Mathematical Sciences: Symplectric Topology and It's Applications to String Theory

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9401367 Ruan The investigator intends to continue his work in the area of symplectic topology and its application to string theory. More precisely, he plans to work on the mathematical foundations of topological sigma model-quantum cohomology theory, to use topological sigma model invariants to study Calabi-Yau manifolds and surfaces of general type, to continue to work on symplectic extremal rays and symplectomophisms, in particular, degenerate fixed points of exact symplectomorphisms. Topological properties are special properties that remain unchanged under the small deformation of objects that the investigator studies. Studying topological properties in all branches of mathematics has been in the center of mathematics for the past century. This project studies some topological properties arising from mathematical physics, in particular, from string theory. But those topological properties are not ordinary topological phenomenon known before. They belong to a new kind of topological theory-Symplectic topology, which the investigator believes will provide much better understanding both for topology theory and for string theory. ***