The combined forces of the Mathematics Department and the Applied Mathematics Program at the University of Arizona provide an environment in the mathematical sciences that offers breadth and excellence in both research and innovative teaching activities. The research spectrum spans a wide range of activities from number theory and geometry to biomathematics and scientific computation; and the teaching program was among the first to embrace technological innovation, outreach, and vertical integration. Over the past several years the Department and the Program have introduced many training innovations, including Research Tutorial Groups, a unique experimental modeling laboratory, a variety of options aimed at improving communication and computational skills, an internship program, and vertically integrated outreach activities. Combined with additional activities especially designed for students who will be supported by VIGRE grants, and the rich research and teaching environment, all these initiatives will provide graduate students, undergraduates, and postdoctoral associates with the training they will need to be successful mathematicians in the coming decades. Central to our proposal are distinguished research groups in Algebra/Number Theory, Biomathematics, Dynamical Systems, Fluid Dynamics, Geometry, Mathematical Physics/Applied Probability, Mathematics Education, Nonlinear Optics, and Partial Differential Equations. Each of these groups has not only an outstanding reputation in research but an excellent track record of integrating graduate students, undergraduates, and postdoctoral fellows into their research activities. We note that the Number Theory group was recently recognized by the award of a Group Infrastructure Grant by the NSF Division of Mathematical Sciences, and the Biomathematics activity is part of a larger enterprise that was recently awarded one of the new IGERT (Integrative Graduate Education and Research Training) grants by the NSF. The interdisciplinary graduate Program in Applied Mathematics adds an extra dimension to this spectrum of research. Established twenty years ago to foster campus-wide cross-disciplinary research and training in the mathematical sciences, the Program connects students with faculty across an array of disciplines, drawn from 15 departments in the Colleges of Science, Engineering, and Medicine. Despite the Program's executive independence it maintains a unique working relationship with the Mathematics Department, many of whose faculty are among its most active members; 31 of the 53 PhDs awarded by the Program in the past five years were supervised by faculty from the Department of Mathematics. Graduate Funding: Except for two first-year fellowships VIGRE funds will be used to support students who have passed Qualifying Examinations. To apply for VIGRE support a student will write a short proposal describing a research strategy and chosen elective activities. The proposal should identify a research group that the student will join and explain the student's intended contribution to the group. A small number of summer support awards will be made to students preparing for examinations. We anticipate a steady state of approximately 18 students supported by VIGRE or matching funds in each academic year semester, and slightly larger numbers having at least partial summer research support. Undergraduate Activities: Engaging the interest of undergraduates in mathematics and encouraging them to pursue careers in mathematics - either as teachers, researchers, or in industry - is a major goal of our proposal. The various training components that will fulfill these goals are a comprehensive range of research experiences, an expanded summer internship program, and opportunities for qualified undergraduates to help in teaching. Under the aegis of VIGRE we will coordinate all undergraduate research initiatives carried out by the Department and the Program. Graduate students and postdoctoral fellows will be encouraged to participate in the supervision of selected undergraduate projects. Participating undergraduate students will prepare written reports about their activities, and at least once each semester the students will present their results at a Saturday conference devoted to undergraduate research. The participants whose presentations are judged the best will be provided with travel funds to present their results at regional or national meetings of appropriate professional organizations such as the Mathematical Association of America and the American Mathematical Society. We anticipate supporting 10-12 REU students each academic year, and slightly smaller numbers in the summers. Postdoctoral Fellows: Our recruitment of VIGRE Postdoctoral Fellows will target recent PhDs who anticipate future careers as faculty members in research universities, and whose research interests have significant intersection with at least one of our research groups. To prepare them for such careers we will provide them with exciting research opportunities, innovative teaching experiences at all levels, and a leadership role in a variety of graduate and undergraduate educational experiences outside the classroom. A Postdoctoral Mentoring Committee will ensure, beginning with the postdoctoral recruiting process, that appropriate faculty mentors will be available for all VIGRE postdoctoral fellows. Postdoctoral fellows will teach one course each semester (in residence). To prepare them properly for university careers, these courses will be at all levels. At least one course will be a graduate course in the fellow's area of research, possibly team-taught with a faculty mentor. Postdoctoral fellows will be encouraged to spend a summer or semester internship at a national laboratory, a research institute such as MSRI, or in industry. We will support four VIGRE postdoctoral fellows in each year after the first, with matching support being provided for two each from the Department and the Program. The two fellows supported and selected by the Department will be chosen to ensure continuing strength and growth in existing research groups. The two supported by the Program will be chosen by a committee of Program faculty to create cross-disciplinary ties with other departments.
