Contact homology and String topology

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The Principal Investigator plans to compute and apply Legendrian contact homology to study Legendrian submanifolds in one-jet spaces. This homology is based on holomorphic curves in symplectic manifolds. In the special case when the Legendrian submanifold is the conormal lift of a smooth submanifold, the Principal Investigator plans to relate the contact homology with an under-construction A-infinity-structure on the open string topology of the smooth submanifold. The next step in the project will be to translate more general structures from open string topology to contact geometry, thereby formulating a relative version of symplectic field theory as a generalization of Legendrian contact homology. Via this conormal construction, the investigator plans to use the homology to study smooth submanifolds, such as knots, in Euclidean or projective space. The Principal Investigator will also apply Legendrian contact homology to study higher homotopy groups of Legendrian submanifolds. The relationship between holomorphic curves and gradient flow trees should facilitate many of the project's computations. Contact geometry makes many appearances in physics, from optics to thermodynamics to classical mechanics. For example, particles obeying the Least Action Principal from mechanics translate into objects in contact geometry (or its closely related field, symplectic geometry) known as holomorphic curves. Some of these connections to physics have been known for centuries; however, only recently has contact geometry benefited from significant advances within the mathematical community. Specifically, studying these holomorphic curves have led to some powerful and sometimes surprising discoveries about contact rigidity and contact dynamics. In the last couple of years, an active area of research has evolved around applying these holomorphic curves towards studying unresolved problems in three and four-dimensional topology. As stated, these problems in low-dimensional topology seem to have nothing to do with holomorphic curves. Yet, the techniques from recent results have firmly established a connection. The Principal Investigator plans to further study the effectiveness of holomorphic curves, with an emphasis on connecting it to the theory of topological knots in three dimensions.