Symplectic Geometry and Applications

辛几何及其应用

• 批准号: 9971914
• 负责人: Jonathan Weitsman
• 金额: $ 7.09万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-15 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971914&HistoricalAwards=false
• 关键词: Symplectic; Geometry; Applications

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.

摘要：dms -9971914主要研究者：Jonathan weitsman本文所关注的辛几何工作大致分为两个重叠的领域。第一组问题侧重于更好地理解辛流形，特别是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几何之间的相似和差异。在这个建议中，我们建议开发研究哈密顿和辛群作用的工具，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。这些问题很大程度上是由量子场论在几何和拓扑学中的应用所激发的，其中许多问题要么直接涉及辛几何，要么涉及辛几何可以帮助提供更好的几何结构视图的领域，这些几何结构必须是这些仍然神秘的物理方法的基础。所涉及的数学领域——辛几何和数学物理——是在物理科学中有其历史起源的领域。因此，他们长期以来一直从自然科学中出现的问题中获得洞察力，并对应用问题做出了重大贡献。这一记录主要是整个领域在几十年的时间里取得的，而不是个人在短期内的贡献。但是，在某种程度上，长期和持续的过去记录可以作为未来的指示，这些核心数学领域的当前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石，这是一个极好的前景。