Five problems in geometry

摘要DMS - 0204651。 PI：大卫奥克兰这个项目是围绕着一系列的问题边界上的拓扑，几何和分析。最不投机的问题是与V. Kapovitch合作的一部分，以证明有拓扑4流形不承认亚历山德罗夫度量。所得结果对S.唐纳森和D.苏利文其中最具投机性的问题是研究Gromov-Witten不变量和有限型不变量之间的相互作用，几何分析中使用了许多不同的思想，这一领域的发展将影响到其他学科的广泛范围。例如，辛几何和规范理论被用来描述力学和电动力学.这些想法发展到微妙的不变量的光滑流形定义通过解决系统的偏微分方程。各种胶合公式计算这些不变量导致了与其他不变量的关系，从组合的角度来看。组合不变量的性质导致光滑不变量的性质，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。

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.