Five problems in geometry

摘要DMS - 0204651。PI: David auckly这个项目围绕着一系列拓扑、几何和分析方面的问题展开。最不具思考性的问题是与V. Kapovitch合作证明存在不允许Alexandrov度量的紧拓扑4流形。该结果应遵循对S. donaldson和D. Sullivan先前工作的适当修改。列表中最具思考性的问题是研究Gromov-Witten不变量与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，该领域的发展将影响其他学科的广泛范围。例如，发展了辛几何和规范理论来描述力学和电动力学。这些思想被发展成光滑流形的微妙不变量，通过对偏微分方程组的解来定义。为了计算这些不变量而开发的各种粘合公式导致了与其他不变量的关系，这些不变量是从组合的角度产生的。组合不变量的性质导致对光滑不变量的猜想，反之亦然。正是这种相互作用使得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.