience by decreasing teaching loads for selected instructors, thereby releasing more time for them to devote to research, and by creating opportunities for all postdocs to participate in a rich array of vertically integrated professional development activities. For example, virtually all of the graduate and undergraduate VIGRE activities indicated earlier will find VIGRE postdocs involved in an integral way. Our goal, and the goal of this VIGRE project, is to help our postdocs, as well as graduate and undergraduate students, grow intellectually and mathematically in a way that allows them to take advantage of a mathematical world enriched and energized by its connections to the larger world of science, engineering, and society.
