Pseudoholomorphic curves, orbifolds, and group actions

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DMS-0603932Weimin Chen Gromov's pseudoholomorphic curve theory has been at the center of many great advances in mathematics in the past twenty years. This project seeks to apply Gromov's ideas and techniques of pseudoholomorphic curves in the context of group actions on manifolds, which technically amounts to studying pseudoholomorphic curves in the quotient space of the group action. Such a quotient space, called an orbifold, may have singularities in general, which correspond to the fixed points of the group action. Particularly, this project proposes to systematically study a certain class of smooth finite group actions on four-dimensional manifolds, which includes finite order automorphisms of nonsingular algebraic surfaces, a subject which has been long studied in algebraic geometry. The project also seeks to classify a certain type of circle actions on the five-dimensional sphere involved in the study of Einstein metrics on the sphere, which is a subject of central importance in Riemannian geometry.The importance of symmetry in mathematics has been long recognized. Acrucial issue in the study of symmetry is to understand the structureof the set of points in the space which are fixed under the symmetry.The central new idea in this project is the observation that when studying symmetries of a four-dimensional space (for instance, the universe in which we live), one can often extract useful information about the fixed points of a symmetry by studying a certain type of rigid two-dimensional subspaces (like a soap bubble) in the four-dimensional space.