Symplectic Topology

DMS-0072512 Dusa McDuff McDuff将继续研究辛流形的结构以及在给定流形M上保持辛结构的双同态群G（M）的结构。与弗朗索瓦Lalonde一起工作，她最近发现，这些同胚不扭曲流形拓扑非常。实际上，他们利用量子同调的思想，证明了G（M）的哈密顿子群H的同伦群平凡地作用于基础流形M的有理同调。 证明了任何具有结构群PH的纤维丛，只要其有理上同调是乘积。 她还计划研究量子同源性存在的其他拓扑含义，特别是它对通量猜想的影响。辛结构是一种非常基本的空间结构，它是经典物理学方程的基础。 最近，由于它们出现在现代对偶理论中，特别是在高能物理的“镜像对称”现象中，它们变得非常突出。 在数学上，它们非常有趣，有助于理解四维空间（弯曲的时空）。 最近，麦克达夫把她的注意力集中在研究不同的方式，点的空间可以移动，同时仍然保持这种结构。 她与一位合作者合作，发现弦理论（在该领域被称为“量子同源性”）的一些观点表明，具有辛结构的空间比没有辛结构的空间刚性得多，不能移动和扭曲得太多。 她打算在赠款期间继续提出这一系列问题。 许多非常有趣的辛刚性现象已经被发现，但仍然有很多悬而未决的问题。

DMS-0072512Dusa McDuffMcDuff will continue studying the structure of symplectic manifolds and of the group G(M) of diffeomorphisms that preserve the symplectic structure on a given manifold M. Working together with Francois Lalonde, she recently discovered that these diffeomorphisms do not twist the manifold topologically very much. Indeed, using ideas from quantum homology, they have shown that the homotopy groups of the Hamiltonian subgroup H of G(M) act trivially on the rational homology of the underlying manifold M. Theconjecture is that any fiber bundle with structural groupH is a product as far as its rational cohomology is concerned. She also plans to study other topological implications of theexistence of quantum homology, in particular its consequences for the Flux conjecture. A symplectic structure is a very basic structure on space that underlies the equations of classical physics. They have become very prominent recently because of their appearance in modern theories of duality, especially in the "mirror symmetry" phenomena of high energy physics. Mathematically, they are very interesting and help in understanding 4 dimensional spaces (curved space-times). Recently McDuff has focussed her attention on studying the different ways that the points of a space canmove while still preserving this structure. Working with a collaborator, she has found that some ideas coming from string theory (known in the field as "quantum homology") show that spaces with symplectic structures are much more rigid than ones without, and cannot be moved and twisted upvery much. She intends to pursue this line of questioning during the period of the grant. Many very interesting symplectic rigidity phenomena have been discovered, but there are still plenty of open questions.