Symplectic Geometry and Applications

AbstractAward：DMS-9971914首席研究员：Jonathan Weitsman本提案所关注的辛几何工作大致分为两个重叠的领域。 第一组问题的重点是发展一个更好的理解辛流形，特别是在拓扑和几何辛流形配备了群作用。在过去的五年中已经看到了显着增加的作用辛范畴在拓扑和几何，以及更好的理解之间的类比和辛几何和Kahler几何的差异。 在这一建议中，我们建议开发工具，研究哈密顿和辛群作用，以及拓扑和几何的辛共轭。 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。 这些问题主要是由量子场论在几何学和拓扑学中的应用引起的，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，这些几何结构必须是这些仍然神秘的物理方法的基础。are areas领域which哪一个have their其historical历史origins起源in the physical物理sciences科学. 正因为如此，他们有着从自然科学中出现的问题中获得洞察力的长期记录，以及对应用问题做出重大贡献的记录。 这一记录主要是由整个领域几十年来取得的，而不是由短期内的个人贡献。 但是在某种程度上，一个长期和一致的过去记录可以被用来作为未来的指示，有极好的前景，目前在这些领域的核心数学的工作将提供不可或缺的基石，为科学和技术的进步，在未来几十年。

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.