**Mathematical Structures in String Theory**

#### in partnership with the Clay Mathematics Institute.

**Coordinators:** Robbert H. Dijkgraaf, Michael R. Douglas, David Alexander Ellwood, Maxim Kontsevich, Gregory Moore, Nikita Nekrasov, Hirosi Ooguri

Ever since the "first superstring revolution" and the compactification of the heterotic string on Calabi-Yau manifolds, interaction with mathematics has been one of the primary forces driving progress in superstring theory. On the one hand string theory has generated many new mathematical concepts; and on the other hand new ideas from mathematics have often found their first applications in string theory. These topics include vertex algebras, conformal field theory, mirror symmetry, topological field theory and string theory, exact solutions of supersymmetric gauge theory, and noncommutative field theory, just to list some highlights. Recent exciting developments include the matrix model approach to N=1 gauge theory, open string mirror symmetry, the derived category approach to D-branes on Calabi-Yau, geometric transitions, proof of the N=2 Seiberg-Witten solution by instanton methods, and indications of integrable structures in super Yang-Mills theory and AdS string theory.

Besides these developments, some areas which are gaining prominence and may well dominate discussion in the years 2005--2006 include the computation of brane superpotentials, the study of orientifolds, the classification of boundary states, the statistics and global structure of the landscape of vacua, strings on twistor spaces, quiver gauge theories, and tensor categories. We also expect further development of connections with number theory(rational CFT; the attractor mechanism), and of the applications of topology to diverse problems in supergravity, string theory and M-theory.

While the program will cover all areas of interaction between string theory and mathematics, to provide additional focus we will emphasize particular subareas during different parts of the workshop, such as mirror symmetry and Calabi-Yau geometry; geometry and topology of gauge theory; vector bundles and representation theory; matrix models and combinatorics; and number theory.

At present, our tentative schedule for focus periods is as follows:

Besides these developments, some areas which are gaining prominence and may well dominate discussion in the years 2005--2006 include the computation of brane superpotentials, the study of orientifolds, the classification of boundary states, the statistics and global structure of the landscape of vacua, strings on twistor spaces, quiver gauge theories, and tensor categories. We also expect further development of connections with number theory(rational CFT; the attractor mechanism), and of the applications of topology to diverse problems in supergravity, string theory and M-theory.

While the program will cover all areas of interaction between string theory and mathematics, to provide additional focus we will emphasize particular subareas during different parts of the workshop, such as mirror symmetry and Calabi-Yau geometry; geometry and topology of gauge theory; vector bundles and representation theory; matrix models and combinatorics; and number theory.

At present, our tentative schedule for focus periods is as follows:

- Gromov-Witten theory and instanton calculus: August
- recent development in AdS/CFT correspondence: September
- topological aspects of string theory and M-theory: Oct. 16 - Nov. 4
- matrix models and large N: November