Nonlinear Problems in Symplectic Geometry and Complex Geometry

点击翻译按钮获取中文摘要

AbstractProposal: DMS-9802479Principal Investigator: Gang TianThe project addresses some fundamental problems from symplectic geometry and Kaehler geometry. The first part concerns the existence of Kaehler-Einstein metrics with positive scalar curvature. These metrics provide solutions of the Einstein equation on Riemannian manifolds. In the second part, the principal investigator (PI) intends to further study new invariants, often refered as GW-invariants, of general symplectic manifolds and find more applications to Hamiltonian systems, symplectic topology and geometry. This part also contains very basic problem of understanding structure of GW-invariants and its possible relation to particular integrable systems. The PI also suggests to study some related nonlinear PDE problems which arise from physics and geometry.Problems in this project were motivated by our desire of probing mathematics of basic physical laws under suitable conditions. For example, GW-invariants were inspired by a sigma model theory coupled with gravity in mathematical physics, and they extend classical enumerative geometry which involves counting curves through a number of points or intersections of curves. The resolutions of these problems will provide new mathematical insights of Einstein equation in general relativity, quantum field theory, mirror symmetry phenomenon in string theory. Our study will also deepen our understanding properties of symplectic manifolds, algebraic manifolds and Calabi-Yau spaces.