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

DMS-0305825Michael G. Sullivan自从在辛流形中发现全纯曲线以来，辛几何和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：切触流形中Legendrian子流形的切触同调和Riemann曲面同态的周期Floer同调.前者是辛场论的一个特例.虽然辛场论还没有严格定义，Sullivan等人已经完成了对他们的接触同调的反版本的基础分析。Sullivan将把这种接触同调的替代版本推广到其他流形。最终目标是发展一个完全的Legendrian合痕类。后一个不变量被证明是一个完全的Legendrian合痕类。与Seiberg-Witten-Floer同源性一致。该项目将发展这一理论的基础，并扩大现有的计算集。研究人员还希望致力于周期性Floerhomology计算的应用，解决4-流形上辛结构的存在和分类问题。辛几何和接触几何解释了某些动力系统的物理学，例如行星绕太阳的轨道，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量原理，这意味着它们的总能量或动量必须守恒。许多辛几何学家研究全纯曲线，这是对最小作用量原理的重新解释，它把几个独立的很好地联系在一起-发展了数学领域，如复分析和微分拓扑学。全纯曲线的研究导致了其他“物理”结果，如海森堡的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学的其他领域，比如弦理论。

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.