Symplectic Topology

最近使辛几何研究发生革命性变化的主要工具之一是解析方法（柯西-黎曼方程的非线性版本）的使用。在过去的一年里，这个理论中的一个严重的技术问题已经被克服，所以它现在适用于所有的流形。在与Lalonde和Polterovich的合作中，McDuff正在应用这一改进的理论来研究流形的微分同态群的性质，以保持其复数结构。这个群有一个由哈密顿函数生成的有限余维子群，但一般不知道这个群在满群中是否总是闭的。如果是这样的话，我们就可以为一个函数（而不是单一形式）生成一个复形态给出一个拓扑原因。McDuff（与Abreu合作）正在研究的另一个问题是，随着底层辛结构的变化，辛形态群的拓扑结构会发生多大程度的变化。当底层流形是两个二维球体的乘积，其中一个改变球体的相对大小时，这个问题的一个特别简单的版本就出现了。他们证明了存在一个偶数维上同类，每当较大的球体的大小增加一个单位，它的维数就会增加4个。他们正在研究产生这种行为的机制。辛结构是一种非常基本的几何结构，它是几乎所有经典和量子物理方程的基础。在过去的15年里，新的工具被开发出来，使数学家们第一次对这种结构的整体意义有了一些了解。过去几年最重要的发展，一方面是对四维辛空间结构的阐明，另一方面是对人们在辛空间中移动的方式的新理解。McDuff与Lalonde一起完成了一种特别简单的称为直纹曲面的4流形的分类，并正在研究更复杂的例子的结构。她也在研究实现一个特定的空间运动需要多少能量的问题，并找到在弯曲流形上估计这种能量的方法。＊＊＊

9704825 McDuff One of the main tools that has recently revolutionized the study of symplectic geometry is the use of analytic methods (non-linear versions of the Cauchy-Riemann equations). In the past year, a serious technical problem in this theory has been overcome, so that it now applies to all manifolds. In collaboration with Lalonde and Polterovich, McDuff is applying this improved theory to study properties of the group of diffeomorphisms of a manifold that preserve its symplectic structure. This group has a subgroup of finite codimension that is generated by Hamiltonian functions, but it is not known in general whether this group is always closed in the full group. If it were, one might be able to formulate a topological reason for a symplectomorphism to be generated by a function (rather than a one-form). Another question that McDuff is working on (in collaboration with Abreu) is the extent to which the topological structure of the group of symplectomorphisms changes as the underlying symplectic structure changes. A particularly simple version of this problem appears when the underlying manifold is the product of two two-dimensional spheres and one changes the relative size of the spheres. They have shown that there is an even-dimensional cohomology class whose dimension jumps up by four each time the size of the larger sphere increases by another unit. They are working on understanding the mechanism that produces this behavior. A symplectic structure is a very basic kind of geometric structure that underlies almost all the equations of classical and quantum physics. In the last fifteen years, new tools have been developed that allow mathematicians for the first time to gain some understanding of the global meaning of this kind of structure. The most significant developments of the past few years concern, on the one hand, the elucidation of the structure of four-dimensional symplectic spaces and, on the other, new understanding of the wa ys that one can move around in a symplectic space. Together with Lalonde, McDuff completed the classification of a specially simple kind of symplectic 4-manifold called a ruled surface and is in the process of studying the structure of more complicated examples. She is also investigating the question of how much energy it takes to achieve a particular movement of space, and to find ways of estimating this energy on curved manifolds. ***