Symplectic Geometry and Applications

摘要奖：DMS-9971914主要研究人员：乔纳森·魏茨曼本提案关注的辛几何工作大致分为两个重叠的区域。第一组问题集中于发展对辛流形的更好的理解，特别是关于具有群作用的辛流形的拓扑和几何。在过去的五年中，辛范畴在拓扑和几何中的作用显著增加，并且更好地理解了辛几何和Kahler几何之间的相似和区别。在这个建议中，我们建议开发工具来研究哈密顿和辛群作用，以及辛商的拓扑和几何。我们关心的第二组问题是将辛几何的思想应用于数学和数学物理的其他领域中出现的问题。这些问题在很大程度上是由量子场论在几何学和拓扑学中的应用所推动的，其中许多要么直接涉及辛几何，要么涉及辛几何可以帮助更好地了解几何结构的领域，而这些几何结构必须是由物理产生的仍然神秘的方法的基础。所涉及的数学领域-辛几何和数学物理--是历史上起源于物理科学的领域。因此，他们在从自然科学中出现的问题中获得洞察力以及在应用问题上做出重大贡献方面有着长期的记录。这一纪录主要是由整个领域几十年来创造的，而不是通过个人在短期内做出的贡献。但是，就长期和一致的过去记录可以作为未来的预测而言，这些核心数学领域目前的工作将为未来几十年的科学和技术进步提供不可或缺的基础，这一前景非常好。

AbstractAward: DMS-9971914Principal Investigator: Jonathan WeitsmanThe work in symplectic geometry on which this proposal focuses isdivided roughly into two overlapping areas. The first set ofproblems is focused on developing a better understanding ofsymplectic manifolds, and in particular on the topology andgeometry of symplectic manifolds equipped with group actions.The past five years have seen a significant increase of the roleof the symplectic category in topology and geometry, as well as abetter understanding of the analogies and differences betweensymplectic geometry and Kahler geometry. In this proposal wepropose to develop tools for the study of Hamiltonian andsymplectic group actions, as well as of the topology and geometryof symplectic quotients. The second set of problems we areconcerned with is the application of ideas from symplecticgeometry to problems arising in other areas of mathematics andmathematical physics. These problems are largely motivated bythe applications of quantum field theory to geometry andtopology, many of which either involve symplectic geometrydirectly or else involve areas where symplectic geometry can helpto provide a better view of the geometrical structure that mustunderlie these still-mysterious methods arising from physics.The areas of mathematics involved---symplectic geometry andmathematical physics---are areas which have their historicalorigins in the physical sciences. As such, they have had a longrecord of gaining insight from problems arising in the naturalsciences, as well as of making significant contributions toapplied problems. This record is one attained mostly by thewhole field over decades, rather than by individual contributionsover the short term. But to the extent that a long and consistentpast record can be used as an indication of the future, there areexcellent prospects that current work in these areas of coremathematics will provide indispensible building blocks forscientific and technological advances in the decades to come.