Holomorphic Curves and Low-Dimensional Topology

点击翻译按钮获取中文摘要

This project focuses on the interplay between low-dimensional topology and knot theory on one side, and symplectic and contact geometry on the other. It has recently been demonstrated that counting holomorphic curves in symplectic manifolds is a powerful tool in the topology of three- and four-manifolds. In particular, the Symplectic Field Theory associated to cotangent bundles shows great promise in yielding information about smooth structures on low-dimensional manifolds. This project proposes to find combinatorial and computationally accessible formulations for Symplectic Field Theory in this circumstance, with applications to areas such as knot theory and the (still unsolved) smooth four-dimensional Poincare conjecture. The project will also investigate the relationship between these symplectic-flavored smooth invariants and other recently developed geometric invariants such as Heegaard Floer theory and string topology. A closely related project will formulate Symplectic Field Theory combinatorially for Legendrian submanifolds, addressing a significant open problem in contact geometry.This proposal studies problems in topology such as the conjectured existence of an "exotic" smooth structure on four-dimensional space (the "smooth four-dimensional Poincare conjecture"). Although the problems are classical, the proposed approach using Symplectic Field Theory is quite new. It involves recently developed techniques from the interrelated fields of symplectic geometry, algebraic geometry, and string theory. Such techniques have been spectacularly successful in the last decade and show great promise in topology for understanding the classification of spaces in three and four dimensions, as well as the classification of knots. This will likely have interesting applications to related aspects of physics such as string theory and mirror symmetry.