Symplectic homology and Stein manifolds

项目编号：dms -1005365项目负责人：Mark mclean本课题的研究领域是Stein流形的辛几何。Stein流形是复仿射空间中适当嵌入的复子流形。这是由仿射空间中的标准形式导出的辛形式。如果我们取一个大的球体并与这个斯坦因流形相交然后我们得到另一个流形叫做斯坦因可填充接触流形。Eliashberg和Gromov在辛研究Stein流形方面取得了重要进展。这个项目的主要目的是寻找奇异的斯坦结构和斯坦可填充的接触结构。本文旨在证明有无数个辛不同的Stein结构微分同构于一个偶维流形，该流形承认一个自下有界的固有莫尔斯函数，且至多在其一半维上只有有限个指标临界点。本文还证明了在5维及以上的奇维球上存在无限多的Stein可填充接触结构，并且更普遍地证明了由仿射变体得到的Stein可填充接触流形上存在无限多的Stein可填充接触结构。PI将使用斯坦因流形的不变量辛同调来区分它们。PI还旨在表明，没有一种算法可以告诉你一个仿射空间的Stein流形对另一个复维数大于6的仿射空间的Stein流形是否辛态。PI还旨在证明所有大于13维的奇维球面上的接触结构的类似的不可判定结果。PI将使用一个称为辛同调的增长率的不变量来实现这一点。PI将使用增长率来证明某些共切束有许多里布轨道（甚至是简并轨道）。这推广了Gromoll-Meyer定理。PI将表明合理双曲流形的余切束对使用增长率的光滑仿射变体不是辛态的。如果我们有一些经典的系统，比如钟摆，那么在任何时间点，它都有一个特定的位置和动量。如果这个系统有很多运动的部分，比如一个双摆或者一个自由粒子集合，那么它就有很多位置和动量。所有这些位置和动量的集合可以被编码成一个叫做辛流形的对象。例如，与钟摆相关的辛流形是一个圆柱体。辛流形在物理学的许多领域都很重要，例如量子力学和弦理论。PI将研究从被称为Stein流形的物体中得到的一大类辛流形。PI将构建一个很大的斯坦因流形列表称为外来斯坦因流形，它们看起来非常类似于来自一组自由粒子的辛流形，但如果我们观察它们各自经典系统的运动，它们实际上是不同的。PI的目的是证明没有计算机算法能告诉你两个给定的奇异斯坦流形是否来自同一个经典系统。这个结果很有用，因为它告诉我们，一般来说，某些经典系统很难研究。

AbstractAward: DMS-1005365Principal Investigator: Mark McLeanThe subject area of this project is the symplectic geometry of Stein manifolds. A Stein manifold is a properly embedded complex submanifold of complex affine space. This has a symplectic form induced from the standard one in affine space. If we take a large sphere and intersect it with this Stein manifold then we get another manifold called a Stein fillable contact manifold. Important progress in studying Stein manifolds symplectically was achieved by Eliashberg and Gromov. The primary aim of this project is to find exotic Stein structures and Stein fillable contact structures. The PI intends to prove that there are uncountably many symplectically different Stein structures diffeomorphic to an even dimensional manifold admitting a proper and bounded from below Morse function with only finitely many critical points of index at most half its dimension. The PI also intends to prove that there are infinitely many Stein fillable contact structures on each odd dimensional sphere of dimension 5 and higher, and more generally on Stein fillable contact manifolds obtained from affine varieties. The PI will use an invariant of Stein manifolds called symplectic homology to distinguish these. The PI also aims to show that there is no algorithm to tell you whether one Stein manifold diffeomorphic to affine space is symplectomorphic to another one diffeomorphic to affine space of complex dimension greater than 6. The PI also aims prove a similar undecidability result for contact structures on all odd dimensional spheres of dimension greater than 13. The PI will use an invariant called the growth rate of symplectic homology to achieve this. The PI will use growth rates to show that certain cotangent bundles have many Reeb orbits (even degenerate ones). This generalizes the Gromoll-Meyer theorem. The PI will show that the cotangent bundle of a rationally hyperbolic manifold is not symplectomorphic to a smooth affine variety using growth rates.If we have some classical system such as a pendulum then at any point in time it has a particular position and momentum. If this system has many moving parts such as a double pendulum or a collection free particles then it has many positions and momenta. The set of all such positions and momenta can be encoded in an object called a symplectic manifold. For example the symplectic manifold associated to a pendulum turns out to be a cylinder. Symplectic manifolds are important in many areas of physics such as quantum mechanics and String theory. The PI will study a large class of symplectic manifolds obtained from objects called Stein manifolds. The PI will construct a large list of Stein manifolds called exotic Stein manifolds which look very similar to the symplectic manifold coming from a set of free particles but are actually different if we look at the motion of their respective classical systems. The PI intends to show that there is no computer algorithm telling you if two given exotic Stein manifolds come from the same classical system. This result is useful because it tells us that certain classical systems are very hard to study in general.