This program harnesses the special resources at Stony Brook to train students and post-docs in geometry (broadly construed) and its interaction with mathematical physics. It involves a radical shift from standard curricula and focuses on training students, from the beginning, in the dual cultures of mathematics and physics. This NSF-funded program entails substantial student-faculty interaction and intensive mentoring. It has a series of interactive learning seminars, and incorporates Stony Brook's major math-physics programs, including the annual workshop in String Theory organized under the leadership of Cumrun Vafa. There are many areas of mathematics connected to this project, including: differential geometry, algebraic topology, algebraic geometry and representation theory, complex analysis, symplectic topology, and dynamical systems. Young people interested in any of these or related fields should find the program appealing.

Stony Brook supports one of the world's best endowed centers for the study of geometry and physics. Its resources, particularly with the recent addition of the highly successful Simons Workshop in Mathematics and Physics, are without parallel in this field.

The Stony Brook Mathematics Department has a large group of geometers whose current research involves many aspects of physics-related mathematics, from algebraic geometry to string topology. The department also has a long list of seminars, many with physics-related themes, and a rich atmosphere of research activity. Seminars in recent years include:

• The Algebra/Geometry/Physics Seminar,
• The Symplectic Topology Seminar,
• The Sullivan String Seminar,
• The Dynamical Systems Seminar,
• The Geometry-Topology Seminar,
• The Mathematical Physics Learning Seminar,
• The Friday Theoretical Graduate Physics Seminar (at YITP)
• Graduate Algebraic Geometry Seminar - Joint with Columbia,
• The General Relativity Seminar.

The C. N. Yang Institute for Theoretical Physics is a center of activity for many branches of theoretical physics. Particularly pertinent to the RTG program are the efforts in string theory. Four faculty members, Peter Van Nieuwenhuizen, Warren Siegel, Leonardo Rastelli, and Martin Rocek, are active researchers in this area. Rocek, in particular, is involved in studying geometric aspects of string theory. There are also several post-docs and a large group of active physics Ph.D. students working on topics that impinge mathematics.

For many years there has been a close collaboration between Stony Brook's Math Department and the Yang Institute for Theoretical Physics. This has involved joint seminars, jointly mentored graduate students, and successful collaborative research projects.

There are also two major programs in mathematics and physics. The first is the Simons Math-Physics Endowment run jointly by the Institute for Mathematical Sciences and the Yang Institute for Theoretical Physics. This program, now over ten years old, has brought a steady stream of visiting scholars and postdocs to the University. Fellows in Residence in this program have included: I. Singer, E. Witten, N. Seiberg, C. Vafa, J. Harris, N. Hitchin, D. Ruelle, K. Fukaya, G. Perelman, W. Fulton, and many others.

The second program, now in its third year, is the highly successful Simons Workshop in Mathematics and Physics. Organized under the leadership of Cumrun Vafa, its scientific advisor, the workshop brings a large number of active researchers of all ages, including world leaders in string theory and related mathematics, for an intense five week program. The second year's program lead to the astonishing discovery (in the midst of the meeting) of a profound relation between topological strings on Calabi-Yau manifolds and the statistical mechanics of crystal melting. This led to new insights into the underlying structure of quantum strings. Current and recent participants include top graduate students, post-docs, junior faculty, and senior scientists such as: Vafa, Witten, Yau, Dijkgraaf, Nekrasov, Pandharipande, Ooguri, Berkovits, Katz, Moore, Okounkov, Hollowood and others. Stony Brook students, post-docs and faculty participate actively. This past summer there was an intensive Physics Clinic for mathematics graduate students and faculty, which ran parallel to, and in direct relation to the Simons workshop.\medskip

In recent years Stony Brook has run a large number of high-level conferences in areas of geometry and physics. Among them were: Connections in Modern Mathematics and Physics, 1998, Laminations and Foliations in Dynamics, Geometry and Topology, 1998, Graphs and Patterns in Mathematics and Theoretical Physics, 2001, Minimal Varieties in Geometry and Physics, 2002, and The Geometry Festival, 2004.

The program entails a full-scale, intensive mentoring program. Many courses will be team taught, with one member of the team representing the physical side of the topic. Some courses have an attached training seminar. Talented undergraduates shall be brought into the program with a high degree of mentoring to guide their participation.

More specifically the program involves:

1. A ``bipolar'' curriculum of basic courses that present material from both the core mathematical and the physical (i.e., physically motivated) viewpoints. Many of these courses will be team-taught by a mathematician and someone with training in physics.
2. Bringing students into the life of workshops, seminars, and invited lectures at an early stage.
   1. Upon arriving each student will be assigned two advisors; one with mathematical and one with physical perspectives. These advisors will meet regularly with the student to discuss upcoming lectures, seminars and workshops and to devise a plan of preparation. They will also discuss past lectures, answer the student's questions, and direct them to further reading on topics that have aroused interest.
   2. For major workshops and lecture series, preliminary working groups will be organized. Also, "post-mortem" groups will be formed to review and absorb what took place.
   3. When appropriate and agreed to by the speaker, lecture notes will be prepared by students and post-docs following major lectures and lecture series.
3. A large seminar component where students will be trained to be proactive -- to think forwardly and critically when learning mathematics.