Flexibility in Symplectic and Contact Geometry

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

This project undertakes a study of flexibility phenomena in symplectic and contact geometry. Symplectic and contact manifolds represent the kinds of geometric objects that connect abstract geometry to mathematical physics, and they give an ideal setup for studying classical mechanics, quantum physics, and string theory from a geometric perspective. Most work done in symplectic geometry today is in the direction of symplectic rigidity, meaning that we have very strong tools for detecting when this geometry obstructs you from doing something, for example, moving some subset of this geometric space somewhere else in the space. In contrast, symplectic flexibility aims to find geometric situations when we have constructive tools. Based on work of the past 40 years it seems that symplectic geometry is a very delicate thing, meaning that by making a small geometric alteration almost all geometric rigidity completely falls apart, allowing the geometry to become extremely malleable and unrestricted. The project aims to study what these geometric alterations are, how they cause the geometry to become flexible, and how these phenomena can be applied to symplectic geometry as a whole. This contrast of rigidity and flexibility should be thought of as measuring the difference between geometry and topology. If we are looking for a geometric structure or equivalence between objects, it certainly must exist as a topological object if it has any hope to exist geometrically. But when does the topology determine the geometry? In symplectic geometry this question can be rephrased as the existence of a solution to an underdetermined PDE, or a partial differential inequality. An h-principle is a systematic way to construct solutions to such equations. The project is intended to study how h-principles relate to symplectic and contact geometry. Depending on the context this has many different facets. In some situations the PI intends to construct new h-principles designed to solve specific problems in symplectic geometry, but it also involves finding new geometric methods to detect when an existing h-principle applies, or extending existing h-principles to a wider context. The type of h-principle needed also depends heavily on the geometry available, such as whether the object in question is symplectic or contact, whether it is the ambient space or a submanifold, whether it is low or high dimensional, and other factors.