Symplectic Structures on Closed Manifolds

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

DMS-0604748 Tian-Jun Li The 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 general shape of closed symplectic manifolds. Four avenues of investigation are addressed. 1) The PI, joint with M. Furuta, is developing a comprehensive treatment of a refined invariant associated with a proper non-linear Fredholm map, which involves a twisted Pontrjagin-Thom construction. 2) Such a construction applied to the Seiberg-Witten map turns out to be particular useful in the classification of symplectic 4--manifolds with torsion symplectic canonical classes. The PI has made progress bounding their Betti numbers and is hoping to fully determine their homoeomorphism types. 3) The PI searches for a simple characterization of the symplectic cone of 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 the somewhat surprising interplay between symplectic forms and embedded symplectic surfaces with negative self-intersections. 4) Joint mainly with Y. Ruan, the PI is investigating the properties of uniruled symplectic manifolds in dimension 6 and above, using relative invariants, their gluing formula, and localization techniques. We also search for the criterion such that symplectic blow-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 proposed activity raises several fundamental questions in this field and lays out plans to answer them. It also creates original concepts in this field and reveals connections with other fields. This proposed activity will advance knowledge in symplectic topology as well as other areas including differential topology, 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 symplectic manifold. Studies of symplectic manifolds, such as proposed here, will thus enhance our understandings of mathematics, physics, and science in general.