Five problems in geometry

点击翻译按钮获取中文摘要

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.