Five problems in geometry

几何中的五个问题

• 批准号: 0204651
• 负责人: David Auckly
• 金额: $ 10.61万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204651&HistoricalAwards=false
• 关键词: Five; problems; geometry

ABSTRACT DMS - 0204651. PI: David AucklyThis project is centered around a list of problems on the border oftopology, geometry and analysis. The least speculative problem is partof a collaboration with V. Kapovitch to prove that there are compacttopological 4-manifolds that do not admit Alexandrov metrics. The resultshould follow by a suitable modification of previous work of S. Donaldsonand D. Sullivan. The most speculative problem in the list is to study theinteraction between Gromov-Witten invariants and finite type invariants.Many different ideas are used in geometric analysis, and developmentsin this field will effect a broad range of other disciplines. For examplesymplectic geometry and gauge theory were developed to describe mechanicsand electrodynamics. These ideas were developed into subtle invariantsof smooth manifolds defined via the solution to systems of partial differential equations. Various gluing formulae developed to compute these invariantsresulted in relationships with other invariants that arise from a combinatorialpoint of view. Properties of the combinatorial invariants lead to conjecturesabout the smooth invariants, and vice versa. It is exactly this interplaythat makes geometric analysis such an important field.

摘要DMS-0204651。PI：David Auckly这个项目围绕着拓扑学、几何学和分析的一系列问题展开。最不具投机性的问题是与V.Kapovitch合作证明存在不允许Alexandrov度量的紧拓扑4-流形的一部分。这一结果之后应该对S.Donaldson和D.Sullivan之前的工作进行适当的修改。最具投机性的问题是研究Gromov-Witten不变量和有限类型不变量之间的相互作用。几何分析中使用了许多不同的想法，这一领域的发展将影响到广泛的其他学科。例如，发展了辛几何和规范理论来描述力学和电动力学。这些思想被发展成通过解偏微分方程组定义的光滑流形的微妙不变量。为了计算这些不变量而开发的各种胶合公式导致了与其他从组合观点出发的不变量之间的关系。组合不变量的性质导致了关于光滑不变量的猜想，反之亦然。正是这种相互作用使几何分析成为一个如此重要的领域。