Symplectic Structures on Closed Manifolds

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DMS-0604748Tian-Jun LiThe PI proposes to develop new techniques and apply existing methods in gauge theory, pseudo-holomorphic curve theory and equivariant stable homotopy theory to gain some fundamental understanding of the generalshape of closed symplectic manifolds. Four avenues of investigationare addressed. 1) The PI, joint with M. Furuta, is developing acomprehensive treatment of a refined invariant associated with aproper non-linear Fredholm map, which involves a twistedPontrjagin-Thom construction. 2) Such a construction applied to theSeiberg-Witten map turns out to be particular useful in theclassification of symplectic 4--manifolds with torsion symplecticcanonical classes. The PI has made progress bounding their Bettinumbers and is hoping to fully determine their homoeomorphism types. 3)The PI searches for a simple characterization of the symplectic coneof a K\"ahler surface. The case of geometric genus 0 has been completely settled by the PI and A. Liu. Joint with M. Usher, promising progress in the case of positive geometric genus has been made, and more is expected by exploring thesomewhat surprising interplay between symplectic forms and embeddedsymplectic surfaces with negative self-intersections. 4) Jointmainly with Y. Ruan, the PI is investigating the properties ofuniruled symplectic manifolds in dimension 6 and above, usingrelative invariants, their gluing formula, and localization techniques. We also search for the criterion such that symplecticblow-downs can be performed in dimension 6.This project belongs to the relatively new and increasingly important subject the PI has pursued a successful line of research. The proposedactivity raises several fundamental questions in this field and laysout plans to answer them. It also creates original concepts in thisfield and reveals connections with other fields. This proposed activity will advance knowledgein symplectic topology as well as other areas including differentialtopology, mathematical physics and algebraic geometry.An n manifold is a space that locally looks like the Euclidean space of dimension n. For example, the space-time universe we live in is a 4--manifold. A symplectic structure is a very basic structure that underlies almost all the equations of classical and quantum physics. A manifold equipped with a symplectic structure is called a symplecticmanifold. Studies of symplectic manifolds, such as proposed here,will thus enhance our understandings of mathematics, physics, andscience in general.