Computations and Applications of Periodic Floer Homology and Contact Homology in Symplectic Geometry

自从辛流形中的全纯曲线被发现以来，辛几何以及最近的接触几何都取得了很大的进展。Sullivan计划继续计算和应用基于这些曲线的两组不变量：接触流形中Legendrian子流形的接触同调和riemann曲面微分同调的周期flower同调。前者是辛场论的一个特例。虽然辛场论仍然没有严格定义，但Sullivan和其他人已经完成了他们版本的接触同调的基础分析。沙利文将把这个替代版本的接触学扩展到其他流形。最终目标是开发一个完整的Legendrian同位素类障碍。后一个不变量被推测为符合Seiberg-Witten-Floer同调。这个项目将发展这一理论的基础，并扩大现有的计算集。研究者也希望研究周期花同调计算的应用，解决4流形上辛结构的存在性和分类问题。辛几何和接触几何解释了某些动力系统的物理学，如行星绕太阳运行的轨道，陀螺的旋转，或带电粒子在磁场中的运动。这样的系统服从最小作用原理，这意味着它们的总能量或动量一定是守恒的。许多辛几何学者研究全纯曲线，这是对最小作用原理的重新解释，它将几个独立发展良好的数学领域联系在一起，如复分析和微分拓扑。对全纯曲线的研究导致了其他“物理”结果，如海森堡不确定性原理的推广。最近，这些曲线被认为出现在理论物理的其他领域，如弦理论。

DMS-0305825Michael G. SullivanMuch progress has been made in symplectic geometry, and more recently contact geometry, since the discoveryof holomorphic curves in symplectic manifolds.Sullivan plans to continue calculating and applying two sets of invariants based on these curves:the contact homology of Legendrian submanifoldsin contact manifolds and the periodic Floer homology ofRiemann surface diffeomorphisms.The former is a special case of symplectic field theory.Although symplectic field theory is still not rigorouslywell-defined, Sullivan and others havecompleted the foundational analysis for theirversion of contact homology.Sullivan will extend this alternative version of contacthomology to other manifolds.The ultimate goal is to develop a complete obstructionof Legendrian isotopy classes.The latter invariant is conjectured to agree with Seiberg-Witten-Floer homology. This project will develop the foundationsof this theory, as well as broaden the existing set of computations.The investigator also hopes to work on applications of the periodic Floerhomology computations, addressing the problems of existenceand classification of symplectic structures on 4-manifolds.Symplectic and contact geometry explain thephysics of certain dynamical systems, such asthe orbits of planets around the sun, the spin of a top, or the motion of a charged particle in a magnetic field.Such systems obey the least action principal, whichamong other things can mean that their total energyor momentum must be conserved.Many symplectic geometers study holomorphic curves,a reinterpretation of the least action principal,which has linked together several independently well-developedmathematical fields such as complex analysis and differential topology.The study of holomorphic curves has led to other``physical" results, such as a generalization of Heisenberg's uncertainty principal.More recently, these curves are thought to appear inother areas of theoretical physics, like string theory.