The Department of Mathematics at the University of Iowa and the closely allied Program in Applied Mathematical and Computational Sciences (AMCS) will work closely together on this VIGRE (Vertical Integration of Research and Education) project. They will train high-quality U. S. mathematicians through successful integration of cutting-edge research and education in the mathematical sciences. The project builds on the successes that the Department of Mathematics and AMCS have had in the identification and professional development of highly talented U. S. undergraduate and graduate students, in particular students from underrepresented minority groups. The Department and AMCS will bring their successful techniques to a broader range of students in combination with further vertical and horizontal integration of the Department and AMCS. Parts of this project are in cooperation with a coalition, The Heartland Mathematics Partnership, of the Department of Mathematics and AMCS at the University of Iowa and the departments of mathematics at twelve other area colleges and universities. This will allow the Department and AMCS to increase the number of highly talented and well-prepared U. S. students who enter a career in the mathematical sciences. This VIGRE project includes vertically integrated research groups in areas of research strength in the Department and AMCS. It also includes research experiences for U. S. undergraduates from the University of Iowa and schools in the Partnership. Finally, it includes Graduate Traineeships, Postdoctoral Associates, and Workshops. Results of the project will be widely disseminated. Assessment will facilitate continuous improvement of the project. This VIGRE project has been carefully designed to provide a model for mathematical sciences departments at top research universities.