The Surveys in Differential Geometry are supplements to the Journal of Differential Geometry, which are published by International Press. They include significant invited papers combining original research and overviews of the most current research in specific areas of interest to the growing Journal of Differential Geometry community. The survey volumes serve as continuing references, inspirations for new research, and introductions to the variety of topics of interest to differential geometers. These supplements are published annually since 1999.

This volume arises out of the conference sponsored by the Journal of Differential Geometry and held at Harvard University to honor the four mathematicians who founded Index Theory. A large number of geometers gathered for this historic occasion which included numerous tributes and reminiscences which will be published in a separate volume. The four founders of the Index Theory - Michael Atiyah, Raoul Bott, Frederich Hirzebruch, and Isadore Singer - are sources of inspiration, mentors and teachers for the other speakers and participants at the conference. The larger-than-usual size of this volume derives directly from the tremendous respect and admiration for the honorees.

Table of Contents: 1. Projective planes, Severi varieties and spheres - M. Atiyah and J. Berndt 2. Degeneration of Einstein metrics and metrics with special holonomy - J. Cheeger 3. The min-max construction of minimal surfaces - T. H. Colding and C. De Lellis 4. Universal volume bounds in Riemannian manifolds - C. B. Croke and M. Katz 5. A Kawamata-Viehweg vanishing theorem on compact Kahler manifolds - J.-P. Demailly and T. Peternell 6. Moment maps in differential geometry - S. K. Donaldson 7. Local rigidity for cocycles - D. Fisher and G. A. Margulis 8. Einstein metrics, four-manifolds, and differential topology - C. LeBrun 9. Topological quantum field theory for Calabi-Yau threefolds and $G_2$-manifold - N. C. Leung 10. Geometric results in classical minimal surface theory - W. H. Meeks III 11. On global existence of wave maps with critical regularity - A. Nahmod 12. Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks - E. Viehweg and K. Zuo 13. Geometry of the Weil-Petersson completion of Teichmüller space - S. A. Wolpert