Symplectic homology and Stein manifolds

辛同调和斯坦因流形

• 批准号: 1005365
• 负责人: Mark McLean
• 金额: $ 13.01万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005365&HistoricalAwards=false
• 关键词: Symplectic; homology; Stein; manifolds

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.

Abstractaward：DMS-1005365原理研究者：Mark McLeanthe该项目的主题区域是Stein歧管的符合性几何形状。 Stein歧管是一个正确嵌入的复杂仿射空间的复杂子手机。这具有仿射空间中标准形式引起的符号形式。如果我们占据一个大球并将其与此Stein歧管相交，那么我们将得到另一个称为Stein填充的接触歧管的歧管。 Eliashberg和Gromov在研究Stein歧管的研究中取得了重要的进展。该项目的主要目的是找到异国情调的Stein结构和Stein可填充的接触结构。 PI打算证明，有许多符合性不同的Stein结构与均匀的歧管差异，从莫尔斯（Morse）函数下方，只有一半的索引界限有限的许多关键点。 PI还打算证明，在尺寸5及更高的每个奇数尺寸球体上有无限的Stein填充接触结构，更通常是从仿射品种获得的Stein填充接触歧管上。 PI将使用称为Symblectic同源性的Stein流形的不变性来区分这些歧管。 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实现这一目标的同源性。 PI将使用增长速度表明某些cotangent束具有许多Reeb轨道（甚至是变质的轨道）。这概括了gromoll-meyer定理。 PI将表明，合理双曲线歧管的cotangent束并非使用生长速率与光滑的仿射品种符号型。如果我们有一些经典系统，例如摆锤，那么在任何时间点，它具有特定的位置和动量。如果该系统具有许多运动部件，例如双摆或无收集粒子，则具有许多位置和动量。所有此类位置和动量的集合可以在称为符号歧管的对象中编码。例如，与钟摆相关的符号歧管原来是一个圆柱体。在量子力学和弦理论等物理学的许多领域中，符号歧管很重要。 PI将研究从称为Stein歧管的对象获得的大量符号歧管。 PI将构建一大批称为异国斯坦歧管的Stein歧管列表，看起来与来自一组自由粒子的符号歧管非常相似，但是如果我们查看其各自的经典系统的运动，则实际上是不同的。 PI打算表明，没有计算机算法告诉您是否有两个给定的异国风歧管来自同一经典系统。该结果很有用，因为它告诉我们某些经典系统通常很难研究。