Five problems in geometry

Kansas State University, Manhattan KS

Investigators

Abstract

ABSTRACT DMS - 0204651. PI: David Auckly This project is centered around a list of problems on the border of topology, geometry and analysis. The least speculative problem is part of a collaboration with V. Kapovitch to prove that there are compact topological 4-manifolds that do not admit Alexandrov metrics. The result should follow by a suitable modification of previous work of S. Donaldson and D. Sullivan. The most speculative problem in the list is to study the interaction between Gromov-Witten invariants and finite type invariants. Many different ideas are used in geometric analysis, and developments in this field will effect a broad range of other disciplines. For example symplectic geometry and gauge theory were developed to describe mechanics and electrodynamics. These ideas were developed into subtle invariants of smooth manifolds defined via the solution to systems of partial differential equations. Various gluing formulae developed to compute these invariants resulted in relationships with other invariants that arise from a combinatorial point of view. Properties of the combinatorial invariants lead to conjectures about the smooth invariants, and vice versa. It is exactly this interplay that makes geometric analysis such an important field.