This project will establish a Research Training Group in geometry as a collaborative effort between the Stony Brook Mathematics Department and the Simons Center for Geometry and Physics. It unites a long commitment to training young mathematicians at a traditional mathematics department with the exceptional scientific opportunities presented by a major international research center located next door. The overall aim of the program is to train a group of young mathematicians who will combine expertise in a specific geometric field with a serious appreciation of the other subfields of geometry and of the connections between geometry and mathematical physics. The RTG will have three main components. It will fund nine Postdoctoral Fellows who will be full participating members of both the mathematics department and the Simons Center. Besides engaging in a full range of traditional mathematical activities, the Fellows will participate in joint seminars bringing together postdocs in mathematics and theoretical physics. The program will also provide partial support to a cadre of graduate trainees in mathematics, who will have an affiliation in the Simons Center and will participate fully in activities in the Center. Finally, the program will put in place a number of new activities designed to enrich the experiences of Stony Brook mathematics undergraduates, including those interested in teaching careers, and to introduce them to the scientific focus of the Simons Center.